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

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

PHYS 122 Notes

Table of Contents

1 Simple Harmonic Motion

2 Electricity

Electric Potentials

Calculating Capacitance of Spheres/Cylinders

Capacitance of Circuits

3 Magnetism

Magnetic Forces in a Magnetic Field

Magnetic Field Sources