Formula Sheet

Phys121 Formula Sheet

Electrostatics $\vec{F}=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r_{12}{}^2}\hat{r}\oint \overrightarrow{\mathrm{E}}\cdot \mathrm{d}\overrightarrow{\mathrm{A}}=\frac{{\mathrm{Q}}_{\mathrm{encl}}}{{\epsilon }_0}{V}_B-V_A=-{\int }_A^B\overrightarrow{\mathrm{E}}\cdot \mathrm{d}\stackrel{\rightharpoonup }{\ell }V(r)=\frac{1}{4\pi {\epsilon }_0}\frac{q}{r}$

Capacitors and RC Circuits

Simple Circuits

$V=IRR=\frac{\rho L}{A}P=IV=I^{2}R$

Magnetism, Induction and RL Circuits

$\vec{F}=q\vec{E}+q\vec{v}\times \vec{B}d\vec{F}=Id\vec{\ell }\times \vec{B}d\vec{B}=\frac{{\mu }_0}{4\pi }\frac{Id\vec{\ell }\times \hat{r}}{r^2}$

$\vec{\mu }=NI\vec{A}$

$\vec{\tau }=\vec{\mu }\times \vec{B}$

$U=-\vec{\mu }\cdot \vec{B}$

$B={\mu }_{0}nI$

$\epsilon =-\frac{d{\Phi }_{\mathrm{B}}}{dt}{\Phi }_{\mathrm{B}}=\int \overrightarrow{\mathrm{B}}\cdot \mathrm{d}\stackrel{\rightharpoonup }{\mathrm{A}}{\Phi }_L=L\frac{dI}{dt}$

$U_L=\frac{1}{2}L{I}^{2}I(t)=I(\infty )\left(1-e^{-t/\tau }\right)I(t)=I(0)e^{-t/\tau }\tau =\frac{L}{R}$

$u_B=\frac{1}{2}\frac{B^2}{{\mu }_0}\oint \vec{B}\cdot d\vec{\ell }={\mu }_{0}I+{\mu }_0{I}_D{I}_D={\epsilon }_0\frac{d{\Phi }_{\mathrm{E}}}{dt}$

Electromagnetic Waves

$E=E_1\cos (kx-\omega t+\varphi );\omega =2\pi f;v=f\lambda =\frac{\omega }{k};v=\frac{c}{n};k=\frac{2\pi }{\lambda }$.

$E=cBu={\epsilon }_0{E}^2\vec{S}=\frac{\vec{E}\times \vec{B}}{{\mu }_0}{f}^{\prime }=f\sqrt{\frac{1\pm v/c}{1\mp v/c}}{f}^{\prime }\approx f\left(1\pm \frac{v}{c}\right)$

Geometric Optics

$n_1\sin {\theta }_1=n_2\sin {\theta }_2\sin {\theta }_c=\frac{n_2}{n_1}\tan {\theta }_B=\frac{n_2}{n_1}$

$\frac{1}{s}+\frac{1}{s^{\prime }}=\frac{1}{f}M=-\frac{s^{\prime }}{s}\frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)f=\frac{R}{2}$

Wave Optics

Resolution limited by a circular aperture: $\theta =\frac{1.22\lambda }{D}$

Constants

$\mathrm{k}=8.99\times 10^9\frac{\mathrm{N}\cdot {\mathrm{m}}^2}{{\mathrm{C}}^2}{\epsilon }_0=8.85\times 10^{-12}\frac{{\mathrm{C}}^2}{\mathrm{N}\cdot {\mathrm{m}}^2}{\mu }_0=4\pi \times 10^{-7}\frac{\mathrm{T}\cdot \mathrm{m}}{\mathrm{A}}$

$c=\frac{1}{\sqrt{{\epsilon }_0{\mu }_0}}=3.0\times 10^8\mathrm{m}/\mathrm{s}e=1.60\times 10^{-19}\mathrm{C}{m}_e=9.11\times 10^{-31}\mathrm{kg}\mathrm{g}=9.81\mathrm{m}/{\mathrm{s}}^2$

Circle: Circumference $=2\pi R$, Area $=\pi R^2$

Sphere: Area $=4\pi R^2$, Volume $=\frac{4\pi }{3}{R}^3$

$\underline{\text{ Basic Integrals }}$

(1) $\int x^{n}dx=\frac{1}{n+1}{x}^{n+1},n\ne -1$

$$\begin{array}{c}\int {\sec }^{2}xdx=\tan x\end{array}\text{\hspace{1em}(11)}$$ $$\begin{array}{c}\int \frac{1}{x}dx=\ln |x|\end{array}\text{\hspace{1em}(2)}$$

(3) $\int udv=uv-\int vdu$

(4)

$$\begin{array}{c}\int e^{x}dx=e^x\\ \int a^{x}dx=\frac{1}{\ln a}{a}^x\\ \int \ln xdx=x\ln x-x\\ \text{ (6) }\int \sin xdx=-\cos x\\ \text{ (8) }\int \tan xdx=\sin x\\ \text{ (9) }\int \sec xdx=\ln |\sec x+\tan x|\end{array}\text{\hspace{1em}(14)(5)}$$

(15) $\int \frac{1}{\sqrt{a^2-x^2}}dx={\sin }^{-1}\frac{x}{a}$

(16) $\int \frac{a}{x\sqrt{x^2-a^2}}dx={\sec }^{-1}\frac{x}{a}$

(17) $\int \frac{1}{\sqrt{x^2-a^2}}dx={\cosh }^{-1}\frac{x}{a}$ $=\ln \left(x+\sqrt{x^2-a^2}\right)$

$$\int \frac{a}{a^2-x^2}dx=\frac{1}{2}\ln \left|\frac{x+a}{x-a}\right|$$