Magnetostatics

What is the Bio-Savart law?

d\underline{B}=\frac{\mu_0}{4\pi}\frac{I\underline{dl}\times\underline{\hat{r}}}{r^2}

What is the superposition principle for magnetic flux?

\underline{B}=\int_{conductor}d\underline{B}

What is the magnetic field for a long straight wire given by?

\underline{B}=\frac{\mu_0I}{2\pi r}\underline{e_{\theta}}

Rules for drawing magnetic field lines

• Magnetic field lines form closed loops

• (magnets) Magnetic field lines ‘exit’ at north poles and ‘enter’ south poles

• (currents) Their direction is given by I\underline{dl}\times\underline{r}

• The density of field lines is proportional to the strength of the field

• Field lines never cross

right hand rule (cross product)

right hand rule (field lines and current)

and vice versa.

Conservation of magnetic flux

\oiint\underline{B}\cdot\underline{dS}=0

Symmetry rules for magnetostatics

• the magnetic field’s direction belongs to the planes of anti-symmetry (so must find at least two planes)

• the magnetic field’s direction is perpendicular to the planes of symmetry (only find one field)

What is Ampere’s law?

\oint_{C}\underline{B}\cdot\underline{dl}=\mu_0I_{enc}

Rule for invariances magnetostatics

The magnetic field due to a current distribution inherits the same invariances

Equation for the Lorentz force

\underline{F}_{L}=q\left(\underline{E}+\underline{v}\times\underline{B}\right)

Equation for electric force

\underline{F}_{E}=q\underline{E}

(split the Lorentz force)

Equation for magnetic force

\underline{F}_{M}=q\left(\underline{v}\times\underline{B}\right)

(split the Lorentz force)

Left hand rule

FBI (thumb, pointer, middle)

What is meant by the cylcotron frequency/gyrofrequency?

A charged particle subject to a constant, uniform magnetic field gyrates around the magnetic field. It does this at a specific frequency, the cyclotron/gyrofrequency.

What is the equation for cyclotron frequency?

\omega_{c}=\frac{qB}{m}

What is meant by the gyroradius (Larmor radius) ?

The radius of gyration of a particle around a constant, uniform magnetic field.

Equation for Larmor radius?

R_{L}=\frac{mv}{qB}

Harmonic oscillator equation

\frac{d^2x}{\differentialD t^2}+\omega^2x=0 or

\frac{d^2x}{\differentialD t^2}+\frac{k}{m}x=0

relating dl, I, dQ, v

Id\underline{l}=dQ\underline{v}

Equation for Laplace force

\underline{F}_{Laplace}=\int_{wire}I\underline{dl}\times\underline{B}=\oint_{wire}I\underline{dl}\times\underline{B}

Equation for magnetic torque

\underline{G}=\underline{\mu}_{B}\times\underline{B}

where mu is the magnetic moment and B is magnetic flux.

Definition of torque

\underline{G}=\underline{r}\times\underline{F} where r is the position vector, F is the force applied on the object.

Equation for magnetic moment

\underline{\mu}_{B}=I\underline{S}

Where I is a uniform current, and S is a surface vector whose magnitude is equal to the area of the current loop.