Magnetism

Magnetism Equations

Point Charge

$\vec{B}=\frac{{\mu }_0}{4\pi }\frac{qv\sin (\theta )}{r^2}$

$\vec{B}=\frac{{\mu }_0}{4\pi }\frac{q\vec{v}\times \hat{r}}{r^2}$

1 Tesla $=N/Am$

Infinite Wire

$B=\frac{{\mu }_0}{2\pi }\frac{I}{r}$

Current Loop

$B=\frac{{\mu }_0}{2}\frac{NI}{R}$

Solenoid

$B={\mu }_0\frac{N}{L}I$

Current Segment

$\vec{B}=\frac{{\mu }_0}{4\pi }\frac{I\Delta \vec{s}\times \hat{r}}{r^2}$

Long, Straight Wire

$B=\frac{{\mu }_0}{2\pi }\frac{I}{r}$

Coil Center

$B=\frac{{\mu }_0}{2}\frac{NI}{R}$

Dipole

$\vec{B}=\frac{{\mu }_0}{4\pi }\frac{2\vec{\mu }}{z^3}$

$\vec{\mu }=AI$

Current is in the direction of curled fingers.

Ampère's Law

${\int }_{surface}\vec{B}\cdot d\vec{s}={\mu }_0{I}_{through}$

Positive currents are ones flowing in direction of thumb when curled with current.

Field Inside Current-Carrying Wire

$B=\frac{{\mu }_{0}I}{2\pi R^2}r$

Solenoid

$B=\frac{{\mu }_{0}NI}{l}={\mu }_{0}nI$

Magnetic Force

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

Notice force is in the opposite direction if the charge is negative

This also implies cyclotron motion by showing that the force is perpendicular when initially traveling straight in a field.

Cyclotron

Radius

$r_{cyc}=\frac{mv}{qB}$

Frequency

$f_{cyc}=\frac{qB}{2\pi m}$

Hall Effect

Voltage

$\Delta V_H=\frac{IB}{tne}$

Forces Between Wires

$\vec{F}=I\vec{l}\times \vec{B}$

Parallel Wires

$F=I_{1}l{B}_2=\frac{{\mu }_{0}l{I}_1{I}_2}{2\pi d}$

d is distance between them.

Same direction attract. Different repel.

Torque

Current Loop

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

$\mu$ is the vector defined earlier.

Induced Currents

Motional EMF

$ϵ=vlB$

Induced Current In Circuit

Current Created

$I=\frac{ϵ}{R}=\frac{vlB}{R}$

Power Dissipated

$P=I^{2}R=\frac{v^2{l}^2{B}^2}{R}$

Magnetic Flux

${\Phi }_m=\vec{A}\cdot \vec{B}$

${\Phi }_m=AB\cos \theta$

Units of weber: $1T{m}^2$

Lenz's Law

The direction of induced current opposes the change in magnetic flux through a loop.

Faraday's Law

$I=\frac{ϵ}{R}$

$ϵ=|\frac{d{\Phi }_m}{dt}|$

${ϵ}_{coin}=N|\frac{d{\Phi }_{percoil}}{dt}|$

This can also be used to describe flux in a circuit by subbing in $\Phi$ for the flux equation and then realizing change in area is related to velocity of movement.

Induced Electric Fields

${\int }_{surface}\vec{E}\cdot d\vec{s}=A|\frac{dB}{dt}|$

So $d\vec{s}$ is length of loop, then solve for $\vec{E}$.

$d\vec{s}$ also ensures that this vector is tangent to the curve.

Inside a solenoid

$E=\frac{r}{2}|\frac{dB}{dt}|$

Electromagnetic Wave Velocity

$v=\frac{1}{\sqrt{{ϵ}_0{\mu }_0}}$

Transformer

Coil ratio:

$V_2=\frac{N_2}{N_1}{V}_1$

Inductor

Henries

$L_{solenoid}=\frac{{\mu }_0{N}^{2}A}{l}$

Potential

$ϵ=L|\frac{dl}{dt}|$

Potential across inductor is opposite of circuit polarity.

Energy

Stored

$U_L=\frac{1}{2}L{I}^2$

Density

$u=\frac{1}{2{\mu }_0}{B}^2$

LC Circuits

Oscillation frequency:

$\omega =\sqrt{\frac{1}{LC}}$

$f=\omega /2\pi$

or

$I=I_{max}\sin (\omega t)$

LR Circuits

Discharging

$I=I_0{e}^{-t/\tau }$

$\tau =L/R$