AP Physics C: E&M Formulas

Electric force formula for two point charges?

$$\vec{F}_e=\frac{kq_1q_2}{r²}$$

Note that this force is either attractive OR repulsive

E-field formulas?

$$\vec{E}=\frac{\vec{F}_e}{q_0}=\frac{k_eq}{r²}=k\int \frac{1}{r²}\hat{r}dq$$

Electric force in terms of the E-field?

$$\vec{F}_e=q\vec{E}$$

What are E-fields?

imaginary lines that show how a positive test charge would behave if placed in the field

What are some E-field properties?

1: They point in the direction a positive charge would go in

2: They start at positives and terminate at negatives

3: When Q>0, E-field and force point in the same direciton

4: When Q<0, E-field and force point in the opposite direction

5: E-field lines never cross each other

6: Uniform E-fields have uniform acceleration

E-field for long distances of lines of charge?

$$\vec{E}_x=\frac{2k\lambda}{D}$$

E-fields for lines of charge perpendicular of the center?

$$\vec{E}_x=\frac{kQ}{D\sqrt{D²+\left(\frac{L}{2}\right)²}}$$

E-fields for a ring of charge?

$$\vec{E}_x=\frac{kQD}{(R²+D²)^{\frac{3}{2}}}$$

E-fields for an arc of charge?

$$\vec{E}_x=\frac{k\lambda}{R}\int_{-\theta}^{\theta}\cos\theta \hat{i}d\theta$$

Electric potential energy for two particles?

$$U_e=\frac{kq_1q_2}{r}$$

What does it mean for electric potential energy to be positive or negative?

Positive: Work has been done to move two objects together. This means a force had to act against a resistive electric force.

Negative: Work has been done to move two objects away from each other. This means a force had to act against an attractive electric force

Electric potential of a singular charged particle moving through a potential difference?

$$U_e=q\Delta V$$

Potential formula?

$$V=\frac{kq}{r}$$

E-field and voltage formula?

$$\vec{E}_x=-\frac{dV}{dx}$$

Integral potential formula?

$$\Delta V = -\int \vec{E}\cdot d\vec{s}$$

What are some properties of equipotential lines?

1: They are always perpendicular of the E-field

2: E-field lines point in decreasing potential

Work done by an electric force?

$W=q\Delta V=q\int_{A}^{B}\vec{E}\cdot d\vec{s}$ =$-\Delta U_e$

Integral change in electric potential energy formula?

$$\Delta U_e=U_B-U_A=-q\int_{A}^{B}\vec{E}\cdot d\vec{s}$$

E-fields and work?

$$E=\frac{W}{q\Delta d}$$

Properties of potential and electric potential energy?

1: Potential is only for fields

2: Potential energy is only for field-charge systems

3: Potential is independent of the test charge

4: Potential has a value everywhere in the field

5: Electric potential is not electric potential energy

Capacitance formula?

$C=\frac{Q}{\Delta V}$ $=\frac{\epsilon_0 A}{d}$

Combining capacitance in series? In parallel?

$Parallel: \; C_eq=C_1+C_2+C_3+…+C_n$
$Series: \; \frac{1}{C_eq}=\frac{1}{C_1}+\frac{1}{C_2}+…+\frac{1}{C_n}$

E-field and potential formula?

$$E=\frac{\Delta V}{d}\Rightarrow \Delta V= Ed$$

Charge distribution properties for non-similar conductors?

1: Both spheres will have equal voltage but not charge

2: The charge on both spheres is equal to $\frac{q_1}{R_1}=\frac{q_2}{R_2}$

Potential energy for capacitors formula?

$$U_C=\frac{Q²}{2C}=\frac{1}{2}C\Delta V² = \frac{1}{2}Q\Delta V$$

Capacitance with dielectrics?

$$C=\kappa \epsilon_0 \frac{A}{d}$$

Properties of dielectrics?

1: They always increase capacitance

2: They always decrease E-field strength and potential difference

3: Charge stored on the plates is unaffected

Dielectric constant formula

$$\kappa =\frac{E_0}{E}$$

How is voltage distributed across capacitors in series? In parallel?

$$Series: \; V_{tot}=V_1+V_2+V_3+…$$

$$Parallel: \; V_{tot}=V_1=V_2=V_3=…$$

Charge density equation for current?

$$I=nqv_dA$$

Differential formula for current?

$$I=\frac{dq}{dt}$$

Current ratio formula?

$$J=\frac{I}{A}$$

Dot-product and integral formula for current?

$$I=\vec{J}\cdot \vec{A}=JA\cos\theta = \int\vec{J}\cdot d\vec{A}$$

E-field formula for resistivity and current density?

$$\vec{E}=\rho \vec{J}$$

Resistance formula?

$$R=\rho \frac{L}{A}$$

Power formula?

$$P=IV=I²R=\frac{V²}{R}$$

Ohm’s Law?

$$V=IR$$

Kirchhoff’s Junction Rule?

States that current leaving and entering a junction must always be the same

Kirchhoff’s Loop Rule?

In a closed loop of circuit, the voltage drops are always equal to zero.

Voltage for an RC circuit derivation?

$$\epsilon-IR-V_C=0$$

$$Q=CV_C, \; I=\frac{dq}{dt}=C\frac{dV_C}{dt}$$

$$-RC\frac{dV_C}{dt}=V_C-\epsilon$$

$$-RCdV_C=(V_C-\epsilon)dt$$

$$\frac{dV_C}{V_C-\epsilon}=-\frac{dt}{RC}$$

$$\int_{V_C0}^{V_C}\frac{dV_C}{V_C-\epsilon}=\int_{0}^{t}-\frac{dt}{RC}$$

$$\ln \left| \frac{V_C(t)-\epsilon}{V_{C0}-\epsilon}\right|=-\frac{t}{RC}$$

$$\frac{V_C(t)-\epsilon}{V_{C0}-\epsilon}=e^{-\frac{t}{RC}}$$

$$V_{C0}=0$$

$V_C(t)=\epsilon (1-e^{-\frac{t}{RC}})$
Note: this is the equation for voltage across a CHARGING capacitor

Charge over time on a charging capacitor in an RC circuit?

$$Q(t)=C\epsilon (1-e^{-\frac{t}{RC}})$$

Voltage across a resistor over time in an RC circuit? Current over time?

$$V_R(t)=\epsilon e^{-\frac{t}{RC}}$$

$$I(t)=\frac{\epsilon}{R}e^{-\frac{t}{RC}}$$

Voltage and charge over a discharging capacitor in an RC circuit?

$$V_C(t)=\epsilon e^{-\frac{t}{RC}}$$

$$Q(t)=Q_0 e^{-\frac{t}{RC}}$$

Voltage and current over a discharging resistor in an RC circuit?

$$V_R(t)=\epsilon e^{-\frac{t}{RC}}$$

$$I(t)=-\frac{\epsilon}{R}e^{-\frac{t}{RC}}$$

Time constant for an RC circuit?

$$\tau = R_{eq}C_{eq}$$

Equation that proves magnetic monopoles cannot exist?

$$\oint \vec{B}\cdot d\vec{A}=0$$

Relative permeability formula?

$$\mu_r = \frac{\mu}{\mu_0}$$

Biot-Savart’s Law?

$$d\vec{B}=\frac{\mu_0}{4\pi}\frac{Id\vec{s}\times \hat{r}}{r²}$$

Cross-product method for finding magnetic field?

$$\vec{B}=\vec{v}\times \hat{r}$$

Cross-product method for magnetic force?

$$\vec{F}_B=q\vec{v}\times \vec{B}$$

Properties of the M-field different from the E-field?

1: Magnetic forces always act perpendicular to M-field

2: Magnetic forces only act on moving particles

3: Magnetic forces in steady M-fields do zero work

Radius of a particle in steady M-fields?

$$r=\frac{mv}{qB}$$

Angular frequency for particles in M-fields? Period?

$$\omega = \frac{v}{r}=\frac{qB}{m}$$

$$T=\frac{2\pi r}{v}=\frac{2\pi }{\omega}= \frac{2\pi m}{qB}$$

Sum of forces for particles in both M-fields and E-fields?

$$\vec{F}=q\vec{E}+q\vec{v}\times \vec{B}$$

Velocity needed for particles to move in a straight line in mass spectrometry?

$$v=\frac{E}{B}$$

mass-to-charge ratios in mass spectrometry?

$$\frac{m}{q}=\frac{rB_0}{v}=\frac{rB_0B}{E}$$

Kinetic energy for a cyclotron?

$$KE=\frac{1}{2}\frac{q²B²R²}{m}$$

Magnetic force on a segment of wire?

$$\vec{F}_B=I\vec{L}\times \vec{B}=(q\vec{v}_d\times \vec{B})nAL=ILB\sin\theta$$

Infinitesimal magnetic force on a infinitesimal segment of wire?

$$d\vec{F}_B=Id\vec{s}\times \vec{B}$$

Total magnetic force for a wire of arbitrary shape?

$$\vec{F}_B=I\int_{a}^{b}d\vec{s}\times \vec{B}$$

Torque for objects in M-fields?

$$\tau = I\vec{A}\times \vec{B}=IAB\sin\theta$$

Torque in terms of magnetic dipole moment?

$$\tau = \vec{\mu}\times \vec{B}$$

Potential energy for a system in an M-field in terms of magnetic dipole moment?

$$U=-\vec{\mu}\cdot \vec{B}$$

$$U_{max}=\mu B$$

$$U_{min}=-\mu B$$

What is the hall effect?

When an M-field causes charges to move in such a way that a potential difference is created in a conductor

M-fields for long, straight wires?

$$|\vec{B}|=\frac{\mu_0 I}{2\pi r_{\perp}}$$

M-fields for rings of wire at the center?

$$|\vec{B}|=\frac{\mu_0 I}{2r}$$

M-fields for rings of wire not at the center?

$$|\vec{B}|=\frac{\mu_0 IR²}{2(x²+r²)^{-\frac{3}{2}}}$$

M-fields for arcs of wire?

$$|\vec{B}|=\frac{\mu_0 I\theta}{4\pi r}$$

Ampere’s Law formula?

$$\oint \vec{B}\cdot d\vec{s}=\mu_0 I_{enc}$$

Enclosed current formula for Ampere’s Law?

$$I_{enc}=I\cdot n\cdot w$$

n=loops per unit length

w=width of amperian loop

M-fields for solenoids?

$$B_{sol}=\mu_0 nI=\frac{\mu_0 NI}{L}$$

Flux formula?

$$\Phi_B=\int \vec{B}\cdot d\vec{A}$$

Faraday’s Law?

States a change in flux generates an induced emf within a loop of wire represented by

$$\epsilon =-\frac{d\Phi_B}{dt}$$

Lenz’s Law?

States that induced emfs generate currents that create an opposing M-field of equal magnitude when flux changes

Faraday’s law for solenoids?

$$|\epsilon|=N\left| \frac{d\Phi_B}{dt}\right |$$

Inductance formula?

$$L=\frac{N\Phi}{I}$$

Inductance in solenoids?

$$L_{sol}=\frac{\mu_0 N²A}{l}$$

What is inductance?

The property for a conductor to resist a change in current

Induced emf due to inductance formula?

$$\epsilon_L=-L\frac{dI}{dt}$$

Power and inductance formula?

$$I\epsilon = I²R+L\frac{dI}{dt}$$

1: The first term represents the power supplied by the emf source

2: The second term represents the power dissipated by a resistor

3: The third term represents the power dissipated by an inductor

Potential energy in inductors?

$$U_L=\frac{1}{2}LI²$$

Time constant formula for LR circuits?

$$\tau = \frac{L}{R}$$

Current over time in a charging inductor?

$$I(t)=\frac{\epsilon}{R}\left (1-e^{-\frac{t}{\tau}}\right)$$

Voltage over time in a resistor in an LR circuit?

$$V_R(t)=\epsilon \left( 1-e^{-\frac{t}{\tau}}\right)$$

Voltage over time in an inductor in an LR circuit?

$$V_L(t)=\epsilon e^{-\frac{t}{\tau}}$$

Discharging current in an LR circuit?

$$I(t)=\frac{\epsilon }{R}e^{-\frac{t}{\tau}}$$

Formulas for energy within an LC circuit?

$$U_{C,max}=U_{L,max}=U_C+U_L$$

$$\frac{1}{2}C\Delta V_{max}²=\frac{1}{2}LI_{max}²$$

Why do LC circuits behave as if they’re in SHM?

The capacitor and inductor will take turns discharging and charging. One component will discharge, the other charges, then the current and fields are swapped and the parts change roles.

Differential equation formula for LC circuits that describes their SHM behavior?

$$\frac{d²q}{dt²}=-\frac{1}{LC}q$$

Angular frequency in LC circuits?

$$\omega² = \frac{1}{LC}$$

$$\omega_{LC}=-\omega² q$$

Period in LC circuits?

$$T=\frac{2\pi}{\omega}=2\pi \sqrt{LC}$$