1

Electrodynamics Notes

Coordinate Systems Cartesian Cylindrical Spherical

Gradient

Divergence

Norm

Differential length

Differential normal area

Differential volume

Useful math results

2

Fields and Potentials

Coulomb’s Law The force on test charge Q due to a single point charge q is given by:

Electric Field The electric field at of the sources charges located at points

is given by:

For a single particle at the origin the field is:

For continuous charge distributions this becomes:

Electric Potential The potential at of point charges located at points

is given by:

For continuous charge this becomes:

Potential relates to the electric field by:

3

The induced surface charge on a conductor with a potential immediately outside of is:

Biot-Savart law The magnetic field at produced by a steady line current is given by:

Currents Surface current in terms of charge velocity and surface charge density :

Volume current in terms of charge velocity and volume charge density :

The continuity equation expresses local charge conservation:

The relationship between currents and fields is given by:

In the low velocity limit we write this as simply:

Ohm’s law is thus typically written as:

Maxwell’s equations

Gauss’ Law

Faraday’s Law

Gauss’ Law for

magnetism

Ampere’s Law

4

Lorentz force law The electromagnetic force acting on moving charge in fields and is:

Force per unit charge is generally proportional to current density , so we write:

Where is the conductivity

Separation of variables General solution in Cartesian coordinates with an open side in the x-direction:

General solution in spherical coordinates with azimuthal symmetry:

General solution in cylindrical coordinates with no z dependence:

Materials

Dielectric materials Induced electric dipole on an atom/molecule is generally proportional to the external field:

The dipole moment per unit volume in linear dielectrics is proportional to total field:

It is often easier to calculate electric displacement , which depends only on free charge:

For linear media this relates back to total field as:

The dielectric constant is defined as:

5

The polarisation determines the bound surface charge density:

As well as the volume bound charge density:

We thus have:

Magnetic materials Separating charge into bound and free, and using Ampere’s law we get:

Total current in a material is equal to:

The dipole moment per unit volume in linear magnetic materials is proportional to :

For linear media this relates back to total field as:

The field produced by any magnetised object is always the sum of the sum of the fields that would be

produced by a volume current and a surface current , with:

Thus we have the potential given by:

6

Electric dipoles The electric dipole moment between two charges of magnitude separated by distance is defined as:

Atomic dipole moment due to an electric field:

The polarisation is the dipole moment per unit volume:

The potential of a single electric dipole is given by:

The electric field of a pure dipole is given by:

Magnetic dipoles We define the magnetic dipole moment to be a vector pointing out of the plane of the current loop and

with a magnitude equal to the product of the current and loop area.

The magnetisation is the magnetic moment per unit volume:

The magnetic potential of a single magnetic dipole is given by:

The magnetic field of a pure magnetic dipole is given by:

The torque on a dipole in a magnetic field is given by:

The potential energy of a dipole in a magnetic field is given by:

7

Electric boundary conditions The general boundary conditions for an electric field are:

The boundary conditions for the field of a dielectric are:

The boundary conditions for a linear dielectric field are:

In terms of the potential, these boundary conditions become:

Magnetic boundary conditions The boundary conditions for a magnetic field are:

In the case of magnetic materials these become:

For the case of a divergenceless magnetic field the magnetic potential has BCs:

Energy and Momentum

Deriving the Poynting vector The total energy stored in electromagnetic fields per unit volume is:

From the Lorentz force law, the work done in interval moving a given set of charges is:

8

Substituting in Ampere-Maxwell law we have:

Using the vector identity we have:

Substitute in Faraday’s law:

Applying the divergence theorem to the final term we get:

This says that the rate at which work is done by fields on electric charges is equal to the rate of change

in internal energy of the fields, minus the energy transmitted outwards by the fields as electromagnetic

waves. This latter contribution motivates the definition of the Poynting vector:

Interpreting the Poynting vector The Poynting vector points in the direction of propagation of EM plane waves. It points radially away

from current sources (like a battery), and radially towards resistors. It represents energy transmitted by

fields, but not by moving charges. Power transmitted by the fields through a surface A is given by:

The linear momentum density is also given by the Poynting vector:

9

Angular momentum density can then be found by:

Maxwell stress tensor The Maxwell stress tensor is defined as:

The force on a given charge distribution can be calculated by integrating the Maxwell stress tensor

over a surface containing that charge distribution. In general we write this as:

So for example:

Waves and Optics

Wave equations The generic form of the wave equation is

We can combine Maxwell’s four equations into two wave equations:

Away from any free currents or charges, these simplify to:

In potential form these become:

10

Plane waves The solutions to the wave equations are known as plane waves. These are waves travelling in the z-

direction with no variation in the x or y directions, and extending in the x and y directions to infinity.

They propagate with a fixed frequency . They have the form:

Wavenumber, frequency, and speed

Impedance For a dielectric this is given by:

It connects E and H in a linear medium:

The more general form for a conductor is:

Energy and momentum for waves Average energy density:

Power transferred per unit surface area:

Energy transferred per unit area across a surface with normal is this:

Average momentum density:

11

Polarisation of light

Waves in a conductor In a conductor we have , and thus the wave equations have an extra attenuation term:

Since the free volume current density in a conductor dissipates exponentially, we can approximate at

timescales longer than as:

This will have the same plane wave solutions as before, but this time we will have complex

wavenumbers:

With , and hence:

12

The imaginary part corresponding to the attenuation of the amplitude as the wave propagates.

The relationship between and is given by:

Note that B and E are no longer in phase, but are phase shifted by:

The Poynting vector is:

Energy is dissipated as the wave propagates a distance .

Skin depth In conductors the wave amplitude decays over a characteristic distance called the skin depth . It is the

distance it takes to reduce the amplitude of the wave by a factor of . It thus measures how far the

wave penetrates into the conductor.

For a good dielectric this takes the form:

For a good conductor this takes the form:

Another useful relation for a good conductor:

13

Complex permittivity and refractive index When light passes through a conductive medium, some part of it will be attenuated. This can be taken

into account by defining a complex refractive index, wavenumber, and permittivity.

The real part n is the refractive index and indicates the phase velocity, while the imaginary part κ is

called the extinction coefficient, and indicates the amount of attenuation as waves pass through the

material.

Phase and group velocity in general The phase velocity is the speed at which individual wavefronts with a single frequency travel; it can be

greater than c. The group velocity is the speed at which a pulse comprised of a combination of

frequencies travels.

Group velocity is the carrier of information because it is the rate at which frequency changes.

Dispersion Dispersive media are those in which the phase velocity of light depends on the light’s frequency. Waves

travelling in dispersive media experience ‘pulse broadening’, as different frequencies get further and

further apart from each other over time.

14

The oscillator model of dispersion

Usually the index of refraction decreases with increasing wavelength. In some cases when the index

increases with increasing wavelength, the medium is said to have anomalous dispersion. This can

happen when the frequency of radiation in question corresponds to a resonance frequency of the

electrons in the material.

Reflection at arbitrary incidence Consider an example of incident light parallel to the plane of incidence. Then we have:

All three waves are at the same angular frequency (assuming a linear medium), so we must have:

By the boundary condition at the reflecting surface, we have:

Since we are dealing with plane waves these coefficients don’t depend on x or y or z, and thus the

exponents must be separately equal. This implies:

For this to hold overall it must also hold for each component of . Of course at the surface of reflection

, so we need only consider the x-component of k:

15

And thus we get the laws of reflection and refraction of classical optics:

Now that we have determined that the exponential factors always cancel, we are left to determine our

coefficients for the electric field using the boundary conditions. In this case the plane of incidence is in

the xz plane and as stated before the incident electric field is polarised parallel to this plane. This means

that the y-component of the field is zero, while the x-component satisfies the perpendicular boundary

condition since it will have a component directed inwards toward the interface:

Thus we can write (adjusting for the fact that the light is coming in at an angle to the interface):

Using the laws of classical optics we simplify these to get:

Rearranging we get:

We solve these two equations to arrive at Fresnel’s equations:

16

From the first of these equations it is evident that there will be an angle at which and hence the

reflected wave is cancelled out. This is called Brewster’s angle.

Using Snell’s law:

We write this as:

Intensities are given by:

Reflection and absorption proportions are given by:

In the alternative case, the electric field may be polarised perpendicular to the plane of incidence,

meaning that its x-component is zero and so it has no component that is perpendicular to the interface.

So instead we have to use the parallel boundary condition for what is now the y-component:

17

Total reflection and evanescence Total internal reflection occurs when light from a more optically dense medium strikes the boundary

with a less optically dense medium at an angle greater than the critical angle. For , the critical

angle is given by:

If we consider a wave striking a boundary parallel to the plane of incidence which is in the xy plane, the

transmitted field is given by:

The transmitted angle is then found as:

If then must be a complex angle. The imaginary component is:

The transmitted wave then becomes:

This exponentially decaying solution in the z-direction is known as an evanescent wave. An evanescent

wave is an oscillating electric and/or magnetic field which does not propagate as an electromagnetic

wave but whose energy is spatially concentrated in the vicinity of the source (oscillating charges and

currents). They do not transmit energy. We can see this by calculating the time-averaged normal

component of the Poynting vector:

18

Reflection by conductors When light is incident on the boundary of a conductor we have:

For good conductors, we have:

And hence:

Then:

Thus in very good conductors almost all of the energy is reflected.

Waveguides A waveguide is a hollow pipe of some shape surrounded by a conductor, which directs the flow of

electromagnetic waves. Since in the interior there is no current or charge, Maxwell’s equations become:

We are looking for wave solutions of the general form:

19

The boundary conditions are:

The solutions will not generally be transverse waves, so we need to consider the general form for the

plane wave coefficients:

Substituting these into the above equations we find two uncoupled equations for and :

And similarly we have expressions for the other components in terms of and :

The solutions will depend upon the additional boundary conditions given by the shape of the specific

waveguide in question. For shorthand we also use the paramaterisation .

TM mode These are defined by the condition that everywhere, with boundary condition . The

solution takes the form:

Neither m nor n can be zero.

Transverse fields then take the form:

20

Where

TE mode

These are defined by the condition that everywhere, with boundary condition

. The

solution takes the form:

Only one of m or n can be zero.

Transverse fields then take the form:

Cutoff frequency The magnitude of the wavenumber of a wave mode travelling down a waveguide is given by:

This will only have imaginary solutions if:

21

We thus define this cutoff frequency as:

For a given mode , the cutoff frequency represents the smallest frequency waves that can travel.

Frequencies lower than this will have an imaginary wavenumber, meaning that they will be

exponentially suppressed and will not propagate down the waveguide.

For a rectangular waveguide the lowest cutoff frequency of all occurs for mode :

Frequencies less than this will not travel down the waveguide at all. This means that small waveguides

can support fewer modes, and very small ones won’t support any at all. This is not an issue for coaxial

waveguides which can support TEM modes (see below).

Phase and group velocity in waveguides Group velocity carries information and cannot be greater than . Phase velocity is not capable of

carrying information (it is just the wavefront of a single frequency component), so can be greater than .

Thus the phase and group velocities of node at frequency with cutoff is given by:

Power in waveguides Power delivered through the waveguide can be calculated by integrating the time-averaged Poynting

vector across the inner surface of the waveguide.

TEM mode Hollow waveguides cannot support TEM modes, so we need a coaxial waveguide. By definition these are

waves where:

So our trial solutions become:

22

The solution is the same as that for the electro- and magnetostatics problems in cylindrical coordinates.

Thus:

The constant A depends upon the potential across the gap between surfaces. If we have:

Then we get:

We note that

, with wavenumber given by:

There is no cutoff wavelength for TEM modes.

Relativity

Key Four-Vectors

Continuity Equation

23

Field Transformation Rules For in the x-direction fields transform as follows:

Electromagnetic Field Tensor

Lorenz-Lorentz Gauge and Potentials We have:

If we implement the Lorentz gauge condition:

24

Then we can simplify this to:

Velocity Transform Equations Four vectors transform as:

Derivatives transform as:

In a frame moving with velocity with respect to frame , we have the transform:

Writing out all three we have:

Or in vector form:

Force Transformation

25

Helical Motion

If energy is constant that means is constant, and so

, hence:

This has the solution:

Noting that

, we have:

With the magnitude of a given by:

Lorentz Invariance of Maxwell’s Equations Gauss’ Law and the Ampere-Maxwell law are expressed as:

Faraday’s law and Gauss’ law for magnetism are expressed as:

Lorentz force law

26

Radiation

Electric dipole radiation Consider an oscillating dipole with frequency . It’s retarded dipole moment is given by:

We use this to find the vector potential as follows:

In spherical polar coordinates for a dipole aligned parallel to the z-axis this becomes:

We find the electric field using:

In spherical polar coordinates with dipole aligned parallel to the z-axis this is (with ):

The magnetic field is then:

27

True radiation is energy that is propagated out to infinity, and so must originate from terms of , as

others all decline to zero at very large distances. Dropping such terms we are left with:

The Poynting vector is found by the usual method:

Averaging over time we get the average power by unit surface area:

This has an frequency dependence, so radiation is much more pronounced in blue wavelengths. This

is the major reason why dipoles in the atmosphere preferentially scatter (absorb and rereadiate) blue

rather than red light.

The total power radiated is found by integrating the time-averaged Poynting vector over a sphere with

radius r:

28

Real antenna A center-fed linear antenna receives current in and out through a gap in its centre. Let’s consider a

simple case of length and no gap. We can still find the magnetic potential using the expression:

If is large then , so we have:

We use the far-field approximation

TO solve this we need a model for the current – one possibility is to use a standing wave pattern of the

form

. This can be solved for the potential giving:

For a ‘half-wave’ antenna, the length of the antenna is equal to a half-wavelength of the frequency of

operation, i.e. . In this case the radiation Poynting vector is actually very similar to that of a

dipole antenna.

29

Nonrelativistic radiation by charged particle The fields of a point charge in arbitrary motion are given by:

Where and . The Poynting vector is:

Only the dependent acceleration terms will contribute to radiation, so ignoring the velocity part of

the electric field we find:

Integrating over the solid angle yields the total power emitted in the form of the Larmor formula:

This is an approximation valid for .

Bremstrahlung and synchrotron radiation

The more general relativistic expression for the power radiated per solid angle

is:

We now consider two special cases, first that where acceleration is in the direction of travel, so

, and so we have for an angle between and the direction of observation :

30

This describes a cone highly concentrated in the forwards direction for high .

In the second case, acceleration is purely orthogonal to the velocity vector, so we have

,

and so we get:

Note that when this replicates the Larmor formula, so at low speeds synchrotron radiation

resembles dipole radiation rotating about a circle. As velocity increases, the radiation projects

increasingly in the forward direction of instantaneous velocity

31

Additional Notes and Issues Always calculate fields and potentials separately inside and outside, even if only one is

ultimately needed, as boundary conditions are often required to solve for the half you want

Only Cartesian coordinate vectors come out of integrals. Other coordinate vectors need to be

converted to Cartesian first

Use the integral form of Maxwell’s equations for calculations

Charge density

is current density, so

;

32

When using to transform a field etc, need also to transform coordinate vectors!

and use

For circular acceleration

Power dissipated per unit volume is

Electric boundary conditions: ,

Magnetic boundary conditions: ,

For angular velocity problems:

Flux:

Take the curl of the curled equations to get the wave equations