# FCU Physics Unit 9-14

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## Electric Charge and Field 電力與電場

1. The magnitude of force, between two electrical point charges separated from distance d, is F. If both of the electrical charges are now reduced to 0.25 of their original quantity and the distance between them is now 0.5d, find the current electrical force in terms of F.
相距d的兩帶電物體之間受力為F，如果每一物體的帶電量變為原帶電量的1/4倍，且相距d變為d/2。則其力變為：
**ANS: $\dfrac{F}{4}$**

2. Regarding “Charge Quantization,” which of the following statements is INCORRECT?
下列有關「電荷量子化的」的敘述，何者『不正確』？
**ANS: None of the above.**

3. What is the static electric force magnitude between two charges of 5.0 C and − 2.0 C separated from 10 meters away? Is the force attractive or repulsive?
5.0庫倫的電荷與-2.0庫倫的電荷相距10m。靜電力為：
**ANS: $9.0×10^8$N 吸引**

4. What is the unit of electrical field strength in the SI unit system?
**ANS: N/C**

5. A point charge of 0.02 μC under the interaction of a uniform electrical field of 4000 N/C. Find the magnitude of force acting on the point charge.
**ANS: $8\times10^{-5}$**

6. We have 4 pairs of plates. A, B, D are electrically charged plastic plates. C is a conductive plate with no electric charge. The electric forces between each pair of plates are shown in figure.
What kind of electric force it is between the fourth pair of plates C and D?
**ANS: mutually attractive**

7. We have 4 pairs of plates. A, B, D are electrically charged plastic plates. C is a conductive plate with no electric charge. The electric forces between each pair of plates are shown in figure.
What kind of electric force it is between the fourth pair of plates B and D?
**ANS: mutually attractive**

8. There are two identical spherical conductors A and B. The electrical charge on A is −50e and on B is 20e. Right after A and B make contact with each other, what is the quantity of the electrical charge on A?
**ANS: -15e**

9. There are two identical spherical conductors A and B. The electrical charge on A is −50e and on B is 20e. Right after A and B make contact with each other, what is the quantity of the electrical charge on B?
**ANS: -15e**

10. There are two identical spherical conductors A and B. The electrical charge on A is 50e and on B is 20e. Right after A and B make contact with each other, what is the quantity of the electrical charge on B?
**ANS: 35e**

11. Three point charges of +q or −q are placed as the figure shown.
d1<d2. Draw a diagram and indicate the direction of the total electrical force acting on −q.
**ANS: $\leftarrow$**

12. “Charge Conservation” means that before and after an arbitrary reaction in a closed system,
**ANS: The total electric charge quantity after the positive and negative charges canceling each other will remain the same.**

13. Two point charges are 8.0 mm away and the magnitude of force between is 3.36 N. Find the magnitude of new electrical force when the distance between those two charges is now 12.0 mm.
**ANS: 1.49 N**

14. A uniformed charged and infinitely long straight wire produces electrical field E at a distance r from the central axis of the wire. Find the mathematical relationship between E and r.
在一無限長帶均勻電荷圓柱導線周圍的電場與距離 r 的關係是?

**ANS: $E=\frac{1}{r}$**

15. A point charge +Q is 3 cm on the left of the other point charge −4Q. Draw a diagram and indicate the position of the third point charge +Q under 0 total force.
**ANS: 3 cm to the left of electric charge +Q**

16. A point charge −Q is 3 cm on the left of the other point charge −4Q. Draw a diagram and indicate the position of the third point charge +Q under 0 total force.
**ANS: 1 cm to the right of the −Q charge**

17. Two point charges A of +9e and B of −4e are separated by 2.0 cm. Draw a diagram of a straight line through A and B to indicate the exact location at which the electrical field is zero.
**ANS: 4.0 cm to the right of B on the extension line of AB**

18. Three point charges of +q or −q are placed as the figure shown.
Draw a diagram and indicate the direction of the total electrical force acting on −q.
**ANS: $\searrow$**

19. As the four figures shown, the distance of all point charges from the origin are the same.

**ANS: A=B=C=D**

20. The surface charge density of a spherical conductor is −85 nC/m2. Find the magnitude of the electrical field e outside, right on the upper top of the sphere.
**ANS: $9.6 k \ N/C, \ \downarrow$**

21. 下列敘述，何者正確?
A. 一般帶正電的物體，係由於加上帶正電的質子所形成。
B. 若兩帶電體互相排斥，則兩者必帶同性電。
C. 若兩物體互相吸引，則兩者必帶異性電。
D. 帶電之電中性物體，其內部不含有任何電荷。
E. 使一物體帶正電,須使其內部的負電荷全部移出物體外。
**ANS: B. 若兩帶電體互相排斥,則兩者必帶同性電。**

22. 若一帶負電的塑膠棒，能吸引一極輕的金屬小球，則此小球:
**ANS: 帶正電或不帶電。**

23. 若一帶正電的玻璃棒，曾排斥一金屬小球，則此金屬小球:
**ANS: 帶正電**

24. 已知物體甲排斥物體乙，物體甲吸引物體丙，物體丙排斥物體丁。若測得物體丁帶負電，則物體乙帶有何種電荷?
**ANS: 帶正電**

25. Two uncharged metal spheres, “1” and “2” are mounted on insulating support rods. A third metal sphere “3”, carrying a positive charge, is then placed near “1”. Now a copper wire is momentarily connected between “1” and “2” and then removed. Finally, sphere “3” is removed. In this final state
**ANS: sphere “1” carries negative charge and “2” carries positive charge.**

26. A tiny styro-foam ball is suspended on a thread. Tests show it gets attracted to a positively charged rod, and also attracted by a negatively charged rod. What can you can conclude about the tiny styro-foam ball?
**ANS: (B) It has zero net charge.**

27. A balloon acquires a positive charge when rubbed on a sweater. As a result, the charge acquired by the sweater is
**ANS: negative**

28. A balloon acquires a positive charge when rubbed on a sweater. “Electrons were lost by the sweater in the balloon charging process.”
**ANS: False**

29. A metal plate is connected by a conductor to the ground through a switch S. The switch is initially closed. A charge +Q is brought close to the plate without touching it, and then the switch is opened. After the switch isopened, the charge +Q is removed. What is the final charge on the plate?
**ANS: (A) The plate is negatively charged**

30. Consider the following procedural steps:
1. ground an electroscope.
2. remove the ground from the electroscope.
3. touch a charged rod to the electroscope.
4. bring a charged rod near, but not touching, the electroscope.
5. remove the charged rod.

To charge an electroscope by induction, use the sequence:

**ANS: 4, 1, 2, 5**

31. 10 C (Coulomb) of charge are placed on a spherical conducting shell. A particle with a charge of −3 C is placed at the center of the cavity. The net charge on the outer surface of the shell
**ANS: +7 C**

32. Which of the following are not action-reaction pairs?
(A) Gravitational force by a book on earth, weight of the book.
(B) Electric force of attraction between an electron and a proton.
(C\) Force exerted by rocket on its rocket fuel, Force that provides upward thrust on rocket.
(D) Weight of a book, normal contact force on the book.
**ANS: (D) Weight of a book, normal contact force on the book**

33. 『複選』下列有關電力線的敘述,何者正確?
(A) 兩電力線不可相交
(B) 靜電平衡的導體，其電力線必垂直於表面
(C\) 電力線上各點電場相等
(D) 兩甚靠近的正、負帶電體，正帶電體的電力線必收攏於負帶電體
(E) 電力線即為電荷在電場中運動的軌跡。
**ANS: AD**

34. 電力線『不能』彼此相交的原因是:

**ANS: (D) 若可相交則在其相交之點上，一可動電荷所受之淨力必同時具有兩個方向，但力是向量, 一向量不能有兩方向，故電力線不能相交**

35. 下列有關電力線的敘述何者『錯誤』?
(A) 電力線為帶正電荷質點在電場中受力方向的連線
(B) 電力線上某點的法線方向即為該點的電場方向
(C\) 電力線為平滑的曲線，彼此不相交
(D) 電力線愈密集處，電場強度愈大

**ANS: B**

36. As the figures shown, there are three cases of electric-field-line distribution. Which one of them can at point B detect the greatest magnitude of electric field?
**ANS: (a)**

37. 
The Figure on the right shows the electric field lines for two point objects separated by a small distance. The charges q1 and q2 can be identified as
**ANS: q1 is negative; q2 is positive; $\frac{q1}{q2}=-\frac{1}{3}$**

38. For perfect charged conductor, the electric field on its surface is:
**ANS: perpendicular to it**

39. For perfect conductor which of the following properties is INCORRECT:
**ANS: none of above.**

40. We increase electric charges on the outer surface of a hollow metal block. The electric field inside now:
中空金屬塊的表面之電量增加，則內部電場強度

**ANS: remains unchanged 不變**

41. An aluminum block has a cavity within it which is completely closed. The block is placed in a region permeated by a uniform electric field which is directed upwards. Which of the following is a CORRECT statement describing conditions in the interior of the block’s cavity?
**ANS: There is no electric field in the cavity.**

42. 下列何處的電場強度一定為零？
(A) 帶電體質量中心
(B) 載流導體的內部
(C\) 帶靜電的金屬材料內部
(D) 帶電絕緣體的內部
(E) 兩個同電量的點電荷連線中心處
**ANS: C**

43. There is a negative surface charge density in a certain region on the surface of a spherical solid conductor. Just beneath(under) the surface of this region, the electric field
**ANS: not enough information given to decide.**

44. 一個測試電荷電量為+3 μC 位於P 點，該處的電場大小為$4×10^6$ N/C, 方向指向右方。若測試電荷現在改為−3 μC, 則位於P 點的外加電場為
**ANS: 不變**

45. Experimenter “A” uses a test charge 2q0 and experimenter “B” uses a very small test charge q0 to measure an electric field produced by stationary charges. Experimenter “A” finds a field that is
**ANS: the same as the field found by experimenter “B”.**

46. Consider the electric charges A, B, C shown in the figure, where q is a positive number.
Which answer correctly describes the magnitude of the net force experienced by the charges?
**ANS: $F_B > F_A > F_C$**

47. As the figure shown, there are two electricbpoint charges on a straight line.
Which part on the line can most likely have no electric field at all?
**ANS: Section A**

48. As the figure shown,
which of the 4-charge combinations can make the middle black point with no electric field at all?
**ANS: (a)**

49. Three ping-pong balls are electrically charged and are arranged in the plane of the page in an equilateral triangle as shown below.
What is the direction of the force acting on the ping-pong ball charged with Q3 = −10μC?
**ANS: Towards the bottom of the page. $\downarrow$**

50. Two point charges −2Q and +Q are placed on the x-axis with −2Q at x = 0 and +Q at x = a. Which of the following statements is true?
**ANS: There is a point on the x-axis, at x > a, where the electric field is zero.**

51. Which of the following statements implies a distribution of non-zero, uniform and isotropic electric field?
**ANS: a pair of charged parallel plates.**

52. Two particles with the same positive charge are 100 cm apart. What happens to the electric force between the two particles when the charge on each particle doubles?
**ANS: The repulsive electric force becomes four times as strong.**

53. What would happen to the electrostatic force between a pair of charged particles if both charges were doubled and the distance between them were also doubled?
**ANS: It would remain unchanged.**

54. Two equal but opposite point charges are fixed in position on the x and y axes, as shown in the figure above.
Which of the following arrows best illustrates the direction of the resulting electric field at the origin, O?
**ANS: $\searrow$**

55. The electric field strength at a point some distance away from a source charge does NOT depend on
**ANS: the sign of the source charge**

56. Two charged point particles are located at two vertices of an equilateral triangle and the electric field is zero at the third vertex. We conclude:

**ANS: At least one other charged particle is present.**

57. A positive point charge is released from rest in an electric field experiencing only the electric force. At any later time, the acceleration of the point charge
**ANS: is in the direction of the electric field at the position of the point charge.**

58. Now take a positively charged regular hexagon wire with a side length of L. It is placed horizontally on a table. The positive charges at the corners repel each other and create a tension on the wire.
(a) Express the tension T on the wire in terms of Q and length L
**ANS: $\frac{kQ^2}{12L^2}(4\sqrt3+15)$**
(b) Assume that the metal wire barely holds against the electric force. If one doubles the side lengths of the loop, what is the maximum total charge that one can have on the wire without breaking it?
**ANS: 12Q**

59. The diagrams show four possible orientations of an electric dipole in a uniform electric field $\vec{E}=E\hat{i}$
**ANS: 1, 2 and 4 tie, then 3.**

60. A dipole is released from rest and allowed to rotate in an electric field. Which of the following is CORRECT regarding the motion of the dipole?
**ANS: The magnitude of the total torque acting on the dipole is QℓE sinθ.**

61. 如右圖所示，
電荷+Q 與−Q 分別均勻分布於半圍環的上半部與下半部。則位於此半圍環環心的P 點，該處電場方向為:
**ANS: $\downarrow$**

62. As the figures shown,
in each quadrant, the electric charge Q uniformly distrubuted along the segment. At the point of origin, which of them can detect the greatest magnitude of electric field?
**ANS: (b)**

63. Which of the following charge distribution produces an electric field with direction parallel to the horizontal (水平) axis at the origin O?
**ANS:**

64. A line of charge centered at point O and oriented along the y-axis has charge +Q distributed uniformly between y = 0 and y = +a and charge −q distributed uniformly between y = 0 and y = −a (see figure).
In this situation, the electric field at point P located on the x-axis as shown in the figure is directed:
**ANS: Along the negative y-axis**

65. A solid sphere has a volume charge density $ρ(r)=\frac{A}{r+c}$ where A and c are two constants, r is radical distance from the center of the sphere. R is the radius of the sphere.
(a) What is the unit of A?

**ANS: C/m$^2$**

(b) If the total charge is Q, what is the magnitude of the electric field at r > R?

**ANS: $\frac{Q}{4\pi\epsilon_0r^2}$**

c) If the total charge is Q, what is the magnitude of the electric potential at r > R?

**ANS: $\frac{Q}{4\pi\epsilon_0r}$**

(d) Express the total charge Q in terms of A, R, c.

**ANS: $2\pi AR^2 [1 - \frac{2c}{R} + \frac{2c^2}{R^2} \ \ln(1 + \frac{R}{c})]$**

(e) Express the total charge at Q in the case where c ≫ R

**ANS: $\frac{4\pi}{3c}AR^3$**

66. A circular loop(ring), with a radius a is charged with +Q at the upper part and −Q at the lower part as shown in the figure.
(a) What is the direction of the electric field at point P?
**ANS: -y**
(b) Calculate the potential $V(r)$ at point P.
**ANS: $\frac{2kQ}{\sqrt{a^2+x^2}}$**
c) Calculate the magnitude of the electric field at the middle of the ring.
**ANS: $\frac{2kQ}{\pi a^2}$**

67. 一個氫原子是由一個核子內含單一質子，單一電子環繞其周圍。兩粒子間的電力為萬有引力的2.3 倍。假設我們可以調整兩粒子之間的距離，能找到一個距離使得電力和萬有引力相等嗎？
**ANS: 否，任何距離都無法**

68. 兩個均勻電荷分佈的球體，絕緣且固定在空氣桌的橡膠墊上。2號球上的電荷是1號球的3倍。關於靜電力的大小及方向，下列哪一個圖是正確的？
**ANS:**

69. 假定圖內之範圍有電場但沒有電荷存在，則下列哪一個是合理的靜電場分布圖？
**ANS: (b)**

70. 一個中性的電偶極置於一外加電場中，如圖，四個情況中，哪一個或數個會使作用在電偶極上的淨力為零？請選出正確選項。
**ANS: c 與 d**

71. 如圖，一個中性的電偶極置於一外加場中，該電偶極所受合力方向為
**ANS: $\nwarrow$**

72. 一個中性的電偶極置於一外部場中，此時作用在電偶極上的合力矩方向為？
**ANS: 順時針 （TBC）**

73. 兩個測試電荷被移至一電荷＋Q附近。起初，測試電荷＋q先被移至距＋Q距離r的A點處。再來將＋q移除，把一個測試電荷＋2q移至距＋Q距離2r的B點處。
(a) 比較測試電測試荷在A點及B點的靜電位
**ANS: $V_A > V_B$**

(b) 比較測試電荷在A點及B點的電場

**ANS: $E_A > E_B$**

74. 兩個測試電荷被移至一電荷＋Q附近。起初，＋q先被移至距＋Q距離r處。再來將＋q移除，把一個測試電荷－q移至相同位置。哪一個情況的靜電位能（U）比較大？
**ANS: $U_{+q} > U_{-q}$**

75. 一電子被推入電場中某處，該處電位為1V。假設換成兩個電子在相同電場中，被推動相同距離到達相同的位置上，當兩個電子佔據該位置時，該位置的電位是？
**ANS: 1V**

76. 一環形電荷半徑為r，電量為Q，以原點o為中心，置於yz平面，
(a) 在原點的電場
**ANS: $E_0=\frac{kQ}{r^2}$**
(b) 在原點的電位
**ANS: $V_0=\frac{kQ}{r}$**
(c) 已知環上每點電荷至P點之距離為d，則P點處之電壓與電場各為何?
**ANS: $E_p=\frac{kQ}{d^2}$, $V_p=\frac{kQ}{d}$**

#### Added Questions

1. The magnitude of the electric force between two protons is 2.30×10$^{-26}$ N. How far apart are they?

**ANS: 0.100 m**

2. In any reaction involving charged particles, the total charge before and after the reaction is always the same. This relationship is known as

**ANS: conservation of charge.**

3. A very small ball has a mass of 5.00×10$^{−3}$ kg and a charge of 4.00 μC. What magnitude electric field directed upward will balance the weight of the ball so that the ball is suspended motionless above the ground?

**ANS: 1.22×10$^4$ N/C**

4. A circular ring of charge with radius b has total charge q uniformly distributed around it. What is the magnitude of the electric field at the center of the ring?

**ANS: 0**

5. Three charged particles are arranged on corners of a square as shown in Figure OQ19.14, with charge −Q on both the particle at the upper left corner and the particle at the lower right corner and with charge +2Q on the particle at the lower left corner.

**ANS: It is downward and to the left. $\swarrow$**

6. A material that permits electric charge to move through it is called a(n)

**ANS: conductor**

7. Which statement is not true?

a. The tangent to an electric field line at a point gives the direction of the field at that point.

b. The density of electric field lines is directly proportional to the strength of the field.

c. The electric field obeys the principle of superposition.

d. Negative charges are sources of electric field lines and positive charge sinks.

**ANS: d**

8. Which of the following is not true? The electric force

**ANS: between two protons separated by a distance d is larger than that between two electrons separated by the same distance d.**

9. The electric field shown in Fig. 21-13
FIGURE 21-13

**ANS: increases to the right.**

10. 一點電荷自靜止經電位差 V 加速後，射入一均勻電場區域，射入的方向垂直電場，則此電荷離開電場時， 其偏向位移的大小為 d. 現如將加速電壓 V 加倍，其它不變，求新狀態的偏向位移。

**ANS: 0.5d**

11. Two point charges attract each other with an electric force of magnitude F. If the charge on one of the particles is reduced to one-third its original value and the distance between the particles is doubled, what is the resulting magnitude of the electric force between them?

**ANS: $\frac{1}{12}F$**

12. When the electric charge on each of two charged particles is doubled, the electric force between them is

**ANS: quadrupled**

13. An object with negative charge is placed in a region of space where the electric field is directed vertically upward. What is the direction of the electric force exerted on this charge?

**ANS: It is down $\downarrow$**

14. Assume the charged objects in Figure OQ19.15 are fixed.


Notice that there is no sight line from the location of q2 to the location of q1. If you were at q1, you would be unable to see q2 because it is behind q3. How would you calculate the electric force exerted on the object with charge q1?

**ANS: Add the force that q2 would exert by itself on charge q1 to the force that q3 would exert by itself on charge q1.**

15. An electric dipole in a uniform electric field experiences

**ANS: only a torque**

16. Two solid spheres, both of radius 5 cm, carry identical total charges of 2 μC. Sphere A is a good conductor. Sphere B is an insulator, and its charge is distributed uniformly throughout its volume. How do the magnitudes of the electric fields they separately create at a radial distance of 6 cm compare?

**ANS: $0 = E_A < E_B$**

17. 帶同性電的 A, B 兩小球，以等長的絕緣細線懸於同一點，平衡時， A, B 兩球的懸線與鉛直線的夾角各為 37$^\circ$、53$^\circ$, 求 A, B 兩球的質量比。

**ANS: 4:3**

18. Fig. 21-12 shows two electric charges of equal magnitudes and opposite signs. Electric field lines surrounding the charges are also shown. Which of the shown arrows correctly represents the electric field vector at point P?
FIGURE 21-12

**ANS: A**

19. Which is (are) true? When the charge distribution on a conductor reaches equilibrium,

a. any electric charge deposited on the conductor resides on the surface.

b. the electric field within the conductor is zero.

c. all of the given choices.

d. the electric field at the surface is perpendicular to the surface.

**ANS: all of the given choices**

20. A point charge +q is moving right from Point 1 to Point 2 and its velocity at Point 1 is greater than at Point 2. The electric force $\vec{F_e}$ decelerates +q.

Now, a point charge −q is moving right from Point 1 to Point 2. Compare the kinetic energy of −q, $E_{k1}$ and $E_{k2}$, at Points 1 and 2.

**ANS: $E_{k1} < E_{k2}$**

21. Which of the arrows shown in Fig. 21-10 represents the correct direction of the electric field between the two metal plates?


**ANS: A**

22. Fig. 21-11 shows 3 electric charges labeled Q1, Q2, Q3 and some electric field lines in the region surrounding the charges. What are the signs of the 3 charges?


**ANS: Q1 is positive, Q2 is negative, Q3 is positive**

## Gauss Law 高斯定律

77. The following figures are 2-dimensional cross-section of 3-dimensional Gauss spheres and cubes. Which one has the greatest electric flux?
**ANS:**

78. One single electric point charge q, is located at a upper back corner of a 6-surface cube, as the figure shown.
(a) Identify those surfaces of the cube intercepted zero electric flux. How many planes of zero electric flux are there?
**ANS: 3**
(b) Assuming the closed surface of the cube is the Gauss surface, calculate the total electric flux $\Phi_E$ over the 6 surfaces of the cube.
**ANS: $\frac{q}{8\epsilon_0}$**

79. Assuming a flat square surface of 2.0 m in length and centered at the origin, the surface is at plane yz, and the normal vector of the given surface is on the x axis. If the electric field at each point on the surface is $\vec{E}=30\hat{j}$ (N/C), find the electric flux $\Phi_E$ over this surface.
**ANS: 0**

80. Assuming a flat square surface of 2.0 m in length and centered at the origin, the surface is at plane yz, and the normal vector of the given surface is on the x axis. If the electric field at each point on the surface is $\vec{E}=30\hat{i}$ (N/C), find the electric flux $\Phi_E$ over this surface.
**ANS: 120 Nm$^2$/C**

81. Assuming a flat square surface of 2.0 m in length and centered at the origin, the surface is at plane yz, and the normal vector of the given surface is on the x axis. If the electric field at each point on the surface is $\vec{E}=(20\hat{i}+20\hat{j})$ (N/C), find the electric flux $\Phi_E$ over this surface.
**ANS: 80 Nm$^2$/C**

82. There is only one electric charge q in space. If its electric flux through a given Gauss spherical surface is not $\frac{q}{\epsilon_0}$, then q should be
**ANS: outside the sphere.**

83. If a bivalent oxygen ion is in a closed Gaussian surface, find the electric flux $\Phi_E$ through this closed Gaussian surface.
**ANS: $3.6\times10^{-8}$ Nm$^2$/C**

84. If a point charge is placed at the center of a cube, and the electric flux through any of the 6 planes is 5 Nm$^2$/C, find the total electric flux $\Phi_E$ on all surfaces of the cube.
**ANS: 30 Nm$^2$/C**

85. An infinitely long straight wire carries a uniformly distributed linear charge density of λ, which is surrounded by a rectangular box of 2-end surface area A and a box length L. Find the total electric flux $\Phi_E$ through the closed box.
**ANS: $\frac{\lambda L}{\epsilon_0}$**

86. Three large parallel plates are each separated with a distance of d, and the charge density on each plate is σ. Find the magnitude of the total electric field outside these plates.
**ANS: $\frac{3\sigma}{2\epsilon_0}$**

87. The distance between two parallel plates is d. If the surface charge density of the first plate is σ1, the surface charge density of the second plate is σ2, and σ1>σ2. Find the magnitude of the electric field between the two plates.
**ANS: $\frac{\sigma_1-\sigma_2}{2\epsilon_0}$**

88. The distance between two parallel plates is d. If the surface charge density of the first plate is σ1, the surface charge density of the second plate is σ2, and σ1>σ2. Find the magnitude of the electric field outside the two plates.
**ANS: $\frac{\sigma_1+\sigma_2}{2\epsilon_0}$**

89. The distance between two parallel straight conductive wires is a. The linear charge density of the first wire is λ1. The linear charge density of the second wire is λ2. λ1>λ2. Find the electric field at the mid-point between these two parallel wires.
**ANS: $\frac{\lambda_1-\lambda_2}{\pi\epsilon_0a}$**

90. Four electrons and four protons are arranged as shown in the figure.
A spherical surface encloses some of the electric charges. Find the electric flux on the spherical surface.
**ANS: 0 Nm$^2$/C**

91. A electric charge +q is placed inside a spherical Gaussian surface, The other three charges (one charged +q, two charged −q) are outside the Gaussian surface as shown.
Find the net electric flux through this Gaussian surface.
**ANS: greater than zero and out of the cylinder surface.**

92. There is a imaginary cylindrical Gaussian surface in the region between a pair of parallel plates as shown.
Both plates carry the electric charges with the same magnitude but opposite sign. One plate is positive charged, +q, and the other is negative charged, −q. The net electric flux $\Phi_E$ on this imaginary Gaussian surface is

**ANS: Zero**

93. In comparison with Gauss’s Law or Coulomb’s Law, to describe the electric field generated by an electrically charged particle:

Statement 1. In electrostatics, Gauss’s Law is equivalent to Coulomb’s Law and superposition principle.
Statement 2. Since Gauss’s Law only applies to those highly symmetrical charge distributions and Coulomb’s Law applies to any charge distribution, so Coulomb’s Law is more generalized than Gauss’s Law.
Statement 3. Since Gauss’s Law can describe the electric field generated by a time-varying magnetic field, Gauss’s Law is more generalized than Coulomb’s Law.

Which of the following choices is CORRECT?

**ANS: Both 1 and 3 are correct.**

94. The inner radius of a hollow conductive shell is 10 cm. A point charge of 2.0 (Coul) is at its center. Find the inner surface charge density of the conductor.

**ANS: -16 C/m$^2$**

95. The inner radius of a hollow conductive shell is 10 cm. A point charge of 2.0 (Coul) is at its center. Find the outer surface charge density of the conductor if its thickness of the shell is 1.0 mm.

**ANS: 16 C/m$^2$**

96. 三角形平面如圖10-2 所示，請寫出此三角形平面之面積向量

**ANS: $\vec{S}=\hat{n}A=\pm\frac{\sqrt3}{2}(\frac{1}{\sqrt3})(1, 1, 1)$**

97. Charges of +2q, +q, and −q are distributed in an area as shown in Figure.
Consider a Gaussian surface located around the +2q charge, with a point P located on the surface as shown. Which of the statements below is true?

**ANS: The net flux through the Gaussian surface depends only on the +2q charge.**

98. Choose the INCORRECT statement:
(A) Gauss’s law holds in a vacuum
(B) Gauss’s law states that the net number of lines crossing any closed surface in an outward direction is proportional to the net charge enclosed within the surface
C) Coulomb’s law can be derived from Gauss’s law and symmetry
(D) Gauss’s law applies to a closed surface of any shape
(E) According to Gauss’s law, if a closed surface encloses no charge, then the electric field must vanish everywhere on the Gaussian surface.

**ANS: (E) According to Gauss’s law, if a closed surface encloses no charge, then the electric field must vanish everywhere on the Gaussian surface.**

99. The electric flux through the surface of a sphere is not $\frac{Q}{\epsilon_0}$ if Q is

**ANS: outside the sphere**

100. A charge q sits at the back corner of a cube, as shown in the right figure. The length of the edge of the cube is L. The permittivity constant is ε0. What is the flux of electric field through the shaded surface of the figure?

**ANS: $\frac{q}{8\epsilon_0}$**

101. Consider a spherical Gaussian surface of radius R centered at the origin. A charge Q is placed inside the sphere. To maximize the magnitude of the flux of the electric field through the Gaussian surface, the charge should be located
**ANS: The charge can be located anywhere, since flux does not depend on the position of the charge as long as it is inside the sphere.**

102. A cubical Gaussian box contains a negatively charged particle with charge −Q, and another positively charged particle with charge +Q lies outside the box. What can you say about the net electric flux through the box?
**ANS: The net electric flux is less than zero**

103. (TBC)The equilateral triangular pyramid (or regular tetrahedron) in the figure is filled with a constant charge density of $\frac{1}{\pi}$nC/m$^3$. Each edge of the pyramid is $\sqrt2$ m thus, its volume is 1/3 m$^3$.

(a) Determine the electric flux through the bottom surface of the pyramid.

**ANS: 3V m**

(b) What is the value of $\oint\vec{E}\ d\vec{A}$ over the whole surface of the pyramid (dA is an infinitesimal element on the surface)?

**ANS: 12V m**

104. For which of the following charge distributions would Gauss’s law NOT be useful for calculating the electric field?

**ANS: a right circular cylinder of radius R and height h with charge uniformly distributed over its surface**

105. Consider a spherical insulator of radius R that has uniform charge distribution. Find the flux through a spherical surface of radius R/2 centered at the center of the sphere in terms of total charge Q of the sphere.

**ANS: $\frac{Q}{8\epsilon_0}$**

106. One of two parallel metallic plates is uniformly charged with charge +q, and the other one is charged with charge −q. In this case, the electric field between them is E. When the negatively charged plate is discharged then recharged with a positive charge 4q, the electric field between the plates becomes:

**ANS: 1.5E**

107. The surfaces of two thin nonconducting large planes are charged uniformly with different surface charge densities as shown in the figure.

(a) Find the magnitude of the electric field at point A

**ANS: $\frac{3\sigma}{2\epsilon_0}$**

(b) Find the magnitude of the electric field at point B

**ANS: $\frac{\sigma}{2\epsilon_0}$**

c) Find the magnitude of the electric field at point C

**ANS: $\frac{3\sigma}{2\epsilon_0}$**

(d) What is the flux through a cube (1) if the cube with a side length b is placed between large planes (two surfaces of the cube are parallel to the plane).

**ANS: 0**

(e) What is the flux through a cube (2) if the cube with a side length b is placed on the right plane so that the cube contains both faces of the plane (two surfaces of the cube are parallel to the plane).

**ANS: $\frac{\sigma b^2}{\epsilon_0}$**

108. The right figure shows a infinite sheet with uniform surface charge density σ > 0 and a slab with uniform charged density ρ < 0. Express $\vec{E}(x)=E(x)\hat{i}$.

Which of following diagrams is CORRECT?

109. For a coaxial cable in electrostatic equilibrium carrying equal but opposite charges on its two conductors,

(a) there’s a nonzero electric field

**ANS: TBC**

(b) In electrostatic equilibrium, charge on the shield

**ANS: lies entirely on its inner surface**

c) How does the electric field between the conductors in a coaxial cable in electrostatic equilibrium depend on the radial distance r from the cable’s axis?

**ANS: as 1/r.**

(d) A coaxial cable in electrostatic equilibrium carries charge −Q on its inner conductor and +Q on its shield. If the charge on the shield only is doubled,

**ANS: the magnitude of the electric field at the outer surface of the shield will equal the magnitude of the field at the shield’s inner surface.**

#### Added Questions

1. A particle with charge q is located inside a cubical Gaussian surface. No other charges are nearby. If the particle is at the center of the cube, what is the flux through each one of the faces of the cube?

**ANS: $\frac{q}{6\epsilon_0}$**

2. Which is (are) true?

1. The electric flux through a closed surface whose volume holds a net charge Q depends on both Q and the surface area.
2. For charges at rest, Coulomb’s law and Gauss’ law are equivalent.

**ANS: Statement 2 only**

3. Rank the electric fluxes through each gaussian surface shown in Figure OQ19.7 from largest to smallest.


**ANS: C > A = B > D**

4. Consider Gauss’s law: $\oint{\vec{E}}\ d\vec{A} = \frac{q}{\varepsilon_0}$.

**ANS: if the charge inside consists of an electric dipole, then the integral is zero**

## Electric Potential 電位

110. As the right figure shown, four point charges are respectively on the right angles of a square of 8 (cm) by 8 (cm).
If $Q = 4.0 ×10^{-6}$ (Coul), find the electric potential energy of the system.

**ANS: − 13.1 (J)**

111. The pair of plates of a parallel-plate capacitor are separated by a distance of 0.48 (mm). The electric potential difference is 28 (Volt). The fringe effect is negligible. Find the electric charge density on the plates.

**ANS: 0.52 μC/m$^2$**

112. An electric charge Q = 100 nC is uniformly distributed in a dielectric sphere of R = 0.30 m. Define $V_\infty = 0$. Find the electric potential at the center of the charged dielectric sphere.

**ANS: 4.5 kV**

113. Two solid conductive spheres A and B are separated far away on an insulated table, and the radius of A is 2 times that of B. A and B are respectively charged with +2Q and −Q.
If the two are connected with an ideal thin wire and reach electrostatic equilibrium,
(a) find the ratio between the surface potentials of A and B.

**ANS: 1.0**

(b) find the ratio between the surface electric fields of A and B.

**ANS: 0.5**

c) find the ratio between the values of surface charge density of A and B.

**ANS: 0.5**

114. What is the final speed at which an electron is initially released from a cathode in an electron gun and hits the anode by an accelerating voltage of 2.8 kV?

**ANS: $3.1×10^7$ (m/s),**

115. A uniformly charged conducting sphere of radius r = 15 cm has a surface charge Q = 500 nC. Find the electric potential at the center of the sphere assuming the electric potential at infinity is zero. $V_\infty$ =0.

**ANS: 30 (kV)**

116. During lightning, the electric charge of 30 C moves across a potential difference of 1.0×10$^8$ V in 20 ms, Find the energy released by the lightning bolt.

**ANS: $3×10^9$ J**

117. Which one of the following statements is TRUE?

A) A proton trends to move from a lower electric potential to a higher one.

B) The electric potential of a negatively-charged conductor must be negative.

C) If the electric field at a given point P is zero, then the electric potential at point P must be zero.

D) If the electric potential at a given point P is zero, then the electric field at point P must be zero.

E) none of the given choices.

**ANS: E) none of the given choices**

118. There are 8 identical rain drops. They all carry the same electric potential V. Find the total electric potential in terms of V when all of the eight merge into one rain drop.

**ANS: 4V**

119. (Multiple choices) Two solid conductive spheres A and B are separated far away on an insulated table, and the diameter of A is 2 times that of B.

A is no electrically charged but B is charged with Q. Now, the two are connected with an ideal thin wire and reach electrostatic equilibrium.

A) The electric field on the surfaces of two spheres must be equal.

B) The charge on the greater sphere is twice of the one on the smaller sphere.

C) The electric potential of the greater sphere is twice of the smaller sphere.

D) The electric potential of the smaller sphere is twice of the greater sphere.

E) The electric potential of both spheres must be equal.

**ANS: BE**

120. Which combination of units with respect to its corresponding physics quantities is FALSE?

A) Electric Potential, N⋅ m/C,
B) Electric Potential, V,
C) Electric Potential, eV,
D) Electric Field, N/C,
E) Electric Field, V/m.

**ANS: C) Electric Potential, eV**

121. A electric potential is $V(x, y, z) = −xy−3z^{−2}$. Find the y component of the electric field.

**ANS: $x$**

122. For a charged solid conductor in electrostatic equilibrium, which of the following is(are) TRUE?

A) The electric field inside is zero.

B) The electric potential inside is zero.

C) The surface electric field is perpendicular to the conductor surface.

D) The surface electric field is parallel to the conductor surface.

E) The sharper the surface of the conductor, the higher the electric potential.

**ANS: AC**

123. A parallel plate capacitor is charged and then disconnected from the source. Using insulating handles, the distance between plates is now doubled. Which of the following is CORRECT?

A) The magnitude of the electric field between the plates will now be half.

B) The magnitude of the electric field between the plates will now be doubled.

C) The potential difference between the plates will remain the same.

D) The potential difference between the plates will now be doubled.

**ANS: D) The potential difference between the plates will now be doubled.**

124. Two point charges of +4.0 μC and −1.0 μC are fixed at the origin x = 0.0 cm and x = 6.0 cm, respectively, Assuming the electric potential at infinity is 0, on the x axis, find all the positions where the electric potential is 0.

**ANS: $x=4.8\ cm, x=8.0\ cm$**

125. There is a uniform charge Q on a metal ring of radius a, as shown.

The x axis is perpendicular to the torus and through the center of the torus. Find the electric potential at point P with a displacement a from the center O to P on the x axis.

**ANS: $\frac{k_e Q}{\sqrt2a}$**

126. A coaxial cable consist of a long, straight conductive wire with radius a surrounded by a long, coaxial, cylindrical conducting shell with radius b as shown.
The electric charges per unit length +λ and −λ is respectively accumulated on the wire and on the shell. Assume the gap in between is vacuum. Find the electric potential difference between the wire and the inner surface of the shell.

**ANS: $2k_e\lambda\ln{(\frac{b}{a})}$**

127. For a hollow conductor in electrostatic equilibrium, which of the following statements is (are) always TRUE?

A) The electric field within a conductor must be zero.

B) The potential of the entire conductor must be equal.

C) There is no electrostatic charge on the inner surface of a cavity inside a conductor.

D) There is no electric field in a cavity inside a conductor.

**ANS: AB**

128. “Removing electrons from an object will raise its electric potential.”

**ANS: True**

129. “All points of a conductor with static charges are at the same electric potential.”

**ANS: True**

130. “Electric field lines point towards regions of higher electric potential.”

**ANS: False**

131. “An electron-volt is the same as $1.602×10^{−19}$ volts.”

**ANS: False**

132. 下列敘述哪些正確？
(A) 電場中，電力線上每點的切線方向，代表正電荷在該點的運動方向。
(B) 順著電力線方向，電場強度越來越小。
(C\) 順著電力線方向，電位一定越來越低。
(D) 正電荷一定由高電位向低電位運動。
(E) 在等位面上移動電荷，電荷必不受電力作用。

**ANS: (C\) 順著電力線方向，電位一定越來越低。**

133. 將一正電荷自無窮遠處移至電場中某點，所作的功稱為

**ANS: 電位能**

134. 下列何者為向量? (A) 電場 (B) 電位 (C\) 電位能 (D) 功

**ANS: (A) 電場**

135. 使 5 庫侖的正電荷，由A 點移至B 點，作功20 焦耳，則B, A 間電位差$V_B − V_A$ 為若干伏特?

**ANS: 4**

136. Consider the following five graphs.
1, Force (y-axis) against distance (x-axis)
2, Force (y-axis) against time (x-axis)
3, Velocity (y-axis) against time (x-axis)
4, Mass (y-axis) against velocity squared (x-axis)
5, Voltage (y-axis) against charge (x-axis)

For which graphs could the area under the graph potentially be a measurement of energy?

**ANS: 1,5**

137. The graph shows the relationship between the work done on a charged body in an electric field and the net charge on the body. What does the slope of this graph represent?

**ANS: potential difference**

138. In a certain region of space, the electric field is zero. From this we can conclude that the electric potential in this region is:

**ANS: constant**

139. The electric potential inside a charged solid spherical conductor in equilibrium:

**ANS: is constant and is equal to its value at the surface**

140. An electron that accelerates from a point near a collection of negative source charges toward a point near a collection of positive source charges experiences

**ANS: a decrease in electrical potential energy as it moves toward a region at a higher electric potential**

141. For perfect conductor which of the following properties is INCORRECT:
(A) The potential is constant on the surface
(B) There is no charge inside the conductor
(C\) The electric filed is zero inside the conductor
(D) None of above.

**ANS: (D) None of above.**

142. As shown in figure, two opposite charges with same amount are fixed in space. The dash lines represent the equipotential surfaces in space produced by the two charges. We take one extra positive charge and move it very slowly along the solid arrow lines from A to B-C-D-E to F.
Which of the following interval will have the positive and maximum work?

**ANS: CD**

143. An electron and a proton are initially at rest and release freely in a uniform electric field. Therefore,

**ANS: The electric potential of the electron increases when it moves.**

144. An electron and proton are simultaneously released from rest in a uniform electric field. Assume that the only forces exerted on the particles are electrical forces due to the electric field. At a later time when the particles are still in the field, the electron and the proton will have the same

**ANS: magnitude of momentum**

145. Which of the following statement is correct regarding electric field and electric potential?
(A) Where the electric potential is zero, the electric field must be zero.
(B) Where the electric field is zero, the electric potential must be zero.
(C\) Where the electric field is zero, the electric potential must not be zero.
(D) Where the electric field is zero, the electric potential must be a constant.

**ANS: (D) Where the electric field is zero, the electric potential must be a constant.**

146. 圖中的虛線，由 B 到 A, 是一帶電粒子穿過一均勻同向電場 E 之運動軌跡。

不計重力，則下列何者正確？

**ANS: 粒子帶負電**

147. 如圖，將半徑r 之光滑塑膠圓環鉛直固定在水平均勻電場 E 中，環上套有質量 m，帶電量 −q 的小球，

當小球從水平直徑的 A 端由靜止釋放，沿順時針方向滑到B 點的瞬間時，下列敘述何者錯誤？

**ANS: 所受電力為qE, 方向向右**

148. 如附圖，均勻電場E 中， 直角三角形的三邊AB、BC、CA的長度分別為$d_{AB}, \ d_{BC}, \ d_{CA}$,

則A、B 兩點的電位差$V_A - V_B$ 等於:

**ANS: $−Ed_{BC}$**

149. 如圖為某靜電場的電力線分佈，其中 A, B 兩點的電場大小各為$E_A, \ E_B$, 電位各為$V_A, \ V_B$, 則：

**ANS: $E_A > E_B,\ V_A < V_B$**

150. Suppose that an electric field points from the right to the left, diagonally up the page as shown below.
An electron is fired from point B, at potential $V_B$ toward point A, with potential $V_A$. If $V_B < V_A$, which of the following is true as an electron moves from point B to point A?

**ANS: negative kinetic energy difference and negative electric potential energy change.**

151. As more and more negative electric charge is being brought to a conducting sphere, inside the sphere

**ANS: the electric field stays constant and the potential decreases**

152. Two atoms interact with each other according to the following force, F, and potential, V, diagrams.

What is their equilibrium separation?

**ANS: The separation u which is equal to z.**

153. The figure shows two equipotential surfaces whose potentials are $V_1$ and $V_2$.
The lines represent four paths (A → A′, B → B′, C → C′, D → D′) along which equal test charges are moved. The work involved, can be said to be

**ANS: the same for all paths**

154. Two conducting spheres of different radii each has charge −Q. Which of the following occurs when the two spheres are connected with a conducting wire?

**ANS: Negative charge flows from thensmaller sphere to the larger sphere until the electric potential of each sphere is the same.**

155. A particle with charge q and initial speed v is stopped by a potential difference V in a distance d and time t. What was its initial momentum?

**ANS: $\frac{qVt}{d}$**

156. For a uniform electric field, with r being the distance, the electric potential is proportional to:

**ANS: 1/r**

157. An ellipsoid-shaped conductor is negatively charged. Which one of the following diagrams best illustrates the charge distribution and electric field lines?

**ANS:**

158. Two long hollow conducting cylinders, each of height h, are placed concentrically on the ground, as shown in the figure (top view). The outer cylinder is grounded, while the inner cylinder is insulated. A positive charge (the black dot in the figure) is placed between the cylinders at a height h/2 from the ground.

Which of the following figures gives the most accurate representation (top view) of the lines of force?

**ANS:**

159. Which of the following(s) is(are) wrong for a positive unit charge moving radially outward?
I) Potential difference near a positive charge located at the center is negative so electric force does positive work.

II) Potential difference near a positive charge located at the center is positive so electric force does negative work.

III) Potential difference near a negative charge located at the center is negative so electric force does positive work.

IV) Potential difference near a negative charge located at the center is positive so electric force does negative work.

V) Potential difference is zero means that electric force does no work.

**ANS: II and III**

160. Which of the following(s) is/are true?
I) Equipotential surfaces for both positive and negative point charge form a sphere.
II) Equipotential surfaces for only positive point charge form a sphere.
III) Equipotential surfaces are planes for uniform electric field.
IV) On a given equipotential surface, the electric field E has the same value at every point.

**ANS: I and III**

161. Find the potential difference between points A and B, $V_{AB}$ in terms of k, q, and r for a spherical charge distribution with charge q, as shown in figure.

**ANS: 0**

162. 『寶可夢』中的皮卡丘在決鬥中可以釋放十萬伏特的電壓攻擊對方。請依照物理觀念與知識，還有附表資訊，推測皮卡丘的攻擊最大距離。

**ANS: 3cm**

#### Added Questions

1. A spherical metal shell carries a uniform positive surface charge. The potential is the same over the surface of the shell. Which statement is CORRECT?

**ANS: The potential at the center of the shell volume is the same as on the shell surface.**

2. 若一點電荷 +q 在點 1 的速度較點 2 大。由點 1 移至點 2, 電力 $\vec{F}_e$ 使電荷 +q 減速。 由點 1 移至點 2, 此點電荷之『電位能』應:

**ANS: 增加**

3. The amount of energy required to assemble a point charge is called the charge's

**ANS: self-energy.**

4. A charge q is placed a distance r from the origin, and a charge 2q is placed a distance 2r. There is a charge Q at the origin. If all charges are positive, which charge has the higher electrostatic potential energy?

**ANS: The two charges have the same potential energy.**

5. a, b 兩點位於兩個點電荷 ±Q 之間， a 位於中心點，如圖所示。


(a) 在 a 點之電場 $\vec{E}_a$ 是否為零? 其電位 $V_a$ 是否為零?

**ANS: $E_a ≠ 0$ and $V_a = 0$**

(b) 現有一電子 $e^−$ 從 b 點移至 a 點. 在 a 與 b 點, 比較各自對應 $e^−$ 電子電位能 $U_a$ 與 $U_b$ 大小關係。

**ANS: $U_a > U_b$**

(c\) 現有一電子 $e^−$ 從 b 點移至 a 點. 在 a 與 b 點, 比較各自對應之『電位』 $V_a$ 與 $V_b$ 大小關係。

**ANS: $V_a < V_b$**

6. 半徑為 r 的金屬球置於金屬球殼中，二者係同心，已知球殼的內半徑為 2r, 外半徑為 3r, ，內部金屬球帶電電荷量 +Q, 外球殼荷電量 +2Q, 求球心電位。

**ANS: $\frac{3kQ}{2r}$**

7. 相距 r 的兩個點電荷，帶電量分別是 +q 與 −4q, 在兩點電荷的連線上電場為零之點與電位為零之點，求兩點相距距離。

**ANS: $\frac{6}{5}r$**

## Capacitance 電容

163. A capacitor is charged to 780 nC and the potential difference is 31 V. Find its capacitance.
某一電容充電至 780 nC 時,電位差為 31 V ,則此電容之電容值為:

**ANS: 25 nF**

164. For a rectangular parallel-plate capacitor, the length of the plate is 0.45 m, the width is 0.15 m, the distance between the two plates is 0.36 mm, and the dielectric constant of 7.5 is filled between the plates. Find the electric charge on each plate when the potential difference between the two plates is 19 V.
某矩形平行板電容,其平板之長為 0.45 m , 寬為 0.15 m ,兩板之間距為 0.36 mm, 且平板間填入介電常數為 7.5 之介電質。當兩板間之電位差為 19 V 時, 每一平板帶電若干?

**ANS: 240 nC**

165. For a rectangular parallel-plate capacitor, the length of the plate is 0.54 m, the width is 0.25 m, the distance between the two plates is 0.46 mm, and the dielectric constant of 4.6 is filled between the plates. When the potential difference between the two plates is 26 V, find the surface charge density on each plate.
某矩形平行板電容,其平板長為 0.54 m, 寬為 0.25 m, 兩板之間距為 0.46 mm, 且平板間填入介電常數為 4.6 之介電質。當兩板間之電位差為 26 V 時, 每一平板上之面電荷密度為若干?

**ANS: 2.3 μ C/m$^2$**

166. Regarding the capacitance of a parallel-plate capacitor, which of the following statements is correct?
有關平行板電容之電容值,下列敘述何者正確?

**ANS: The capacitance depends on the area of the plate and the distance between the plates.
電容值取決於平板之面積及板間間距**

167. A flat plate of a parallel-plate capacitor is a disk with a radius of 0.21 m. The two plates are each charged with 340 nC, the distance between them is 0.18 mm, and the potential difference is 12 V. Find the dielectric constant of the material between the two plates.
某平行板電容之平板為半徑為 0.21 m 之圓盤,兩板各帶電 340 nC, 間距為 0.18 mm, 電位差為 12 V, 則兩板間物質之介電常數應為:

**ANS: 4.2.**

168. When the potential difference of a given capacitor is 100 V, its electric energy is 1 mJ. Now, the electric potential difference of this capacitor is 200 V, how much will its electric energy increase from its original value?
某電容當電位差為 100 V 時, 電能為 1 mJ, 若此電容之電位差為 200 V 時, 電能較原值增加多少?

**ANS: 3 mJ**

169. For a parallel-plate capacitor, at first charge and then cut off the charging power supply, and finally pull the two plates closer together with insulating tools. As a result,
一平行板電容, 先充電後將充電電源切斷, 然後以絕緣把手將兩板拉近, 則:

**ANS: the potential difference between the plates decreases.
板間的電位差減少**

170. A 20-μF capacitor is charged to 200 V. Find the electric energy stored in this capacitor.
一 20 μF 的電容,充電到 200 V ,此電容儲存的電能是:

**ANS: 0.4 J**

171. The capacitance of a parallel-plate capacitor is
一平行板電容器的電容值

**ANS: proportional to the area of the electrode plate.
正比於電極板面積**

172. How large a potential difference is required to charge a 6-μF capacitor up to 9 μC?
將一 6 μF 的電容充電至 9 μC 需要多大電位差的電源?

**ANS: 1.5 V**

173. A 7-μF capacitor and a 3-μF one are charged to a potential difference 100 V, respectively. Then, the positive terminal of the 3-μF capacitor is connected to the negative terminal of the 7-μF capacitor, and the negative terminal of the 3-μF capacitor is connected to the positive terminal of the 7-μF capacitor. Find the final total energy in Joule stored in the two capacitors.
一個 7 μF 和一個 3 μF 的電容分別被充電到 100 V 電位差。 若將 3 μF 電容的正極連接到 7μF 電容的負極, 3 μF 電容的負極連接到 7 μF 的正極,則最終此兩電容所儲存的總電能是多少焦耳?

**ANS: 並聯(parallel)(8.0×10$^{-3}$) / 串聯(serial) (4.0×10$^{-2}$)**

174. Two parallel capacitors with the same capacitance value have different distances between their electrode plates, but both are filled with air. Charge these two capacitors with the same battery. When both capacitors are fully charged, which of the following physical quantities will have different values?
兩電容值相同之平行電容器, 它們的電極板距離不同, 但皆充滿空氣。 將這兩個電容分別以同一電池充電。當這兩個電容器都完全充電後, 它們有哪些物理量的值會不相同?

**ANS: Electric field between plates 電板間電場**

175. A 1-pF parallel-plate capacitor is filled with air initially. If the distance between the plates of this capacitor is doubled, and then the capacitance becomes 2 pF after the wax dielectric filled between the plates. Find the dielectric constant of wax.
有一由空氣充滿的 1 pF 平行板電容器, 若此電容器的板間距離變成二倍, 並在電板中間充入蠟介電質後電容變成 2 pF, 則此蠟的介電質常數為?

**ANS: 4.0**

176. Ideally, the capacitance is independent of the following properties, with one exception:
理想上, 電容值與下列各性質無關, 除了哪一項例外:

**ANS: Geometric shape of the capacitor 幾何形狀**

177. The capacitance C of a parallel-plate capacitor is related to the distance d between two plates and the plate area A as:
平行板電容器的電容 C 與兩板間距 d, 板面積 A 之關係為:

**ANS: $C∝\frac{A}{d}$**

178. Two capacitors of 6 μF and 30 μF are connected in series or in parallel. Respectively, find the equivalent values in unit μF of capacitance in series and in parallel.
兩個分別為 6 μF 與 30 μF 的電容器,將其串聯或並聯後, 其總電容分別為 (單位 μF ):

**ANS: 5, 36**

179. If the two capacitors are connected in series and in parallel, the connection offers the smaller total capacitance as:
兩電容器分別用串聯與並聯的接法, 則較小總電容值出現在何種接法:

**ANS: in serial 串聯**

180. After a parallel-plate capacitor fully charged by a battery, a terminal voltage difference between the two plates is generated as ∆V. If the battery is still connected and a dielectric slab is inserted between the plates, then each of the following increases except:
一平行板電容器經由一電池充電後, 而使兩板面間產生一端電壓差為 ∆V. 若電池連接著, 再將一介電質插入板面間, 則下列各項皆增加, 除了何者為例外:

**ANS: the electric field between two plates 兩平板間的電場**

181. After a parallel-plate capacitor fully charged by a battery, the battery is disconnected. If a dielectric slab is inserted between the plates of the capacitor, each of following physical quantities deceases except:
一平行板電容器經電池充電後, 將電池拆掉。若此時將一介電質插入電容器, 則下列各物理量皆減少, 除了何者為例外:

**ANS: the electric charge on the plate 板上的電荷**

182. The positive terminals of these two capacitors, C1 = 12 (μF) and C2 = 6 (μF), are connected together, and the negative terminals are also connected together. Find the equivalent capacitance in unit of μF.
兩電容器 C1 = 12 (μF); C2 = 6 (μF), 此兩電容正端接在一起,負端接在一起。 其總電容為多少 μF?

**ANS: 18**

183. “Doubling the plate area will quadruple the capacitance of parallel plates.”

**ANS: False**

184. “Combining two 5.0 F capacitors in series makes a larger capacitance.”

**ANS: False**

185. Which of the following factors will NOT change the capacitance of a parallel-plate capacitor?

**ANS: the quantity of the electric charge on each plate of the parallel-plate capacitor**

186. A parallel plate capacitor is charged and then disconnected from the source. Using insulating handles, the plates are brought closer together. Therefore:

**ANS: the potential difference between the plates decreased**

187. A dielectric slab is slowly inserted between the plates of a parallel plate capacitor, while the electric potential difference between the plates is held constant by a battery. As it is being inserted:

**ANS: the capacitance and the charge on the plate increase but the potential difference between the plates remains the same.**

188. The quantity $\frac{1}{2}\epsilon_0 \vec{E^2}$ has the dimension of:

**ANS: energy/volume**

189. Capacitance is measured in

**ANS: Farads.**

190. 有關電容器電容值(capacitance)，下列何者『錯誤』?

**ANS: 電容值大小跟電容器板的幾何形狀及其間之介質無關**

191. 正常使用下，平行板電容器電容 C 與兩極板間電位差 V 的關係為下列何者？

**ANS: C 與 V 無關**

192. Suppose the current charging a capacitor is kept constant. Which graph below correctly gives the potential difference V across the capacitor as a function of time, t?

**ANS: II**

193. A capacitor is constructed with two conducting plates of equal area A separated by an insulator. The capacitance is measured to be C. The conducting plates are then shrunk to half the original area. What is the capacitance now?

**ANS: $\frac{C}{2}$**

194. Which of the following arguments are TRUE?
I. The external work done to increase the plate separation of a parallel-plate capacitor is positive.
II. If the potential difference across a capacitor is tripled the stored energy decreases to 1/3 of its initial value.
III. The presence of a dielectric increases the maximum operating voltage of a capacitor.

**ANS: I, III**

195. The capacitance of a spherical capacitor with inner radius a and outer radius b is proportional to:

**ANS: $\frac{ab}{b-a}$**

196. The capacitance of a cylindrical capacitor can be increased by:

**ANS: increasing both the radius of the inner cylinder and the length.**

197. The figure shows a circuit with V =10 V, C1 = 2 nF, and C2 = 3 nF. The switch is closed, to A, and the capacitor C1 is fully charged.

(a) Find the energy delivered by the battery.

**ANS: 100 nJ**

(b) Find the energy stored in $C_1$.

**ANS: 100nJ**

(c\) Then the switch is thrown to B and the circuit is allowed to reach equilibrium. Find the total energy stored at $C_1$.

**ANS: 16 nJ**

(d) Find the total energy stored at C2 after the switch is thrown to B and the circuit is allowed to reach equilibrium.

**ANS: 24 nJ**

(e) Find the total energy stored at $C_1 + C_2$ after the switch is thrown to B and the circuit is allowed to reach equilibrium.

**ANS: 40 nJ**

#### Added Questions

1. 考慮一平行金屬板電容器，兩板相距 d. 上板表面電荷密度固定為 +σ, 下板為 −σ. 插入一厚度為 b = ℓ < d 的金屬厚板，且不跟電容板接觸。插入金屬厚板後，電容值會如何？


**ANS: 增加**

2. In order to increase the energy stored in a parallel-plate capacitor when an electric potential is applied, we should:

1. increase the area of the plates.
2. increase the separation between the plates.
3. insert a dielectric between the plates.

Which of the statements is (are) true?

**ANS: Two of the three given statements are true**

3. 考慮一個平行板電容器，兩板帶有等量但相反的電荷，板距為 d, 若帶電量固定, 將兩板拉開一個距離 D>d. 若拉大電容板間距之過程，保持電池之連接(電壓固定)，則儲存在電容器的靜電能相較於被拉開前，則儲存在電容器的靜電能相較於被拉開前

**ANS: 較小**

## Electric Circuit 電流與直流電路

198. Definition of electric current is,

**ANS: the amount of electric charge per unit time**

199. Which of the following is NOT related to the magnitude of resistance?

**ANS: shape of fixed cross-sectional area.**

200. The resistance value of a wire is R. Take another wire of the same material of the length and diameter are half of the original value. Find the resistance value of this new wire.

**ANS: 2R**

201. The resistance value of a cylindrical copper resistor is R. Find the new resistance value if its length is doubled to the original length but the total volume remains the same.

**ANS: 4R**

202. There is a steady current I = 5.0 (A) on a 20-Ω resistor. How many charges in Coulombs flow through this resistor in 4 minutes?

**ANS: 1200**

203. The resistance of a conductive wire is 1 m long and the cross-sectional area is 1 mm$^2$. When a voltage of 2 V is applied, the current is 4 A. Find the resistivity of the wire.

**ANS: $5×10^{−7}Ω$ m**

204. The definition of electric power can be expressed as

**ANS: $\frac{E}{t}$**

205. The resistance of a light bulb is 60 Ω. Find the applied voltage if the current on it reaches 0.5 A.

**ANS: 30 V**

206. Which of the following physical quantities correctly corresponds to the assigned unit?

**ANS: Potential difference - J/C**

207. When the power plant is to transmit a fixed electric power of 10000 kW over a long distance, the voltage of 10000 V is better than 220 V. Which of the following is correct?

**ANS: The transmission at high voltage loses less heat energy on the wire.**

208. Find the electric resistance on a light bulb labeled 60 watt, 120 volt.

**ANS: 240 Ω**

209. The electric resistance value of a variable resistor is moving linearly from 1000 Ω to 1500 Ω in 10 seconds. If the externally applied constant voltage on the variable resistor is 10 V, find the electric energy in Joule consumed by the resistor in 10 seconds.

**ANS: 83**

210. Which of the following statements is the most complete regarding the meaning of the electric current on a conductive wire?

1. The direction of the current is the same direction of the drift velocity of positive charge-carriers.
2. The direction of the current is the opposite direction of the drift velocity of negative charge-carriers.
3. The direction of the current is the direction of the electric field in the conductor.

**ANS: statements 1, 2, and 3 are all correct.**

211. Which of the following is a complete description of Ohm’s law?
1. $V=IR$
2. $R=\frac{ρℓ}{A}$
3. $R$ is independent of $V$ or $I$

**ANS: Statements 1 and 3 are both necessary.**

212. A car battery is marked as 80 A-hr. What is the unit of this ampere-hour?.

**ANS: Electric charge.**

213. The unit of resistivity is,

**ANS: Ohm-meter, Ω-m**

214. Two wires of different conductive materials carry the same electric current density. If they carry the same current, they must satisfy that,

**ANS: they have the same cross-sectional area.**

215. A current passes through a conductor, and its electron drift speed will be:

**ANS: far less than the average electron velocity.**

216. In the circuit in the figure, if $I_1=2$ A, find $V_2$.
於圖中之電路，若 $I_1=2$A，則$V_2$是多少？

**ANS: 22 V**

217. In RC serial circuit, if R=2 MΩ, C= 2 μF, connected to the power supply $\epsilon$ = 9 V. Find the initial value of the current.

**ANS: 4.5 μA**

218. A material is a good conductor because

**ANS: it has numerous free electrons.**

219. A conductor of length L has a varying cross section with area 2A at P and A at Q as shown in the adjacent figure.

If it carries a steady current I, then

**ANS: Current per unit area near P is less than current per unit area near Q**

220. Car batteries are often rated in ampere-hours. Does this information designate the amount of _______ that battery can supply.

**ANS: charge**

221. The Coulomb is the same as (A: ampere, s: second, M: mass, T: temperature)

**ANS: A·s**

222. What is the name for the flow of electrons in an electric circuit?

**ANS: Current**

223. “In a good conductor the electric field is zero even when the electric current is nonzero.”

**ANS: False**

224. “Good conductors are materials with low resistivity.”

**ANS: True**

225. “Electric current always corresponds to a flow of electrons.”

**ANS: True**

226. “In a battery that is charging, the electric current flows into the positive terminal.”

**ANS: True**

227. “A 300 mA current through your chest will probably cause ventricular fibrillation.”

**ANS: True**

228. For an ohmic substance, the resistivity is the proportionality constant for:

**ANS: current density and electric field**

229. A battery is replaced by two identical batteries connected in parallel. The combination can deliver

**ANS: the same maximum voltage and a higher maximum current**

230. A battery is connected across two identical resistors in series. If one of the resistors is instantaneously replaced by an uncharged capacitor, what happens to the current in the circuit?

**ANS: It initially rises, but then falls**

231. A real battery has an emf $\epsilon$ and an internal resistance r. A variable resistor R is connected across the terminals of the battery. A current I is drawn from the battery and the potential difference across the terminals of the battery is V . If R is slowly decreased to zero, which of the following best describes I, and V ?

**ANS: $I$ approaches $\epsilon/r$ ; $V$ decreases to zero.**

232. The characteristic graphs i−V (current - voltage) of resistors A and B in an integrated circuit (IC) are plotted as the figure shown.

If we connect these two resistors in parallel, find its equivalent resistance.

**ANS: 0.589 Ω**

233. 圖(a), 圖(b) 為甲導體、乙導體的電流I 與電壓V 的關係圖。


將甲、乙接成圖(c\)時， 安培計讀數恰為0.3 Ampere, 而此時電池本身的內電阻為10 Ω, 求電池電動勢E .

**ANS: 10 (V)**

234. 附圖中每個電阻均為2 歐姆，通過電池的電流為3 安培，則電池的端電壓為多少伏特？

**ANS: 15 V**

235. When Ohm’s Law, V = IR, is applied to a resistor R, the “V” refers to the potential When Ohm’s Law, V = IR, is applied to a resistor R, the “V ” refers to the potential

**ANS: difference between the ends of the resistor.**

236. A circuit has a light bulb connected to a constant voltage source. Which of the following changes must increase the power in the circuit?

**ANS: using a light bulb that draws more current**

237. Two space heaters are made to output their nominal power of 1000 W and 500 W respectively when they are supplied with 120 V. If they are connected in series to a normal (120V) wall outlet, compared to only the 1000 W heater how much heat will be produced?

**ANS: Less**

238. Rank in order, from brightest to dimmest, the identical bulbs A to D.


**ANS: C = D > B = A**

239. 教室內的日光燈管規格為 110 V、100 W， 每間教室內共有20 支燈管，若一天 8 小時的上課時間皆點燈，則一天每間教室必須用電多少度？

**ANS: 16度**

240. An electric car has a battery pack delivering 160 V and 100 A of steady current when moving at 36 km/h. What is the air resistance, assuming 100 % efficiency?

**ANS: roughly 1600 N**

241. An electric motor is driven by a battery of voltage 6 V and draws a current of 1 A. If the motor is used to lift vertically a block of mass 100 g, what is the vertical velocity of the mass?

**ANS: 6 m/s**

242. Which term describes the rate at which electrical energy is used?

**ANS: Power**

243. A kilowatt-hour is a unit of

**ANS: electric energy**

#### Added Questions

1. A resistor and an initially uncharged capacitor arranged in series are charged by a battery, which is connected at t = 0. The current in the circuit

**ANS: decreases exponentially in time**

2. Which statement(s) is(are) true? When a long straight conducting wire of constant cross-section is connected to the terminals of a battery, the electric field

**ANS: both of the given choices**

3. Which statement(s) is(are) true?

The electric potential energy of a charge distribution is:

1. equal to the amount of work required to bring the charges to their final configuration if they are initially separated by large distances.
2. proportional to the square of the electric field generated by the charges.

**ANS: both of the given statements**

4. A 3.6-V mobile phone battery can produce 0.7 A of current for 1 hour. This can be charged using a square solar panel 25 cm on each side. Assuming an efficiency of 10% and an incident solar power density of 1kW/m$^2$ what time is needed to charge the battery?
一只 3.6-V 手機電池可產生 0.7 A 的電流，持續 1 小時。可使用每邊 25-cm 的方形太陽能板充電。假設效率為 10%, 入射太陽能功率密度為 1 kW/m$^2$, 電池充電需要多長時間？

**ANS: 0.40 hours**

5. A 9V battery is connected across a 100 Ω resistor. Given that the charge on an electron is 1.6×10$^{−19}$ C what is the number of electrons passing through the resistor every second?
一個 9-V 電池與一個 100-Ω 電阻器串聯成一直流電路。假設一電子上的電荷為 1.6 × 10$^{−19}$ C，那麼每秒通過電阻器的電子數量為何？

**ANS: $5.6 \times 10^{17}$**

6. A light dependent resistor is connected across an ideal 12V source and placed in the open in the middle of a desert. When is the power dissipated in the resistor highest?
一個光敏電阻連接在理想的 12-V 電源上，並放置在沙漠中部的空地上。此光敏電阻器，何時的消耗功率最高？

**ANS: noon 中午**

7. A cube of metal has sides of length x. The electrical resistance between opposite faces of the cube is:
一個金屬立方體有一固定邊長 x. 求立方體相對面與面之間的電阻是:

**ANS: inversely proportional to x. 與 x 反比**

8. 如圖, A, B 兩電燈泡, 燈絲粗細不同


a) 比較電阻大小

**ANS: $R_A > R_B$**

b) 比較功率大小

**ANS: $W_A < W_B$**

9. Which is(are) true?

The emf of a source of electric potential energy is:

1. the amount of electric energy delivered by the source per coulomb of positive charge as this charge passes through the source from the low to the high-potential terminal.
2. equal in magnitude to the potential drop in the external circuit connected between the terminals of the source of emf.

**ANS: both of the given statements**

10. 標示 60 瓦特的甲燈泡對 100 瓦特的乙燈泡的電阻，何者較大？

**ANS: 60 瓦特的甲燈泡電阻較大**

11. 將某粗細均勻之電阻線接於無內電阻之電池之兩極時，電阻線中電場強度之大小為 $E_1$. 若將該電阻線抽細，使其直徑為原來之 1/3 倍，再接於同一電池之兩極，則電阻線中電場強度之大小變為 $E_2$. 求 $\frac{E_1}{E_2}$

**ANS: 9**

12. 如圖示，若 V = 18 伏特

**ANS:
本組合電路之總電流為 2.5 A
流經 8 歐姆電阻之電流為 1.5 A
流經 4 歐姆電阻之電流為 1.5 A**

13. Which term was not defined in the up-to-date lecture content in the textbook of general physics?

**ANS: impedance**

14. As shown in the figure, a circuit is formed from a battery, electromotive force $\varepsilon=V$, four resistors, of resistance R, 2R, 3R, 4R, and a capacitor of capacitance C. The internal resistance of the battery is negligible.


What is the electrical charge stored in the capacitor after sufficient time elapses?

**ANS: 0.4CV**

15. 如圖，設 A B C D E 五個燈泡具相同結構，由大到小排列。寫出五燈泡之電壓之大小關係：


**ANS: $V_A > V_D = V_E > V_B = V_C$**

16. Three different resistors are all connected in a DC circuit. How many different values of the equivalent resistance of all possible combinations in serial and/or in parallel are there?

**ANS: 8**

17. 如圖，五個燈泡電阻均為 12.0 Ω, 電池之電壓為 9.0 V，當開關接通後, A 燈泡有無電流通過？

**ANS: 沒有電流流過**

18. A Wheatstone bridge is a device used to measure

**ANS: resistance**

19. Two identical resistors are connected first in series and then in parallel. Which combination has the larger net resistance?

**ANS: the pair in series**

## Magnetic Field 磁力與磁場

244. What is the prerequisites for the existence of a magnetic interaction force from the magnetic field on charged particles?

**ANS: It exists only when charged particles pass through magnetic field lines.**

245. An electron is moving eastward $\rightarrow$ in the magnetic field going from east to west, $\leftarrow$. Find the direction of the magnetic force acting on this electron.

**ANS: zero**

246. A proton and a Helium nuclei are injected into the same magnetic field perpendicularly and are all subjected to the same magnetic force. What is the ratio of two speeds of the proton and the helium nuclei?

**ANS: 2**

247. An electron is moving southward, $\downarrow$, through an area filled with uniform and southward-directed magnetic fields, $\downarrow$.

**ANS: It will not be affected by the magnetic field.**

248. A current-carrying coil placed in a uniform magnetic field will be influenced by its torque induced by magnetic forces. Which of the following statements is true?

**ANS: When the normal direction of the coil plane is perpendicular to the magnetic field, there will be a maximum torque.**

249. A current of 3.0 Ampere on a closed-loop coil flows clockwise along the outer edge of the paper. The enclosed area is $5.0 × 10^{−2} \ m^2$. Find the magnetic dipole moment of this coil.

**ANS: 0.15 A⋅ m$^2$, into paper, ⊗**

250. Regarding the formula about magnetic force, $\vec{F_B}=q\vec{v}\times\vec{B}$, which of the following statements is TRUE?

**ANS: The direction of $\vec{F_B}$ must be perpendicular to $\vec{B}$ and the direction of $\vec{F_B}$ must be perpendicular to $\vec{v}$.**

251. The current I on a long straight wire is to the right, $\rightarrow$, as shown.


On point P of the same plane, find the direction of the magnetic field produced by the current on the wire.

**ANS: vertically out of the surface, ⊙**

252. An infinitely long-and-straight cylindrical current wire, with a radius R, carries a uniform current I. Where is the maximum of its magnetic field from its central axis?

**ANS: r = R**

253. An electron of mass m and charge −e is accelerated from rest by an external electric potential difference V and then deflected by a perpendicular magnetic field, B. Find the trajectory radius of the electron.

**ANS: $\frac{1}{B}\sqrt{\frac{2mV}{e}}$**

254. A moving proton entering perpendicularly into a magnetic field performs a circular motion. Find the ratio of the circular time period to the original one, if the speed of protons is now doubled.

**ANS: the same**

255. A cyclotron operates with a given uniform magnetic field and at a constant frequency. If R denotes the curvature radius of the final orbit, the final particle energy is proportional to:

**ANS: $R^2$**

256. A rectangular current loop of 100 windings and the length and width are 0.300 m and 0.400 m respectively. It carries a current of 1.20 A. The angle between the plane of the coil and the x axis is 30.0$^∘$, and the magnetic field strength is 0.400 T in +x direction.


Find the magnetic torque on the coil.

**ANS: 4.99 N⋅m**

257. A current of 8.0 A is applied to a evenly-wound solenoid with a total length of 0.40 m, The magnetic field strength in the center of the solenoid is B=5.0×10$^{−3}$ T. Find the total number of turns of the solenoid.

**ANS: 200**

258. A coaxial cable of inner and outer diameters of a and b, respectively, is carrying inside and outside with currents of the same magnitude I but opposite directions, as shown in the figure. R is the radial distance from the center axis to the point of measurement.


Which of the following regions would have no magnetic field?

**ANS: only III (b < R)**

259. There is an ideal solenoid with current I, as the figure shown.


In the two closed paths shown the figure, which section will experience a non-zero magnetic field?

**ANS: only 1**

260. Two parallel conductive wires carry currents in the same direction. If one current is doubled and the other current remains the same, then the value of magnetic force per unit length between two wires should be

**ANS: doubled**

261. A proton with velocity v = 3.2×10$^6$ m/s passes a uniform magnetic field B = 1.5×10$^{−4}$ T in the direction of 80$^∘$.


Find the pitch of the spiral if its trajectory is a spiral.

**ANS: 240 m**

262. An ideal long straight solenoid with a length of 20 cm and a winding of 2000 turns is carrying a current I = 0.20 A. What is the magnitude of the magnetic field at its center?

**ANS: 2.5×10$^{−3}$ T,**

263. 下面哪一個單位不是磁場的單位？

**ANS: 牛頓/安培**

264. A coil of wire stores energy in the form of: (1) charge; (2) potential difference; (3) magnetic field

**ANS: (3) magnetic field**

265. A charged moving object enters a volume where a uniform field is present. After some time the object moves in a circular orbit. Which field was present in this area?

**ANS: magnetic**

266. Two long parallel wires are placed side-by-side on a level table. The two wires carry equal currents in the same direction. Which of the following is true of the magnetic forces exerted on each wire by the other?

**ANS: The two wires are attracted to one another.**

267. Two parallel conductors carry sinusoidal alternating currents that flow in opposite direction as shown below. The current in wire A is twice that of the current in wire B.


Which of the following statements is true?

**ANS: The magnetic field acting on wire B is stronger than the magnetic field on wire A**

268. The figure below represents four long, straight current-carrying wires P, Q, R, S which are perpendicular to the plane of the paper. The currents all have the same magnitude. The currents in P, Q and R flow into the plane of the paper. The current in S flows out of the plane of the paper.


What is the direction of the resultant magnetic field at the center, O.

**ANS: Toward P**

269. Two parallel conductors carry sinusoidal alternating currents that flow in opposite direction as shown below.


Which of the following graphs shows a possible variation of the force, with respect to time, experienced by one of the conductors?
**ANS:**


270. Three particles travel through a region of space where the magnetic field is out of the page, as shown in the figure below.


Which statement below about their charges is correct?

**ANS: 1 is negative, 2 is neutral, and 3 is positive.**

271. A charged particle accelerated to a velocity v enters the chamber of a mass spectrometer. The particle’s velocity is perpendicular to the direction of the uniform magnetic field B in the chamber. After the particle enters the magnetic field, its path is a

**ANS: circle**

272. A magnetic field cannot:

**ANS: change the kinetic energy of a charge**

273. Two charges q and Q move with nonzero velocities with respect to a fixed reference frame. The magnetic force on q exerted by Q is

**ANS: perpendicular to the velocity of q and depends on both the velocity of Q and that of q.**

274. Which is (are) true? The magnetic field inside a solenoid
(A) is parallel to the axis of the solenoid.
(B) has a magnitude that is proportional to the total number of turns.
(C\) both of the given choices
(D) neither of the given choices.

**ANS: both of the given choices**

275. A sphere of radius R is placed near a long, straight wire that carries a steady current I. The magnetic field generated by the current is B. The total magnetic flux passing through the sphere is

**ANS: $\mu_0I$**

276. Ampere’s law gives the magnetic field produced by a distribution of currents. Which condition(s) must be satisfied?

(A) The distribution of currents must be steady
(B) In order to solve, the distribution must have sufficient symmetry.
(C\) both of the given choices.
(D) neither of the given choices.

**ANS: both of the given choices**

277. In using Ampere’s law, the integral must be evaluated

**ANS: around a closed path**

278. A square conductive frame is moving in the vertical Y X plane at a constant velocity v through a region of uniform magnetic field B directed perpendicular to the plane of the frame as shown in the figure.


Does charge separation occur in the frame?

**ANS: Yes, with the top positive.**

279. The current i is from left to right in the conductor as shown. The magnetic field B is upward and point S is at a higher potential than point T.


The charge carriers are

**ANS: negative**

280. 附圖中，何者為最合理的，帶正電粒子的速度與磁場及磁力的關係？

**ANS: d**


281. 關於「磁」之基本概念，下列敘述何者為誤?

**ANS: 磁為靜止電荷間的一種交互作用**

282. 下列何者為磁通量$\Phi_B$ 的定義表示式？

**ANS: $\int{\vec{B} \ \cdot d\vec{a}}$**

283. 一電荷Q, 以速度v 射入均勻磁場B, 則該電荷受磁場作用下，不可能進行下列何種運動？

**ANS: 變速率螺旋線運動**

284. 甲、乙兩長直導線垂直於紙面，甲電流出紙面，乙電流入紙面，甲、乙兩條導線電流強度相同， A 點感應磁場方向：


**ANS: $\downarrow$**

285. Which graph below correctly gives the magnitude of the $\vec{B}$ field due to an infinitely long straight current carrying wire as a function or the distance r from the wire?

**ANS:**


286. One of Maxwell’s equations is $\vec{\nabla} \cdot \vec{B} = 0$ or $\oint\vec{B} \cdot d\vec{a} = 0$. Which of the following sketches shows magnetic field lines that clearly violate this equation within the region bounded by the dashed lines?

**ANS:**


287. Which combination of units is the odd one out? (Which one is different from others?)

**ANS: Cms$^{-1}$**

288. In the figure shown, the current carrying loop is fixed, where as current carrying straight conductor is free to move.


Then, ignore gravity, straight wire will

**ANS: move towards the loop.**

#### Added Questions

1. We have three statements. Which of the following is(are) TRUE and most suitable?
1. The magnetic field lines of a moving charge form closed loops.
2. The magnetic field obeys the principle of superposition.
3. The magnetic flux through a closed surface is proportional to the total number of magnetic poles enclosed within the surface.

**ANS: Two of the three given statements are true.**

2. The Hall effect

a. can be used to determine the density of free electrons in a metal.

b. both of the given choices.

c. neither of the given choices.

d. provides empirical evidence that the charge carriers in metals are negative.

**ANS: both of the given choices**

3. 一個帶電量為 +q 的質點，垂直射入均勻磁場 B 中

已知質點在磁場中運動的時間為 t, 半徑為 R, 則當質點離該磁場時， 圖中所示的角度 θ 為

**ANS: $\frac{qBt}{m}$**

4. Which of the following is the SI unit of magnetic dipole moment?

**ANS: Ampere ⋅ m$^2$**

5. 如圖所示，在正方形空腔內有均勻磁場（方向垂直紙面向裡），一束具有不同速率的電子，從 a 孔垂直於磁場方向射入室腔，有電子分別從 c, d 兩孔射出，則它們的速率之比 $v_c:v_d$ 和在空腔內運行的時間之比 $t_c:t_d$ 分別為


**ANS: 2:1 與 1:2**

6. A small planar current loop is placed in a uniform magnetic field. The magnitude of the torque on the loop is a maximum when

**ANS: the plane of the loop is parallel to the direction of the field.**

7. 兩平行長直導線相距 d, P 點距兩導線皆為 d. 若兩導線載有大小皆為 i, 而方向相反之電流，則 P 點之磁場強度為 B；若兩導線電流仍各為 i, 但同方向電流時，求位於 P 點之磁場強度。

**ANS: $\sqrt{3}B$**

8. 下列各物理量的國際標準(SI)單位，何者與眾不同？

**ANS: $Cms^{-1}, (Coul-m-s^{-1})$**

9. A long straight wire lies along the x-axis and carries a current of electrons that move in the positive x-direction. The magnetic field due to this current, at a point P on the negative y-axis, points in which direction?

**ANS: +z**

10. 半徑為 R 的圓形線圈，通有電流 I, 置於強度為 B 與線圈面垂直的均勻磁場中， 此線圈平面與磁場垂直，則此線圈的張力大小為何？

**ANS: IRB**

11. 右圖中，將一無限長的導線彎成兩半無限長直導線與半徑為 R, 圓心角為 θ 之圓弧， 且兩直線部分恰與圓弧部分相切。


欲使圓心位置處的合成磁場強度恰為零，求圓心角 θ 之弧度大小。

**ANS: 2**

12. 假設運動與所提及的電流是沿著 x 軸方向，場，則是在 y 方向。電場會施力在一靜止帶電物體嗎？如果是磁場，會嗎？

**ANS: "yes" and "no" 對一電場而言 “會” 。但是，對磁場而言 “不會”**

13. 一個帶電粒子，進入一個均勻磁場，粒子的速度和磁場方向垂直，則粒子的運動是


**ANS: 等速圓周運動，速率大小不隨磁場大小而改變**

14. 假設運動與所提及的電流是沿著 x 軸方向，場，則是在 y 方向。 電場會施力在一道飛行電子束上嗎？如果是磁場，會嗎？

**ANS: "Yes" and "Yes" 都 “會”**

15. 一根 1.00 m 長的細銅棒質量 50.0 g. 假若在 0.100 T 的磁場中此桿離地漂浮空中，求桿中最小電流。

**ANS: 4.90 A**

16. 有一個半徑為 R, 帶電量為 q 的圓盤，繞通過圓心且垂直盤面的軸轉動，已知其轉動的頻率為 f, 且圓盤上的電荷分布均勻，求此圓盤圓心處的磁場。

**ANS: $\frac{\mu_0 qf}{R}$**

17. 兩板相距 d, 其電位差是 V, 所產生的電場與一磁場互相垂直。 若有一質量 m, 動能是 K 的質點，射人該兩互相垂直電場與磁場中， 無偏向時，求此磁場的強度。

**ANS: $\frac{V}{d}\sqrt{\frac{m}{2K}}$**

18. 一質子 ($^1_1H^+$) 與一 α 粒子 ($^4_2He^{2+}$) 以相同的入射速率，射入同一磁場中，入射方向各與磁場夾 60$^\circ$ 及 30$^\circ$ 角，則兩者在磁場中之迴轉週期比為何？

**ANS: 1:2**

19. 將一質子 ($^1_1H^+$) 與一 α 粒子 ($^4_2He^{2+}$)以相同速率及方向，分別射入相同均勻磁場中。由於入射速度不與磁場方向垂直，因此質子及 α 粒子均做螺線形運動。 設質子及 α 粒子所做的螺線運動之螺距分別為 d1 及 d2, 則 d1/d2 之值為：

**ANS: 0.5**