Magnetostatics

The fundamental equations magnetostatics are linear equations,

(SI units)

(Gaussian units)

The principle of superposition holds.

The magnetostaticstatic force on a particle with charge q is

(SI units), (Gaussian units).

Definitions:

Drift velocity: N = number of charge carriers

Current density:

Current:

The continuity equation is In statics .

Amperes law

(SI units)

(Gaussian units)

In situations with enough symmetry Amperes law alone can be used to find the magnitude of B. The flux of B through any closed surface is zero.

The Biot-Savart law

(SI units)

(Gaussian units)

For filamentary currents we have

The magnetic vector potential

.

A is not unique. , with an arbitrary scalar field and C an arbitrary constant vector is also a vector potential for the same field.

In magnetostatics we choose Then

(SI units)

Gaussian units)

The uniqueness theorem:

If A or its normal derivatives are specified at the boundaries of a volume V, then a unique solution exists for A inside V.

Boundary conditions in magnetostatics

(SI units)

(Gaussian units)

is continuous across the boundary. is continuous across the boundary.

The force on a current distribution

(SI units)

(Gaussian units)

For filamentary currents we have

The magnetic dipole moment of a charge distribution

(SI units)

(Gaussian units)

The vector potential of a magnetic dipole at the origin is

The magnetic field of a magnetic dipole at the origin is

The energy of a magnetic dipole in an external magnetic field is

This is the mechanical work done to bring the dipole from infinity to its present position.

The force on a dipole is .

The torque on a dipole is .

Magnetic materials

The magnetization is defined as the magnetic dipole moment per unit volume.

The total current density is due to free and to magnetization current densities.

(SI units)

Gaussian units)

This definition is not unique.

For linear, isotropic, homogeneous (lih) magnetic materials we have:

(SI units)

(Gaussian units)

for diamagnetic materials, for paramagnetic materials, permanent magnets are not lih.

Boundary conditions for H:

(SI units), (Gaussian units),

in general.

Maxwells equations

Maxwells equations are:

(SI units)

(Gaussian units)

or, in macroscopic form:

In lih materials with

we have:

Maxwells equations are linear equations and the principle of superposition holds.

,

(SI units)

(Gaussian units)

Faradays law

(SI units)

Gaussian units)

Define the flux

and the electromotive force

. Then :

Any induced emf tries to oppose the flux changes that produces it. This is Lenzs rule.

Quasi-static situations

Consider N filamentary circuits. Then the flux through the ith circuit is

where (SI units), (Gaussian units).

is the coefficient of mutual induction and is the coefficient of self inductance. We have

.

For a single filamentary circuit we have . To change the current in a

circuit we need an external emf, Vext, to overcome the induced emf .

.

The energy stored in the circuit is . For a system of N circuits we have:

(SI units)

(Gaussian units)

or

or

Energy and momentum in electrodynamics

Poyntings theorem, is a statement of energy conservation.

(SI units), (Gaussian units)

is the energy density and

(SI units), (Gaussian units)

is the energy flux in the electromagnetic field. We define the momentum density as

(in SI and Gaussian units).

The Lorentz gauge

If in electrodynamics we choose the Lorentz gauge defined through

(SI units)

(Gaussian units)

then and each Cartesian component of A satisfy the inhomogeneous wave equation

Choosing is called choosing the Coulomb gauge.