Electric and Magnetic Fields

Guass’ Law in differential form

Written for a charge density $\rho(\vec r)$ as

$$\vec ∇·\vec E = \frac{ρ(\vec r)}{ϵ_0}$$

Gauss’ Law in integral form

Written for a volume, $V$, enclosed by a surface $S$

$$\oint_S\vec E\cdot\vec n\,da=\int_V\frac{\rho}{\epsilon_0}dv=\frac{Q}{\epsilon_0}$$

Lorentz force law

Describes the force exerted by a magnetic and electric field on a charge, $\vec F =q(\vec E+\vec v×\vec B)$.

Biot- Savart Law

For an element of a current-carrying loop, $d\vec l$, carrying current $I$ at $\vec r_2$, the contribution to the magnetic field can be written as

$$d\vec B(\vec r_1)=\frac{\mu_0 I}{4\pi}\frac{d\vec l\times\vec r_{21}}{r_{21}^3}$$

The total magnetic field is given by the integral over the whole loop $C$.

• From this law, can show that $\vec \nabla\cdot\vec B=0$.

Ampère’s Law in integral form

For a current I flowing through a surface $S$ bounded by a loop $C$,

$$\oint_C\vec B\cdot d\vec l=\mu_0I$$

Magnetic flux

The magnetic flux is $\Phi_C=\int_S\vec B\cdot \vec n\,da$ where S is a surface bounded by the circuit C.

EMF induced around a circuit

$$\varepsilon=-\frac{d\Phi_C}{dt}=\oint_C\vec E\cdot d\vec I$$

Electric dipole moment

An electric dipole is a pair of equal but opposite charges separated by a small distance.

Dipole moment is defined as $\vec p=q\vec l$ where the charges are separated by $\vec l$, which moves from negative to positive charge.

Electrostatic potential

$$\vec E=-\vec \nabla V$$

Conductor

The electric field inside a conductor is always zero, and any excess charge resides on the surface.

Polarisation

The average dipole moment per unit volume of the material,

$$\vec P=\frac{\Delta\vec p}{\Delta v}$$

where the dipole moment of a small volume of material $\Delta v$ is given by $\Delta\vec p=\int_{\Delta v}\vec r \,dq=\int_{\Delta v}\vec r \rho(\vec r)\,dv$

Surface and volume polarisation charge densities

Surface: $σ_P=\vec P·\vec n$

Volume: $ρ_P=−\vec \nabla·\vec P$

Charge densities will be zero if there is a uniform polarisation.

Electric displacement

Defined as

$$\vec D=\epsilon_0\vec E+\vec P$$

which can be written as $\vec\nabla\cdot\vec D=\rho$.

Linear dielectric polarisation

Can write $\vec P=\epsilon_o\chi_e\vec E$ if the material is:

• Linear: $\vec P\propto \vec E$ for the material.

• Isotropic: $\vec P$ and $\vec E$ are parallel, so polarisation all points in same direction.

• Homogenous: $\chi_e$ does not vary with position in the material (no variation material's density).

Allows us to write $\vec D=\epsilon_0\vec E+\epsilon_0\chi_e\vec E=\epsilon\vec E$ where \epsilon=\epsilon_\epsilon_r is the permittivity, and $\epsilon_r=1+\chi_e$ is the relative permittivity.

A material with permanent polarisation ($\vec P\neq 0$ outside of an electric field) cannot be linear.

Total energy of a system (electric field)

Given by $U=\frac{1}2\vec D\cdot \vec E$, or, if in a vacuum: $U_\text{vac}=\frac{1}2\epsilon_0 E^2$.

Boundary conditions on electric displacement

Normal components of $\vec D$ are continuous across an interface which has no external charges.

Boundary conditions on electric field

• Tangential components of $\vec E$ are continuous across an interface.

• Field lines of \vec E are not conserved across interfaces in general.

Direction of polarisation

$\vec P$ points in the opposite direction to $\vec E$.

Direction of electric displacement

In a vacuum, $\vec D\propto\vec E$, so can plot the $\vec D$ field outside the material as the same as the $\vec E$ field outside the material.

Inside a material, find $\vec D$ by considering the boundary conditions.

• Can find $\vec E$ inside the material by using $\vec D=\epsilon_0\vec E+\vec P$ and checking answer by using knowledge of boundary conditions.

• $\vec E$ inside the material will point in the opposite direction as it does on the outside if there are surface charges present.

Current densities

Volume current density is written in terms of volume charge density: $\vec J=\rho\vec v$.

Surface current density is written in terms of surface charge density: $\vec K = σ\vec v$.

Vector potential

$$\vec B=\vec\nabla\times \vec A$$

Coulomb guage

$$\vec\nabla\cdot\vec A=0$$

Electric field for a time-varying magnetic field

$$\vec E(t) = −\vec ∇V− \frac{∂\vec A }{∂t}$$

Magnetisation

$\vec M$ describes how materials respond to magnetic fields, and is defined as the magnetic dipole moment per unit volume.

For a small volume $\Delta v$ of magnetised material,

$$\vec M= \lim_{∆v→0} \frac{1}{∆v }\sum_i \vec m_i$$

where $\vec m_i$ is the dipole due to the $i^\text{th}$ atom in $∆v$.

When there is no applied field, the directions of the dipoles are random and $\vec M = 0$.

Magnetic dipole moment

Magnetic dipole moment of a single planar loop of area $A$ is given by $\vec m=I A \hat n$.

Magnetisation current densities

Volume: $\vec J_M=\vec\nabla\times \vec M$

Surface: $\vec j_M=\vec M\times \vec n$

Total current density

Total current density of a system is

$$\vec J = \vec J_M+\vec J_f$$

where $J_f$ the free current density, and is due to the motion of free charges.

Magnetic intensity

Defined as

$$\vec H=\frac{1}{\mu_0}\vec B-\vec M$$

and satisfies $\vec\nabla\times\vec H=\vec J_f$.

Magnetisation in a linear material

$$\vec M=\chi_m\vec H$$

letting us write $\vec B=\mu_0(1+\chi_m)\vec H=\mu_0\mu_r\vec H$.

Conditions on $\chi_m$

If χ_m > 0 the material is paramagnetic

If χ_m < 0 the material is diamagnetic

Diamagnets

No intrinsic magnetic moments.

• No external field: no moments, so $\vec M=0$.

• As $\vec H$ is applied, current loops from electrons orbiting cause a magnetic response, which will oppose change in flux, leading to a small magnetisation, χ_m < 0 and \vec B < µ_0\vec H.

No temperature dependence.

Paramagnets

Some free intrinsic moments.

• No external field: moments are randomly aligned, $\vec M=0$.

• As $\vec H$ is applied, moments align with it, giving χ_M > 0 and \vec B > µ_0\vec H.

Competition between diamagnetic and paramagnetic effects, but at low temperatures, paramagnetic effects dominate.

Alignment is opposed by random thermal fluctuations, giving a temperature dependence via Curie’s law.

Lenz’s law

The direction of an induced electric current always opposes the change in magnetic flux that produced it.

Boundary conditions on magnetic field

Field lines of $\vec B$ are conserved at an interface that has no external charges.

Boundary conditions on magnetic intensity

The tangential components of $\vec H$ are conserved at an interface with no free currents.

The field lines of $\vec H$ are not conserved in general across the interface.