1

EM Theory Lecture Notes

Lectured by Dr Keesing

Notes written by Victor Naden Robinson

vlnr500@york.ac.uk

February 2013

MPhys University of York

Lecture 1: Maxwell

Lecture 2: The Poynting Vector

Lecture 3: Rotation Matrices

Lecture 4: Tensors

Lecture 5: The EM Field Tensor

Lecture 6: The Vector Potential

Lecture 7: In Special Relativity

Lecture 8: Applying the Tensor and Spin

Lecture 9: Maxwell from

Lecture 10: Measuring Moving Charge

Lecture 11: Hand out

Lecture 12: E and B fields of a Moving Charge

Lecture 13: Radiation from an Accelerating Charge

Lecture 14: Some Atomic Physics

Lecture 15: Radiation from a charge where and are parallel

Lecture 16: The Rate of Loss of Energy by the Charge

Lecture 17: Symmetric Radiation

Lecture 18: Frequency Spectra from an Accelerating Charge

2

Lecture 1: Maxwell and the rest, where it all went wrong?

1) a) Maxwell’s equations in vacuum

Recall

Flux,

Total Flux,

∫

[Figure of a spherical charge and the E field as well as flux associated with it]

Taking

∫

∫

Then

∫

∫

∫

∫

Thus

∫

∫

This is Maxwell’s first law (M1)

b) The point charge:

First consider the point charge, which is in essence a concept or model rather than a reality.

3

Taking divergence in spherical polars (about a point charge),

(

)

This is a problem as it breaks M1, also implies

∫

∫

Though I think that tau for point charge needs to be defined.

( )

[Figure of a point charge showing r varying across it, the E field and defining density]

So,

( )

Taking divergence of this in spherical polars:

(

)

This shows that point charges do not make actual sense in physics though they are very

useful.

c) Mass of the electron:

Recall,

[Image of the electron with radius and extension of radius ]

Examine the setup,

4

∫

[

]

Meaning

d) Maxwell’s further proofs:

Recall M2,

Therefore the magnetic field is always in circles or loops, i.e. a monopole.

Recall M3,

Remember that the curl means the line integral per unit area.

The EMF is given by,

∫

∫

But

∮ ∫

Using Stokes theorem,

∫

∮ ∫

So, there are two kinds of E field. One is conservative and one is not. This leads to the

definition of EMF.

5

e) Maxwell’s fourth:

Recall M4,

( )

Amperes circuit theorem,

[Figure of a current with B looping around it]

∮ ∫

Using Stokes theorem,

∫

∮ ∫

Taking the divergence,

But

( )

( )

Lecture 2: The Poynting Vector

Take M3 and M4,

( )

Now,

6

( )

( )

Dividing by ,

Put into space and integrate over all space for physical meaning,

∫

∫

∫

∫

Using the divergence theorem:

∫ ( )

∫( )

Now

( )

This ( ) is the energy density in the magnetic field (should be familiar).

And for the EM field

∫

(

)

∫( )

∫

( )

∫

∫

∫

( )

∫( )

E is force per unit change. B is force per unit current. These are definitions to know.

[Figure of amorphous shape with axes j E B all perpendicular to each other, N points out]

( )

b) The parallel plate capacitor

[Figure of PPC]

7

Looking at the magnetic field using M4,

( )

Applying Stokes Theorem,

∫

∮ ∫

( )

So

[Figure of N in the PCC]

(

) (

)

∫

So

( )

(

)

8

Lecture 3: Rotation Matrices

This lecture introduces the mathematic tool of rotation matrices and applies it to Ohm’s Law.

Acting on Cartesian co-ordinates [Figure of two 2D x-y axes one rotated by theta]:

(

) (

) ( )

[Figure of 3D version, now 3 angles and axes now all denoted by x plus suffix and or prime]

Define all directions with x:

So,

Write for shorthand

(

) (

) (

)

Remembering

Take

This can be written as

∑

Last step done by assuming the summation convention: a repeated index is summed over.

9

Where are dummy matrices

So if you do not want to sum you have to say so – applies to n dimensions.

Term is an orthogonal rotation matrix

One of the fundamental assumptions is that the laws of physics are the same regardless of the

frames of reference, form invariance.

Ohm’s Law:

[Standard resister ammeter voltmeter cell circuit]

These are all scalars, we want to use vectors so instead use

(

) (

)(

)

For an isotropic solid [Figure of J and E in parallel directions on xyz graph]

(

) (

)(

)

(

)

[Figure of J and E pointing in different directions on xyz graph]

∑

Description of a vector in two rectangular Cartesian frames

( ) ( )

Conductivity, note is a tensor, this is physics:

10

Book keeping and physics are very different things even if they look the same.

All physical things are tensors except spinors (Dirac)

Lecture 4: Tensors

This lecture introduces tensors and tensor differentiation

[3D graph from lecture 3 with all axes denoted by x, second figure is graph of a frame with E

pointing in the same direction as a primed axis of the second frame i.e. rotate to E]

Conductivity,

Rotating the frame,

Showing form invariance, changing dummy indices to avoid repetition

Thus (lines linking the l’s and m’s)

This is not a matrix multiplication. General form:

Rank 0 tensor transforms as

Rank 1 tensor transforms as (

) (

) (

)

Rank 2 tensor transforms as

Expanding this general form,

11

( ) ( )

( )

Vector products:

( )

( )

Transform:

}

Thus the outer product of 2 vectors is a rank 2 tensor. [Some figures representing this]

Tensor Differentiation:

Consider a scalar field,

But

Once again applying the summation convention [Figure relating terms with angles]

12

So it is a rank 1 tensor. An object is a scalar / vector / rank 2 tensor if it rotates like one

because of frame invariance.

Consider a vector field,

( )

So the differential of a vector gives a rank 2 tensor. So generally increase the rank of the

tensor by 1 when differentiating. Notation:

NB: this is a covariant differentiation but because in rectangular coordinates it is the same as

any differentiation.

[Problem class on Biot Savart, Ohm’s Law, Lorentz force]

Lecture 5: The EM Field Tensor

Where V is the scalar potential and A is the magnetic vector potential

∫ ∫

13

∮ ∫ ∫

... Where this leads to was forgotten temporally

Take,

( )

(

)

( )

( )

Creating

(

)

The electric field

Remember Minkowski speak,

(

)

(

)

(

)

Where

Follows

14

Replacing

Similarly

So

(

)

Skew symmetry;

(

)

15

Thinking about the Lorentz force:

(

) ( )

(

)

( )

Four force:

(

)

(

)

Four velocity:

( )

(

)

The EM field tensor:

(

)

(

)

(

)

(

)

(

)

The x component of the Lorentz force per unit charge

LHS is a force and RHS is a tensor.

16

Lecture 6: The Vector Potential A

Generally, where is any vector field,

The potentials:

- The Electrostatic field

- and ∮ because static fields are conservative

- For a varying magnetic field, faraday, the EMF

∫ ∫

∫

[Figure of loop with B passing through showing Stokes’ theorem for next step]

∮ ( ) ∫

∫( )

The final field due to and , [Figure of surface with B and a charge q at P]

( )

The Vector Potential of a Current Element: [Figure of current loop in two inertial frames]

Biot Savart:

( ) ( ) ( )

[Figure showing directions of and which are all at right angles ( being the surface)]

17

( )

( ) ( )

But is at the source of the field,

(

)

Now,

Where is a scalar field and is a vector field

( )

(

)

(

)

(

)

So,

( (

)

)

But because is a function of the source and not the field co ordinates

∫

∫ (

)

We can swap the order of differentiation and integrate:

∫

(

)

But

∫

(

)

This is an important proof! Thus the vector potential points in the direction of the

But

18

[Figure of and pointing in parallel directions]

So [Figure of current loop and point and radius r from loop follows this]

∫

Lecture 7: In Special Relativity

( )

(

) (

)

In Minkowski

(

)

(

)

(

)

The four velocity (note is not contain )

( )

(

)

(

)

The four momentum,

( )

(

)

19

(

)

( )

( ) ( )

Examples

(

)(

)

(

)

(

)

(

)

Minkowski Rotation Matrix (MRM)

( )

Four Current:

( )

(

)

Where is the charge density in a stationary frame w.r.t. the charge

( ) (

)

( )( ) ( ( ) ( ))

( ( ))

Transforming from

(

)

(

)

(

)

In a 3D rotation:

(

) (

) (

)

20

Remember Form invariance so relationships in one frame are the same in the other

( )

Four Force:

( )

(

)

(

)

(

)

(

)

(

)

(

)

Lorentz Force as seen in Lecture 5

Transformation of between inertial frames:

Since rank 2 tensor,

The subscripts refer to the MRM

( ) ( )

( )

( )

Which gives 16 terms, but most of them are equal to zero which makes things simpler.

21

Lecture 8: Applying the Tensor and Spin

( ) ( )

( )

( )

[Figure of E and B in two reference frames though one is moving at u]

(

)

(

)

(

)

(

)

( (

) (

))

(

)

Looking at

22

( )

( )

Thus transforms and gives

( )

(

)

( )

(

)

Spin-orbit coupling in H:

[Figure of two reference frames one in motion with velocity u the other originating at p]

In S,

So

( )

( )

(

)

(

)

23

(

)

Re: the Bohr Model – circular orbit [Figure as before with e at origin of moving frame and B field]

Quantisation:

Spin orbit coupling

Lecture 9: Maxwell from

The 4D divergence

In the case of the EM field tensor:

(

)

24

You might see this as turning into components of curl (side note)

( ) ( ) ( )

Consider

(

)

From M4

( ) ( )

( )

So,

For ( ) for example

(

)

But from M1

The Four Current:

( )

( )

(

)

25

( )( ) ( ) ( )

( ) ( )

Combining these

}

The 4D divergence of is the sum of M1 and M4. What about M2 and M3?

Cyclic differentiation:

Form

and cycle as you go

In order to cope with M2 and M3 in 4D ( )( )( )( ) choose any 3 of 4,

Consider

Cyclic differentiation

For ( ):

(

)

x

y

z 1

2

2

26

(

)

M3:

( )

( )

So, describes M2 and M3

Tensor formalism of Maxwell’s equations

Prime denotes true in any frame,

27

Lecture 10: Measuring moving charge - Lienard-Wiechert Potential

[Diagram 1 of a surface at with retarded velocity a distance from point P at ]

[Diagram 2 – handout number 6]

Charge escapes through the “front”. The amount of charge which has escaped detection in the

volume

The charge we measure:

(

)

(

)

But the potential is due to the charge in and not the measured charge.

(

)

And so,

∫

∫

( )

For a small charge assume that is constant. Also is constant.

For an electron (or point charge)

( )

[Diagram 3 vector lines drawn between e and P for velocity and radius (twice)]

(

)

28

And so,

(

) ( )

(So field increases from behind as an electron accelerates away from you?)

The Vector Potential

[Diagram 4 section of current of width loop inside this of , moving at , centre of

which is from P from which points ]

The charge in

29

( )

( )

For a vanishingly small charge:

(

)∫

But ∫

(

)

If we require the field then,

The gradient requires to be evaluated at affixed time at the field point.

Lecture 11: Handout

Mostly done on worksheet – find said worksheet

30

Lecture 12: E and B fields of a charge q

The E and B fields of a charge q in an arbitrary state of motion

(

)

[(

)(

)

( (

) )]

[(

)( ) (

( (

) ))]

For

(

) (

)

(

)( )

For

(

)

( )

Again

(

) (

)

[Complicated diagram with triangles, a point P, angle theta, some trigonometry]

Look at triangle ∆OAP:

So

(

)

31

Consider the term s,

(

) (

) (

)

Note from this point on

∆OAP:

(

)

[Redone figure for a new point on z axis therefore making and new angle from P ]

(

)

(

)

(

)

Looking at B:

(

)( )

( )

(

)

( )

(

)

Cool. For

Then

For then

(

)

(

( ))

(

)

32

[Figure showing pancake like orbits, circle for and shape for ]

[Another few figures showing what a “Polar Plot” is and how this relates to the pancake field]

The pointing vector for :

(

)(

)

(

)( )

The numerator

So, [Figure of sphere (noting surface area) around q with axes of radius r and q moving at ]

∫

Thus no energy is transferred from the charge to (for ).

We call these fields convective.

But at

Lecture 13: Radiation from an Accelerating Charge

Now leaving out for retarded terms

[(

)(

)

( (

) )]

[(

)( ) (

( (

) ))]

Poynting Vector:

( )

( )

Where the subscripts 1 and 2 refer to each additive part of E and B.

Total radiation

33

Total radiation

Similarly Total radiation

Total radiation

Radiation Fields:

( (

) )

[(

( (

) ))]

First we consider

(

)

So,

( ( ))

(

( ( )))

Poynting Vector:

( ( )) ( ( ( )))

We could take ( ) ( ) ( ) but instead we shall do it graphically:

[Figure showing the motion forming a Dipole Pattern]

( ( ))

( ( ))

34

( ( ( )))

( ( ))

Dipole: ( )( )

The total radiation into . The ring element

The rate of radiation

∫

∫

∫

[

]

[(

) (

)]

The total rate of radiation is the Larmor formula:

( )

Example: [Potential of 0 to V of distance d which e crosses and E goes from 0 to V direction]

The total energy lost to radiation:

35

∫

Time of transit:

(

)

(

)

(

)

(

)

The radiation loss K.E. gain

(

)

So most of the radiation energy goes into the change of kinetic energy [Problem Class]

Lecture 14: Some Atomic Physics

The lifetime of excited states in atomic hydrogen:

( )

( )

From (1)

The rate of loss of energy:

[

]

Larmor

36

From (1)

Thus

(

)

(

)

[

]

So

(

)

[

]

(

)

[

]

For H

(

)

[ ] [

]

“You can fall off the cliff but it’s difficult to rise up it” – R Keesing on mathematicians doing

experiments.

Thompson Scattering:

( )

( )

37

The force is the Lorentz Force

( )

But for the Larmor formula to apply but | | | |

(

)

( )

Put e at

( )

Larmor

( )

( )

Poynting Vector

( ) ( )

And | | | |

( )

The scatting cross section is then

( )

( )

And

(

)

38

Lecture 15: Radiation from a charge where and are parallel

The radiation fields:

( (

) )

( ( (

) ))

As and are parallel [Standard (by now) figure, q radiating in xyz with angles]

( ( ))

( ( ( )))

Evaluating terms in and

( ) ( )

( ( )) ( ) ( ) ( )

( ( ( )))

The Poynting Vector:

( )

( )

[Figure of High velocity (more directed in the forward direction) and Low ve locity (more

perpendicular to forward direction {z}) cases of pancake orbits, height determined by theta –

for when ]

Rate of radiation:

∫

∫

( )

The range of is from

39

∫

( )

Trick to solve this, can be solved in Maple:

∫( ) ( )

( )

(

(

)

(

)

)

For

( )

The maxima in radiation pattern:

( )

( )

( )

( )

(

( ))

( )

( )

( )

( )

(

)

In order to evaluate

Going back to special relativity for a moment

40

(

)

(

)

(

)

(

)

(

(

)

)

[Some numerical data giving examples of energies, speeds and ]

The total radiation:

(

(

)

(

)

)

We need (

)

and

in order to eliminate the energy radiation

(

)

(

)

(

)

(

)

(

)

And (

)

41

(

)

(

)

(

)

(

)

[Figure of potential difference and and both travelling in z direction]

Lecture 16: The Rate of Loss of Energy by the Charge

The Poynting Vector, same case as for Lecture 15

(

)

( )

( )

This is the rate of radiation through in the field at ( )

We want

the rate of energy by particle

(

)

(

)

(

)

( )

(

)

( )

(

)

The rate of energy loss into all directions:

42

∫

∫

( ) ( )

( )

As before just one power difference

(

)

(

)

The first factor defines the Larmor Radius, Recall

(

)

(

)

∫

(

)

(

)

(

)

Let us call

∫

(

)

(

)

(

)

[Figure showing why this next step happens as and ]

(

)

(

)

( )

(

)

[Example for numerical data from the LHC]

43

Circular Orbits:

( (

) )

( ( (

) ))

Evaluating terms

( (

) ) ( ) (

) ( )

( ) ( )

( ( ))

( )

( ) (

) ( )

(

)( )

( (

)( )) ( )

(

) ( )

[Problem Class on Poynting vector with energy losses and fundamental problems]

Lecture 17: Symmetric Radiation

Approximations: [New figure, adapted old one with a circle and 3D triangle with and ]

( (

) )

( ( (

) ))

Circular

| (

) | ( ( ))

( ( ))

44

( ( )) ( )

( ( ))

( )

| (

) | (

)( ) (

) ( )

So,

(

)

(

)

( )

( )

( )

( )

( )

( )

There were some corrections to the mathematics if this looks confusing.

For high energy particles assume that there is symmetry about the z-axis

( )

( )

( )

( )

(

)

Radiation into the far field:

∫

( )

( )

If we consider the rate of loss of energy from the charge

45

∫

( )

( )

Note power change. The exact integral:

(

)

And

(

)

(

)

(

)

(

)

(

)

(

)

The acceleration , for [Again standard figure with 3D triangle]

(

)

(

)

(

)

Example for LHC figures, energy loss per orbit ,

and about 0.5% per hour.

46

Lecture 18: Frequency Spectra from an Accelerating Charge

There are lots of figures in this lecture and a large hand out.

( ) ∫ ( )

Fourier Transform (time),

( ) ∫

( )

[ ]

Note (for intensity figure)

( )

( )

[

]

[

]

[ ]

( )

(

)

( )

[Figure of decaying ( ) over ]

Synchrotron Radiation:

( )

( )

(

)

(

)

[Annotation: For a q at a position q like a laser on a turn table all the radiation is seen to lie

in ( )]

47

( ) (

)

(

)

And the Fourier Transform,

( ) ∫(

)

(

)

Where the argument of the integral is ( )

{ ( )

( )

See hand out for time evolution and energy distribution – “angular energy and F.T.”

For greater accuracy need to work out ( (

) ) and so need to cross with

Cartesian for example.

Fin