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# System prepended metadata

title: Syed's EPHY formulae cheatsheet

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# Syed's EPHY formulae cheatsheet

<!-- If you would like to go through topics instead, refer [this](https://hackmd.io/@Zp5jWZOvRA2y4ArFHInpEA/rJYRTZo3t). -->

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|                              Aim                               | Formulae                                                                                                                                                                                                                                                        |                                             Calculator                                              |
|:--------------------------------------------------------------:|:--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |:---------------------------------------------------------------------------------------------------:|
|                 To find magnitude of a vector                  | $\vec{A} = \|A\|$                                                                                                                                                                                                                                               |                    [click](https://www.omnicalculator.com/math/vector-magnitude)                    |
|                  To find dot product (simple)                  | $\vec{A} \cdot \vec{B} = AB\cos{\theta}$                                                                                                                                                                                                                        |                      [click](https://www.omnicalculator.com/math/dot-product)                       |
|                 To find cross product (simple)                 | $\vec{A} \times \vec{B} = AB\sin{\theta}\hat{n}$                                                                                                                                                                                                                |                     [click](https://www.omnicalculator.com/math/cross-product)                      |
|                    To add vector components                    | $\vec{A} + \vec{B} = (A_x + B_x)\hat{x} + (A_y + B_y)\hat{y} + (A_z + B_z)\hat{z}$                                                                                                                                                                              |                    [click](https://www.omnicalculator.com/math/vector-addition)                     |
|                To multiply a vector by a scalar                | $a\vec{A} = aA_x\hat{x} + aA_y\hat{y} + aA_z\hat{z}$                                                                                                                                                                                                            |                  [click](https://mathforyou.net/en/online/vectors/product/number/)                  |
|              To multiply vector by a vector (dot)              | $\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z$                                                                                                                                                                                                              |                                           same as simple                                            |
|            To multiply vectory by a vector (cross)             | $\vec{A} \times \vec{B} = (A_yB_z - A_zB_y)\hat{x} \\ + (A_zB_x - A_xB_z)\hat{y} \\ + (A_xB_y - A_yB_x)\hat{z}$                                                                                                                                                 |                                           same as simple                                            |
|             To find magnitude of a position vector             | $r = \sqrt{\vec{r} \cdot \vec{r}} = \sqrt{x^2 + y^2 + z^2}$                                                                                                                                                                                                     |               [click](https://www.vcalc.com/wiki/vCalc/Magnitude+of+Position+Vector)                |
|          To find direction unit of a position vector           | $\hat{r} = \frac{\vec{r}}{r} = \frac{x\hat{x} + y\hat{y} + z\hat{z}}{\sqrt{x^2 + y^2 + z^2}}$                                                                                                                                                                   |                      [click](https://www.omnicalculator.com/math/unit-vector)                       |
|           To find infinitesimal displacement vector            | $d\vec{l} = dx\hat{x} + dy\hat{y} + dz\hat{z}$                                                                                                                                                                                                                  |     [click](https://www.easycalculation.com/physics/classical-physics/displacement-vector.php)      |
|                   To find separation vector                    | $r = r - r^{\prime}$                                                                                                                                                                                                                                            |                                                                                                     |
|             To find magnitude of separation vector             | $r =\|r - r^{\prime}\|$                                                                                                                                                                                                                                         |                                                                                                     |
|            To find unit vector of separation vector            | $\hat{r} = \frac{r}{r} = \frac{r - r^{\prime}}{\|r - r^{\prime}\|}$                                                                                                                                                                                             |                                                                                                     |
|                  To find ordinary derivative                   | $df = (\frac{df}{dx})dx$                                                                                                                                                                                                                                        |                                                                                                     |
|                  To find gradient of a vector                  | $\nabla H = \frac{\partial H}{\partial x}\hat{x} + \frac{\partial H}{\partial y}\hat{y} + \frac{\partial H}{\partial z}\hat{z}$                                                                                                                                 |                        [click](https://www.omnicalculator.com/math/gradient)                        |
|                 To find divergence of a vector                 | $\nabla \cdot \vec{v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}$                                                                                                                                    |               [click](https://www.easycalculation.com/differentiation/divergence.php)               |
|                    To find curl of a vector                    | $\nabla \times \vec{v} = \hat{x}(\frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z}) + \hat{y}(\frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x}) + \hat{z}(\frac{\partial v_y}{\partial x} - \frac {\partial v_x}{\partial y})$ |                            [click](https://www.geogebra.org/m/jWfTBWWT)                             |
|                  To calculate a line integral                  | $W = \int^{b}_{a\rho} F \cdot dl$                                                                                                                                                                                                                               |               [click](https://www.wolframalpha.com/calculators/integral-calculator/)                |
|                To calculate a surface integral                 | $\int_s = v \cdot da$                                                                                                                                                                                                                                           |                        [click](https://www.desmos.com/calculator/rhcrsvebxy)                        |
|                 To calculate a volume integral                 | $\int_v = T d\tau$                                                                                                                                                                                                                                              |     [click](https://www.wolframalpha.com/widgets/view.jsp?id=a83fc1af67a3fdc3cf56863e7f1b5dda)      |
|                 To find the conservative force                 | $F_c = -\nabla U$                                                                                                                                                                                                                                               |                [click](https://www.calculatorhut.com/physics/potential-energy.html)                 |
|            To use fundamental theorem of divergence            | $\int_v (\nabla \cdot v) d\tau = \oint_s v \cdot da$                                                                                                                                                                                                            |                   [click](https://www.symbolab.com/solver/divergence-calculator)                    |
|                     To use stokes theorem                      | $\int_s (\nabla \cdot v) da = \oint_P c \cdot dl$                                                                                                                                                                                                               |                     [click](https://www.omnicalculator.com/physics/stokes-law)                      |
|               To caclulate velocity (coordinate)               | $\vec{v} = \frac{d\vec{r}}{dt} = \frac{d}{dt} (x\hat{x} + y\hat{y})$                                                                                                                                                                                            |                                                                                                     |
|              To calculate velocity (plane-polar)               | $\vec{v} = \dot{r}\hat{r} + r\dot{\theta}\hat{\theta}$                                                                                                                                                                                                          |                                                                                                     |
|            To calculate line element (cylindrical)             | $\vec{dl} = dr\hat{r} + rd\theta\hat{\theta} + dz\hat{z}$                                                                                                                                                                                                       |                                                                                                     |
|          To calculate surface element with fix r(cyl)          | $\vec{dA} = rd\theta dz\hat{r}$                                                                                                                                                                                                                                 |                                                                                                     |
|               To calculate volume element (cyl)                | $dv = rdrd\theta dz$                                                                                                                                                                                                                                            |                                                                                                     |
|             To calculate line element (spherical)              | $\vec{dl} = dr\hat{r} + rd\theta \hat{\theta} + r\sin{\theta}d \phi \hat{\phi}$                                                                                                                                                                                 |                                                                                                     |
|            To calculate volume element (spherical)             | $dV = dl_r dl_{\theta} dl_{\phi} = r^2 \sin{\theta} drd\theta d\phi$                                                                                                                                                                                            |                                                                                                     |
|      To calculate surface element with fix r (spherical)       | $d\vec{A} = r^2 \sin{\theta}d\theta d\phi \hat{r}$                                                                                                                                                                                                              |                                                                                                     |
|            To calculate force (using coulombs law)             | $F = \frac{kqQ}{r^2} \hat{r} = \frac{1}{4\pi\varepsilon_0}\frac{qQ}{r^2} \hat{r}$                                                                                                                                                                               |    [click](https://www.calculatoratoz.com/en/electric-force-by-coulombs-law-calculator/Calc-513)    |
|          To calculate electric field of a line charge          | $E(r) = \frac{1}{4\pi\varepsilon_0} \int_P \frac{\lambda(r^{\prime})}{r^2} dl^{\prime}$                                                                                                                                                                         | [click](https://www.easycalculation.com/physics/electromagnetism/electric-field-of-line-charge.php) |
|        To calculate electric field for a surface charge        | $E(r) = \frac{1}{4\pi\varepsilon_0} \int_S \frac{\sigma(r^{\prime})}{r^2} da^{\prime}$                                                                                                                                                                          | [click](https://www.calculatoratoz.com/en/electric-field-due-to-infinite-sheet-calculator/Calc-672) |
|        To calculate electric field for a volume charge         | $E(r) = \frac{1}{4\pi\varepsilon_0} \int_V \frac{\rho(r^{\prime})}{r^2} d\tau^{\prime}$                                                                                                                                                                         |                                                                                                     |
|                 To calculate density of lines                  | $\frac{n}{4\pi r^2}$                                                                                                                                                                                                                                            |                                                                                                     |
|              To calculate electric flux (uniform)              | $\Phi_E = EA cos{\theta} = E \cdot A$                                                                                                                                                                                                                           |                      [click](https://www.omnicalculator.com/physics/gauss-law)                      |
|            To calculate electric flux (non-uniform)            | $\Phi_E = \int E \cdot \hat{n}da$                                                                                                                                                                                                                               |                                                                                                     |
|                 To apply gauss law (integral)                  | $\Phi_E = \oint \vec{E}(\vec{r}) \cdot a\vec{A} = \frac{Q_{enclosed}}{\varepsilon_0}$                                                                                                                                                                           |                                                                                                     |
|               To apply gauss law (differential)                | $\vec{\nabla} \cdot \vec{E}(\vec{r}) = \frac{\rho(\vec{r})}{\varepsilon_0}$                                                                                                                                                                                     |                                                                                                     |
|                  To use dirac delta function                   | $\int_{-\infty}^{\infty} \delta(x)dx$                                                                                                                                                                                                                           |                        [click](https://www.desmos.com/calculator/si83byyyrx)                        |
|       To calculate realization of a dirac delta function       | $\delta(x) = \lim_{s \to 0} \frac{1}{\sqrt{2\pi s^2}}exp [- \frac{(x - a^2)}{2s^2}]$                                                                                                                                                                            |                                                                                                     |
|           To calculate electric potential (integral)           | $V(\vec{r}) = -\int_{O_{ref}}^r \vec{E}(\vec{r}) \cdot d\vec{l}$                                                                                                                                                                                                |                                                                                                     |
|         To calculate electric potential (differential)         | $\vec{E}(\vec{r}) = - \vec{\nabla}V(\vec{r})$                                                                                                                                                                                                                   |                                                                                                     |
|                  To apply poisson's equation                   | $\nabla^2 V(\vec{r}) = \frac{\rho_{encl}(\vec{r})}{\varepsilon_0}$                                                                                                                                                                                              |                    [click](https://stattrek.com/online-calculator/poisson.aspx)                     |
|          To calculate potential energy (generalised)           | $P.E. = W = Q_TV(\vec{r})$                                                                                                                                                                                                                                      |                                                                                                     |
|                  To calculate total work done                  | $W_{TOT} = \frac{\varepsilon_0}{2} \int_{V_{all \\ space}} E^2(\vec{r})d\tau$                                                                                                                                                                                   |                                                                                                     |
|         To use Energy property of a conductor (extern)         | $\vec{E}_{ext}(\vec{r}) = E_0 \hat{x}$                                                                                                                                                                                                                          |                                                                                                     |
|         To use Energy property of a conductor (inside)         | $\vec{E}_{induced \\ inside}(\vec{r}) = -\vec{E}_{ext}(\vec{r}) = -E_0 \hat{x}$                                                                                                                                                                                 |                                                                                                     |
|           To use volume free charge density property           | $\vec{\nabla} \cdot \vec{E}_{inside}(\vec{r}) = \rho_{free \\ inside} \frac{(\vec{r})}{\varepsilon_0} \implies \rho_{free \\ inside}(\vec{r}) = 0$                                                                                                              |                                                                                                     |
|               To calculate capacitance (normal)                | $C = \frac{Q}{V}$                                                                                                                                                                                                                                               |               [click](https://www.allaboutcircuits.com/tools/capacitance-calculator)                |
|         To calculate work done in charging a capacitor         | $W = \int_0^Q (\frac{Q}{C})dq = \frac{1}{2} \frac{Q^2}{C} = \frac{1}{2}CV^2$                                                                                                                                                                                    |                                                                                                     |
|           To calculate capacitance (parralel-place)            | $C = \frac{\varepsilon_0 A}{d}$                                                                                                                                                                                                                                 |                                                                                                     |
|                   To calculate dipole energy                   | $E_{dip}(r, \theta) = \frac{P}{4\pi \varepsilon_0 r^3}(2 cos{\theta}\hat{r} + sin{\theta}\hat{\theta})$                                                                                                                                                         |     [click](https://www.calculatoratoz.com/en/electric-potential-of-dipole-calculator/Calc-580)     |
|                To calculate N (dipole context)                 | $N = p \times E$                                                                                                                                                                                                                                                |                                                                                                     |
|                  To calculate polarizability                   | $P = aE$                                                                                                                                                                                                                                                        |           [click](https://www.calculatoratoz.com/en/polarizability-calculator/Calc-4242)            |
|      To calculate surface charge density (dipole context)      | $\sigma_b = P(r^\prime) \cdot \hat{n}$                                                                                                                                                                                                                          |                                                                                                     |
|      To calculate volume charge density (dipole context)       | $\rho_b = -\nabla^\prime \cdot P(r^\prime)$                                                                                                                                                                                                                     |                                                                                                     |
|               To calculate electric displacement               | $D = \epsilon_0 E + P$                                                                                                                                                                                                                                          |           [click](https://physics.icalculator.info/displacement-current-calculator.html)            |
|      Gauss law in presence of a dielectric (differential)      | $\nabla \cdot D = \rho f$                                                                                                                                                                                                                                       |                                                                                                     |
|        Gauss law in presence of a dielectric (integral)        | $\oint D \cdot da = Q_{f enc}$                                                                                                                                                                                                                                  |                                                                                                     |
|      To find dipole moment P (linear dielectrics context)      | $P = \epsilon_0 X_e E$                                                                                                                                                                                                                                          |                                                                                                     |
|               To find permittivity of a material               | $\epsilon = \epsilon_0 (1 + X_e)$                                                                                                                                                                                                                               |     [click](https://www.vcalc.com/wiki/TylerJones/Dielectric+Constant+(Relative+Permittivity))      |
|                To find the dielectric constant                 | $\epsilon_r = \frac{\epsilon}{\epsilon_0} = 1 + X_e$                                                                                                                                                                                                            |                                          ig same as above                                           |
|        To find electric force (magnetostatics context)         | $F_{elec} = QE$                                                                                                                                                                                                                                                 |                                           same as simple                                            |
|            To find magnetic force (magnetostatics)             | $F_{mag} = Q(v \times B)$                                                                                                                                                                                                                                       |                 [click](https://physicscalc.com/physics/lorentz-force-calculator/)                  |
|           To calculate total force (magnetostatics)            | $F = F_{elec} + F_{mag} = Q(E + v \times B)$                                                                                                                                                                                                                    |                                                                                                     |
|           To calculate work done by magnetic forces            | $W_{mag} = \int Q(v \times B) \cdot vdt$                                                                                                                                                                                                                        |                                                                                                     |
|                To find current (charge flowing)                | $I = \frac{dq}{dt} = \frac{\lambda dl}{dt} = \lambda v$                                                                                                                                                                                                         |                                                                                                     |
|       To find magnetic force on a current carrying wire        | $F_{mag} = (v \times B)Q = \int (v \times B)dq = \int (v \times B)\lambda dl = I \int (dl \times B)$                                                                                                                                                            |    [click](https://physicscalc.com/physics/magnetic-force-on-current-carrying-wire-calculator/)     |
|              To find charge flowing on a surface               | $F_{mag} = (v \times B)Q = \int (v \times B)\sigma da = \int (K \times B)da$                                                                                                                                                                                    |                                                                                                     |
|               To find charge flowing in a volume               | $F_{mag} = (v \times B)Q = \int (v \times B)\rho d\tau = \int (J \times B)d\tau$                                                                                                                                                                                |                                                                                                     |
|                 To use the continuity equation                 | $\nabla \cdot J = -\frac{d\rho}{dt}$                                                                                                                                                                                                                            |                                                                                                     |
|          To find total charge crossing (closed surf)           | $I = \oint_S J \cdot da$                                                                                                                                                                                                                                        |                                                                                                     |
|           To find total charge leaving (closed surf)           | $\oint_S J \cdot da = \int_V(\nabla \cdot J)d\tau$                                                                                                                                                                                                              |                                                                                                     |
|                   To use the biot-savart law                   | $B(r) = \frac{\mu_0}{4\pi} \int \frac{I \times \hat{r}}{r^2} = \frac{\mu_0}{4\pi} I \int \frac{dl^\prime \times \hat{r}}{r^2}$                                                                                                                                  |        [click](https://www.easycalculation.com/physics/electromagnetism/biot-savart-law.php)        |
|       To find magnetic field produced by surface current       | $B(r) = \frac{\mu_0}{4\pi} \int \frac{K(r^\prime) \times \hat{r}}{r^2}da^\prime$                                                                                                                                                                                |                                                                                                     |
|      To find magnetic field produced by a volume current       | $B(r) = \frac{\mu_0}{4\pi} \int \frac{J(r^\prime) \times \hat{r}}{r^2}d\tau^\prime$                                                                                                                                                                             |                                                                                                     |
|            Divergence and curl relation (not sure)             | $\oint B \cdot dl = \mu_0 I$                                                                                                                                                                                                                                    |                                                                                                     |
|               To use ampere's law (differential)               | $\nabla \times B = \mu_0 J$                                                                                                                                                                                                                                     |               [click](https://www.calctool.org/CALC/phys/electromagnetism/ampere_law)               |
|                 To use ampere's law (integral)                 | $\oint B \cdot dl = \mu_0 I_{enc}$                                                                                                                                                                                                                              |                                                                                                     |
|             To calculate magnetic vector potential             | $B = \nabla \times A$                                                                                                                                                                                                                                           |                                                                                                     |
|       To calculate vector potential for surface current        | $A(r) = \frac{\mu_0}{4\pi} \int \frac{K(r^\prime)}{r} da^\prime$                                                                                                                                                                                                |                                                                                                     |
|         To calculate vector potential for line current         | $A(r) = \frac{\mu_0}{4\pi} I \int \frac{dl^\prime}{r}$                                                                                                                                                                                                          |                                                                                                     |
|                To calculate monopole potential                 | $A_{mono}(r) = \frac{\mu_0 I}{4\pi} \frac{1}{r} \oint dl^\prime = 0$                                                                                                                                                                                            |                                          ig same as below                                           |
|                 To calculate dipole potential                  | $A_{dip}(r) = \frac{\mu_0}{4\pi} \frac{m \times \hat{r}}{r^2}$                                                                                                                                                                                                  |               [click](https://www.omnicalculator.com/physics/magnetic-dipole-moment)                |
|                 To find magnetic dipole moment                 | $m = I \int da^\prime$                                                                                                                                                                                                                                          |                                            same as above                                            |
|   To find volume current in the field of a magnetized object   | $J_b(r^\prime) = \nabla^\prime \times M(r^\prime)$                                                                                                                                                                                                              |                                                                                                     |
|  To find surface current in the field of a magnetized object   | $K_b(r^\prime) \times \hat{n}$                                                                                                                                                                                                                                  |                                                                                                     |
|   To use ampere's law in magnetized material (differential)    | $\nabla \times H = J_f$                                                                                                                                                                                                                                         |                                                                                                     |
|         | $\oint H \cdot dl = I_{fenc}$                                                                                                                                                                                                                                   |                                                                                                     |
|              To find the magnetic susceptibility               | $M = X_m H$                                                                                                                                                                                                                                                     |                                              see below                                              |
|               To find the magnetic permeability                | $\mu = \mu_0(1 + X_m)$                                                                                                                                                                                                                                          |                [click](https://www.omnicalculator.com/physics/magnetic-permeability)                |
|                  To use ohms law (empirical)                   | $J = \sigma E$                                                                                                                                                                                                                                                  |                                                                                                 |
|             To calculate electromotive force (emf)             | $\epsilon = \oint f \cdot dl = \oint f_s \cdot dl$                                                                                                                                                                                                              |                       [click](https://www.omnicalculator.com/physics/faraday)                       |
|                   To calculate motional emf                    | $\epsilon = \oint f_{mag} \cdot dl = \oint(v \times B) \cdot dl = vBh$                                                                                                                                                                                          |            [clikc](https://www.calculatoratoz.com/en/motional-emf-calculator/Calc-2145)             |
| To calculate electric field induced on changing magnetic field | $\nabla \times E = \frac{\partial B}{\partial t}$                                                                                                                                                                                                               |                                                                                                     |
|                       maxwells equations                       | too lazy to do those                                                                                                                                                                                                                                            |                                                                                                     |

> Empty calculator row means we couldn't find one for that respective aim.

## Credits

- Syed (https://github.com/SyedAhkam)
- Prarthana
- Aryan
- Aarush (Acharya)