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}}

Faraday’s and Lenz’s Law:

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}

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Siujoeng Lau

Liberty will never perish.

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