Unit 1 - Electrostatics
Charging Objects via Induction and Conduction
Conductors
- Inside, $\vec{E} = 0$.
- Outside, $\vec{E}$ behaves like a point charge.
Insulators
- Inside, $\vec{E} \neq 0$, use charge density $\rho$ to find $Q_{enc}$:
- Uniform:
- $Q_{enc} = \rho V$
- $Q_{enc}=Q_{total} (\frac{r^3}{R^3})$
- Non-Uniform:
- $Q_{enc}= \int \rho \,dv$
- Outside, $\vec{E}$ behaves like a point charge.
Grounding
- Grounding connects a system to the Earth.
- Essentially, it acts as an infinite well of electrons that can get taken or added to.
- When a system is grounded, the system is no longer isolated, and the Conservation of Charge law no longer applies.
Coulomb's Law for Discrete Point Charges
$|\vec{F_{E}}|$ $ = \frac{1}{4 \pi \epsilon_{0}} \, \frac{q_1 q_2}{r^2}$
- Magnitude of the electric field of a point charge can be obtained through Coulomb's Law: $|\vec{E}|$ $ = \frac{|\vec{F_{E}}|}{q} = \frac{1}{4 \pi \epsilon_{0}} \, \frac{Q}{r^2}$.
- Used to find forces on charges $q_1$ and $q_2$ when they are a distance $r$ apart from one another.
- Use Coulomb's Law to find the magnitude of the force, then find the direction of the force separately.
- Like charges repel one another, while opposite charges attract.
- Forces can also "add together/subtract against each other" depending on their directions:
- Two different electric forces going in the same direction will cause a net electric force with an equal magnitude of both of the forces added together.
- Two electric forces going against one another will cause a net electric force with a magnitude that is equal to both forces being subtracted against each other.
Electric Field Due to Some Source on a Test Charge
- Test charges are always positive unless otherwise stated.
$|\vec{E}|$ $ = \frac{|\vec{F_{E}}|}{q} = \frac{1}{4 \pi \epsilon_{0}} \, \frac{Q}{r^2}$
- Calculate as magnitude, then use the electric field lines to find direction.
- Positively charged ions follow electric field lines.
- Negatively charged ions oppose electric field lines.
Continuous Line/Arc of Charge
$dE = \frac{1}{4 \pi \epsilon_{0}} \frac{dq}{r^2} \, \hat{r}$
When calculating magnitude, $dE = \frac{1}{4 \pi \epsilon_{0}} \frac{dq}{r^2} \,$.
Gauss's Law
$\oint \vec{E} \cdot d\vec{a} = \frac{q_{enc}}{\epsilon_{0}}$
There are 3 main situations where Gauss's Law can be used:
- Sphereical Electric fields, where $\oint \vec{E} \cdot d\vec{a} = \vec{E} \, (4 \pi r^2)$
- "Very long" Cylinders/Wires (Lines of Charge), where $\oint \vec{E} \cdot d\vec{a} = \vec{E} \, (2 \pi r\ell)$
- Infinite Sheets of Charge, where $\vec{E} = \frac{\sigma}{2 \epsilon_{0}}$