# 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)