# Gen.Phys. (II) Recitation Notes 04/13

## Outline
* Homework Assignment Review
* Electrodynamics
* Ampère's Inconsistency
* (Dis)charging Capacitor
* Finite Wire
* Charge Conservation
* Magnetic Vector Potential

## Electrodynamics
### Ampère's Inconsistency
So far, we have finished our analyses on static electric and magnetic fields. We started from empirical deductions of the fields from its respective sources, i.e. Coulomb's & Biot-Savart Law, and worked our way in formulating the prototype of Maxwell's equations:

$$\begin{cases}
\nabla \cdot\mathbf{E}=\rho/\varepsilon_0, & \nabla\times\mathbf{E}=-\partial_t\mathbf{B},\\
\nabla\cdot\mathbf{B}=0, &\nabla\times\mathbf{B}=\mu_0\mathbf{J}
\end{cases}$$

However, there is an inconsistency here, a subtle problem resultant of Ampère's Law.
A classical example is the one below, an RC circuit during (dis)charge:

#### Example 1: (Dis)Charging Capacitor in RC Circuit


Integrating Ampère's Law along the path indicated, we get
$$\begin{align}
\oint \mathbf{B}\cdot d\mathbf{l}=\mu_0I_\text{enc}=\mu_0I_\text{circuit}
\end{align}$$

However, using Stokes' Theorem, we know that the integral is equivalent to:

$$\begin{align}
\oint \mathbf{B}\cdot d\mathbf{l}=\int (\nabla\times\mathbf{B})\cdot d\mathbf{a}
\end{align}$$

which is invariant under a homeomorphism. For example, we could blow up the loop into a bubble enclosing a single plate, such that it does not enclose any current.
But **WAIT!!** This leads to an inconsistency for the value of the integral for the same Ampèrian loop. But what led to this "paradox" --- is the step from magnetostatics to electrodynamics.

*'Note that this could not be experimentally verified in Maxwell's time, due to the big difference in the order of magnitude of Maxwell's correction term with that of Ampére's. We would see this in work soon.'*

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#### Example 2: The Finite Wire

Actually, this is not our first encounter with such an inconsistency with Ampère's Law, but you may have just been too trusting of me to realize 😜 !

Here's a hint, it is that from the finite current-carrying wire!


If you remember our derivation in a previous recitation, the magnetic field on the perpendicular bisector at a distance $R$ from the finite wire of current $I$ and length $\ell$ is given by

$$B=\frac{\mu_0I\sin\theta_0}{2\pi R}$$

where $\theta_0=\sqrt{(\ell/2)^2+R^2}$ is the angle subtended in the figure above. The result obtained seems sensible, and would converge to the familiar result
$$B=\frac{\mu_0 I}{2\pi R}$$

for an infinite wire as $\theta_0\to\pi/2$.

We obtained the result above using Biot-Savart's Law, and seems satisfactory in the limiting cases. However, a similar result using Ampère's Law would tell a different story. In fact, the calculation via Ampére's Law would yield no difference in the magnetic field of a finite wire or an infinite wire!

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### Charge Conservation

So what's going on?