# Study Notes: AP Physics C Reference Sheet

### Coulomb’s Law

Force between two point charges: $\small F=\frac{Q_{1}Q_{2}}{4\pi \varepsilon _{0}R^{2}}$

### Electric Fields

Definition of Electric Field: $\small E = \frac{F}{Q}=\frac{Q}{4\pi \varepsilon _{0}R^{2}}$
For a single point: $\small F=qE$

### Potential and Voltage

Definition of electric potential V: $\small V = \frac{U}{Q}$
For a constant electric field: $\small E_{x}=-\frac{\Delta V}{\Delta x}$
Potential due to a point charge at distance a: $\small V(a)= \frac{Q}{4 \pi \varepsilon _{0}a}$

### Gauss’s Law

Gauss’s Law: $\small \Phi_{\text{electric}} = \oint E \cdot dA = \frac{Q_{\text{enclosed}}}{\varepsilon {0}}$

Electric Field due to an infinite plane of charge with charge density $\small \sigma$: $\small E = \frac{\sigma}{2 \varepsilon _{0}}$

Electric Field due to an infinite line of charge with charge density $\small \lambda$: $\small E=\frac{\lambda}{2\pi r \varepsilon _{0}}$

### Circuit containing Batteries and Resistors

Definition of the magnitude of current: $\small \frac{dQ}{dt}$
Current density J: $\small J=\frac{I}{A}=nev_{D}$
Ohm’s law: $\small V=IR$
Resistivity: $\small \rho=\frac{J}{E}$
Resistance: $\small R= \frac{V}{I}=\frac{\rho L}{A}$
Conductivity: $\small \sigma=\frac{E}{J}$
Energy: $\small P=IV=I^{2}R=\frac{V^{2}}{R}$

### Capacitors

Definition of capaciance: $\small C=\frac{Q}{V}$
Capaciance of parallel plate with area A and distance d: $\small C=\frac{\varepsilon_{0}A}{d}$
Energy: $\small U=\frac{Q^{2}}{2C}=\frac{1}{2}CV^{2}=\frac{1}{2}QV$
Capaciance with Dielectrics: $\small C=\kappa {D} C_{0}$

### RC Circuits

#### Discharging a Capacitor through a Resistor

Charge: $\small Q(t)=Q_{0}e^{-\frac{t}{RC}}$
Current: $\small I(t)=I_{0}e^{-\frac{t}{RC}}$

#### Charging a Capacitor

Charge: $\small Q(t)=Q_{\text{final}}(1-e^{-\frac{t}{RC}})$
Current: $\small I(t)=I_{0}e^{-\frac{t}{RC}}$

### Magnetic Fields

Force exerted on point charge by magnetic field: $\small F_{\text{magnetic}}=qv \times B$
Force exerted on a straight current-carrying wire: $\small F_{\text{magnetic}}=Il \times B$
Magnetic field due to a point charge: $\small B=\frac{\mu_{0}}{4\pi}\frac{qv \times \hat{r}}{r^{2}}$
Biot-Savart Law: Magnetic field due to a differential length of wire: $\small dB=\frac{\mu_{0}}{4\pi}\frac{I}{r^{2}}dl \times \hat{r}$
Magnetic field due to a infinitiely long wire: $\small B=\frac{\mu_{0}I}{2\pi r}$
Ampere’s Law: $\small \oint B \cdot dl = \mu_{0}I_{\text{enclosed}}$

Magnetic flux: $\small \Phi_{B}=\int B\cdot dA$
Faraday’s Law: $\small \varepsilon = \frac{d \Phi}{dt}$
Farady’s Law and Electric field: $\small V=\oint E \cdot dl=\frac{d \Phi}{dt}$
Lenz’s Law: The induced current produces a magnetic field which in turn produces a flux that opposes the change in flux in the loop caused by the changing external magnetic field.

### Inductors

Self-inductance: $\small L=\frac{\Phi_{B}}{I}$ or $\small \varepsilon=L \frac{dI}{dt}$
Energy: $\small U=\frac{LI^{2}}{2}$

#### RL Circuits(Resistor and Inductor)

Current decay: $\small I(t)=I_{0}e^{-\frac{R}{L}t}$
Voltage decay: $\small V(t)=V_{0}e^{-\frac{R}{L}t}$

Current growth: $\small I(t)=I_{\text{final}}(1-e^{-\frac{R}{L}t})$
Voltage decay: $\small V(t)=V_{battery}e^{-\frac{R}{L}t}$

#### LC Circuit(Capacitor and Inductor)

Capacitor’s charge: $\small Q=K sin(\frac{t}{\sqrt{Lc}}+\Phi)$
Current: $\small I=-\frac{dQ}{dt}=-\frac{K}{\sqrt{LC}}cos(\frac{t}{\sqrt{Lc}}+\Phi)$
Angular frequency: $\small \omega = \frac{1}{\sqrt{LC}}$
Linear frequency: $\small f=\frac{\omega}{2\pi}=\frac{1}{2\pi}\sqrt{\frac{1}{LC}}$
Relationship between energy: $\small \frac{Q_{\text{max}}^{2}}{2C}=\frac{I_{\text{max}}^{2}}{2}$

### Maxwell’s Equations

Gauss’s Law for electricity: $\small \oint E \cdot dA = \frac{Q_{\text{enclosed}}}{\varepsilon {0}}$

Gauss’s Law for magnetism: $\small \oint B \cdot dA=0$

Faraday’s Law: $\small \varepsilon = \frac{d \Phi}{dt}$

Ampere-Maxwell Law: $\small \oint B \cdot dl = \mu{0}I_{\text{enclosed}} + \mu_{0}\varepsilon_{0}\frac{d\Phi_{E}}{dt}$