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# 1 Simple Harmonic Motion

Useful Formula
$\omega=\frac{\Delta\theta}{\Delta t}=\frac{2\pi}{T}=2\pi f$
$x=x_0\cos(\omega t)$ describes the object that is at $x=x_0$ at $t=0$.
General Equation: $x=x_0\cos(\omega t + \phi)$
\begin{align*} \frac{dx}{dt}&=\frac{d}{dt} x_0\cos(\omega t + \phi)\\ v&= -\omega x_0\sin(\omega t + \phi)\\ a&= -\omega^2 x_0\cos(\omega t + \phi)=-\omega^2 x \end{align*}
A simple harmonic oscillator is:
• A projection of an object moving in a circle.
• An object whose acceleration is proportional to its position.
e.g. For an object on a spring undergoing simple harmonic motion, \begin{align*} F&=-kx\\ ma&= -kx\\ a&= -\frac{k}{m}x\\ \therefore \frac{k}{m}&=\omega^2\\ \omega &= \sqrt{\frac{k}{m}} \end{align*}

# 2 Electricity

## Electric Potentials

$$\Delta V=\frac{\Delta U}{q}=-\int_a^b\vec{E}\cdot d\vec{r}$$ $$V_{ab}=V_a-V_b=-\int_b^a\vec{E}\cdot d\vec{r}=\int_a^b\vec{E}\cdot d\vec{r}$$
• $\Delta V_{\text{point}}=\frac{Q}{4\pi\epsilon_0 r}$
• $\Delta V_{\text{line}}=\frac{\lambda}{2\pi\epsilon_0}\ln(\frac{r_0}{r})$, where $r_0$ is the reference point where $V=0$ (normally radius of outer shell)
• $\Delta V_{\text{place}}=-\frac{\sigma}{2\epsilon_0}\Delta r$
• In a conductor/inside a conducting shell, $V$ is constant, but not necessarily $0$. ($\vec{E}=0$)
To find the electric field given Electric Potential at a certain radial distance, $$E_r=-\frac{\delta V}{\delta r}$$

## Calculating Capacitance of Spheres/Cylinders

1. Find Potential Difference between outer and inner shell. Treat cylinders as lines of charge and spheres as a point charge.
2. Use $C=\frac{Q}{\Delta V}$ ($Q$ should cancel out)

## Capacitance of Circuits

• Series: $Q$ is constant.
• Parallel: $\Delta V$ is constant.
• Use $Q=C\Delta V$.

# 3 Magnetism

## Magnetic Forces in a Magnetic Field

• Moving Point Charge: $\vec{F}=q\vec{v}\times\vec{B}$
• Current-Carrying Wire: $\vec{F}=I\vec{l}\times\vec{B}$

## Magnetic Field Sources

• Point Charge: $\vec{B}=\frac{\mu_0}{4\pi}\frac{q\vec{v}\times\hat{r}}{r^2}$
• Current: $\vec{B}=\frac{\mu_0}{4\pi}\int\frac{I d\vec{l}\times\hat{r}}{r^2}$ [Biot-Savart's Law]
• Near a long, straight, current-carrying conductor: $\vec{B}=\frac{\mu_0 I}{2\pi r}$
• In the middle of a current-carrying loop: $\vec{B}=\frac{\mu_0 I}{2r}$
• Solenoid: $B_x=\mu_0 nI$, $n$ is the number of loops per unit length
• Ampere's Law: $\oint \vec{B}\cdot d\vec{l}=\mu_0 I_{\text{encl}}$