Physics 2 Equations

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Math – Physics 1 – Electrostatics – Circuits – Magnetism – Electromagnetic Induction – AC Circuits – Electromagnetic Waves – Optics – Symbols and Units – Constants

Last modified: Thu December 14 2023, 02:06 PM.

Math

\[ \vec{A} = \begin{bmatrix} A_x \\ A_y \\ A_z \end{bmatrix} = A_x \hat{\imath} + A_y \hat{\jmath} + A_z \hat{k} = A\hat{u} \] \[ \hat{u} = \frac{\vec{A}}{ A } \] \[ c\vec{A}+d\vec{B} = \begin{bmatrix}cA_x+dB_x \\ cA_y+dB_y\\ cA_z+dB_z\end{bmatrix} \]

\[ \vec{A} \cdot \vec{B} = A_x B_x +A_y B_y + A_z B_z = A B \cos \theta \] \[ \vec{A}\times\vec{B}=\left (AB \sin \theta \right ) \hat{u}_{\perp} \] \[ \vec{A}\times\vec{B} =\left (A_yB_z-A_zB_y \right )\hat{\imath}+ \left (A_zB_x-A_xB_z \right )\hat{\jmath}+ \left (A_xB_y-A_yB_x \right ) \hat{k} \]

\[ \hat{u}=\hat{\imath} \cos \theta_x + \hat{\jmath} \sin \theta_x \] \[ \sin \theta_x = \cos \theta_y \]

useful approximations:

\[ |x| \ll 1 \] \[ (1+x)^\alpha \approx 1+ \alpha x \] \[ \rm{if}\; \theta \ll 1 \left\{\begin{matrix} \sin \theta \approx \theta \\ \tan \theta \approx \theta\\ \cos \theta \approx 1 \end{matrix}\right. \]

Physics 1

\[ \vec{F} = m \vec{a} \] \[ \vec{F}_{12} = -\vec{F}_{21} \] \[ K = \tfrac{1}{2}mv^2 \] \[ \vec{F}_G = m\vec{g} \] \[ W = \vec{F} \cdot \vec{s} \] \[ F_\mathrm{centrip} = m \frac{v^2}{r} \] \[ \vec{\tau} = \vec{r} \times \vec{F} \]

Electrostatics

\[ \vec{F} = \frac{1}{4 \pi \epsilon_0} \frac{ q_1 q_2 }{r^2} \hat{r} \] \[ k = \frac{1}{4 \pi \epsilon_0} \] \[ \vec{F} = q \vec{E} \] \[ \vec{p} = q \vec{s} \] \[ \vec{E} = \frac{1}{4 \pi \epsilon_0} \frac{ q }{r^2} \hat{r} \] \[ \vec{E} = \frac{1}{4 \pi \epsilon_0} \frac{ 2 \lambda }{r} \hat{r} \] \[ \vec{E} = \frac{ \sigma }{2 \epsilon_0} \hat{n} \] \[ \vec{E} = \frac{ \sigma }{ \epsilon_0 } \hat{n} \] \[ \vec{E}_{\rm{dp}} = \frac{1}{4 \pi \epsilon_0} \frac{ 2 \vec{p} }{r^3} \] \[ \vec{\tau} = \vec{p} \times \vec{E} \] \[ \Phi_{\rm{e}} = \vec{E} \cdot \vec{A} \] \[ \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\rm{in}}}{\epsilon_0} \] \[ V = \frac{1}{4 \pi \epsilon_0} \frac{q}{r} \] \[ U = qV \] \[ U_{\rm{dp}} = -\vec{p} \cdot \vec{E} \] \[ E_s = - \frac{dV}{ds} \] \[ \vec{E} = -\nabla V \] \[ \Delta V = - \int_{i}^{f} E_s \; ds \] \[ C = \frac{Q}{V_C} \] \[ C_{\rm{pp}} = \epsilon_0 \frac{A}{d} \] \[ C = \kappa C_0 \] \[ U = \frac{QV_C}{2} = \frac{CV_C^2}{2} = \frac{Q^2}{2C} \] \[ u_e = \tfrac{1}{2} \kappa \epsilon_0 E^2 \]

Circuits

\[ \frac{1}{C_T} = \frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3} + \cdots \] \[ C_T = C_1 + C_2 + C_3 +\cdots \] \[ J = \frac{I}{A} \] \[ I = \frac{d q}{d t} \] \[ I = n_e e A v_d \] \[ J = \sigma E \] \[ \sigma = \rho^{-1} \] \[ R = \frac{\rho L}{A} \] \[ V = IR \] \[ P = IV = \frac{V^2}{R} = I^2R \] \[ V_T = V_1 + V_2 + V_3 +\cdots \] \[ V_1 = V_2 = V_3 = \cdots \] \[ I_1 = I_2 = I_3 = \cdots \] \[ I_T = I_1 + I_2 + I_3 +\cdots \] \[ R_T = R_1 + R_2 + R_3 +\cdots \] \[ \frac{1}{R_T} = \frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3} + \cdots \] \[ \sum \Delta V_i = 0 \] \[ \sum I_{\rm{in}} = \sum I_{\rm{out}} \] \[ \tau = RC \] \[ X(t) = X_0 e^{-t/\tau} \] \[ X(t) = X_0 \left (1-e^{-t/\tau} \right ) \]

Magnetism

\[ \vec{B} = \frac{\mu_0}{4 \pi} \frac{q\vec{v}\times\hat{r}}{r^2} \] \[ d\vec{B} = \frac{\mu_0}{4 \pi} \frac{I d \vec{\ell}\times\hat{r}}{r^2} \] \[ \vec{\mu} = N I A \hat{n} \] \[ \oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{\rm{thu}} \] \[ B_{\rm{wire}} = \frac{\mu_0 I}{2 \pi r} \] \[ B_{\rm{loop}} = \frac{\mu_0 I}{2 R} \] \[ B_{\rm{sol}} = \frac{\mu_0 N I}{\ell} \] \[ \vec{F} = q\vec{v} \times \vec{B} \] \[ \vec{F} = I\vec{\ell} \times \vec{B} \] \[ \vec{\tau} = \vec{\mu} \times \vec{B} \] \[ F = \frac{\mu_0 I_1 I_2}{2 \pi r} \ell \] \[ r_{\rm{cyc}} = \frac{m v}{qB} \] \[ f_{\rm{cyc}} = \frac{qB}{2 \pi m} \]

Electromagnetic Induction

\[ \mathcal{E} = B l v \] \[ \mathcal{E} = NBA \omega \sin (\omega t) \] \[ \Phi_{\rm{m}} = \int_{\rm{area}} \vec{B} \cdot d\vec{A} \] \[ M = \frac{N_2 \Phi_2}{I_1} = \frac{N_1 \Phi_1}{I_2} \] \[ \mathcal{E}_1 = -M \frac{dI_2}{dt} \] \[ \mathcal{E} = \oint \vec{E} \cdot d\vec{\ell} = - \frac{d\Phi_{\rm{m}}}{dt} \] \[ L = N \frac{\Phi_{\rm{m}}}{I} \] \[ \Delta V_L = - L \frac{dI}{dt} \] \[ \mathcal{E} = - L \frac{dI}{dt} \] \[ V_L(t) = V_0 e^{-t/\tau} \] \[ \tau = L/R \] \[ \tau = \frac{L}{R} \] \[ V_L(t) = V_0 \cos( \omega t + \phi) \] \[ \omega = \sqrt{ \frac{1}{LC}} \] \[ \omega = \frac{1}{\sqrt{LC}} \] \[ U_L = \tfrac{1}{2}L I^2 \] \[ u_B = \frac{1}{2 \mu_0} B^2 \] \[ L_{\rm{sol}} = \frac{\mu_0 N^2 A}{\ell} \]

AC Circuits

\[ \frac{V_2}{V_1} = \frac{N_2}{N_1} \] \[ \omega = 2 \pi f = \frac{2 \pi}{T} \] \[ \mathcal{E}(t) = \mathcal{E}_0 \cos \omega t \] \[ i(t) = I \cos ( \omega t - \phi ) \] \[ V_R = I R \] \[ V_C = I X_C \] \[ V_L = I X_L \]

\[ X_C = 1/\omega C \] \[ X_L = \omega L \] \[ \mathcal{E}_0 = I Z \] \[ Z = \sqrt{R^2 + (X_L-X_C)^2} \] \[ \mathcal{E}_0^2 = V_R^2 +(V_L-V_C)^2 \] \[ \phi = \tan^{-1} \frac{X_L - X_C}{R} \] \[ V_{\rm{rms}} = \frac{V}{\sqrt{2}} \] \[ I_{\rm{rms}} = \frac{I}{\sqrt{2}} \] \[ P_R = I_{\rm{rms}} V_{\rm{rms}} \] \[ P = I_{\rm{rms}} \mathcal{E}_{\rm{rms}} \cos \phi \] \[ \omega_c = \frac{1}{RC} \] \[ \omega_0 = \frac{1}{\sqrt{LC}} \]

Electromagnetic Waves

\[ \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\rm{in}}}{\epsilon_0} \] \[ \oint \vec{B} \cdot d\vec{A} = 0 \] \[ \oint \vec{E} \cdot d\vec{\ell} = - \frac{d\Phi_{\rm{m}}}{dt} \] \[ \oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{\rm{thru}} + \epsilon_0 \mu_0 \frac{d\Phi_{\rm{e}}}{dt} \] \[ \vec{F} = q ( \vec{E} + \vec{v} \times \vec{B} ) \] \[ c = f \lambda \] \[ \frac{E_0}{B_0} = c \] \[ \vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B} \] \[ I = S_{\rm{ave}} = \frac{E_0 B_0}{2 \mu_0} \] \[ I = \frac{P_{\rm{source}}}{4 \pi r^2} \] \[ c^2 = \frac{1}{\epsilon_0 \mu_0} \] \[ I = I_0 \cos^2 \theta \]

Optics

\[ f = \frac{R}{2} \] \[ \frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f} \] \[ \frac{1}{f} = \left (n-1 \right ) \left (\frac{1}{R_1}-\frac{1}{R_2} \right ) \] \[ m = \frac{h_i}{h_o} = -\frac{d_i}{d_o} \] \[ n = \frac{c}{v} \] \[ \frac{n_1}{n_2} = \frac{v_2}{v_1} = \frac{\lambda_2}{\lambda_1} = \frac{\sin \theta_2}{\sin \theta_1} \] \[ \sin \theta_c = \frac{n_2}{n_1} \] \[ d \sin \theta = m \lambda \] \[ a \sin \theta = m \lambda \] \[ D \sin \theta = 1.22 \lambda \] \[ M = \frac{\theta_i}{\theta_o} \] \[ I = I_0 \left ( \frac{\sin (\phi/2)}{\phi/2} \right )^2 \] \[ I = I_0 \left ( \frac{\sin \beta}{\beta} \right )^2 \] \[ \phi = 2 \pi \frac{a \sin \theta}{\lambda} \]

Symbols and Units

Constants