Electromagnetism PHYS20141

Paul.Campbell-3@manchester.ac.uk

Schuster 4-11

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Week 5 Part i)

- Potential formulation

- Lorentz invariance & Lorentz transformations for E & B

- Static fields & special relativity

Important identities:

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Lorenz gauge

As in the static case the choice is not unique

This is called gauge freedom. It is described in the printed

notes that one can always make the choice

which is known as the Lorenz gauge conditionPHYS20141 Electromagnetism4

Not a spelling mistake !

Lorentz transformations (along z)

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Lorentz invariance

On the printed notes it is shown that Maxwell’s

equations are invariant under LT

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Reminder:

Wire in z-direction with static line charge l

Wire in z-direction with current I

In cylindrical polar coordinates

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Consider the x-y plane:

S

No B-field

E-field Const |E|

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Consider the x-y plane:

S S’

No B-field

E-field Const |E| Const |E’|E’-field

B’-field

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3. EM in simple materials

3.1 Conductors

3.2 Method of images

3.3 Capacitance, relative permittivity & dielectrics

3.4 Polarization & electric susceptibility

3.5 Mechanisms for polarization

3.6 Electrostatics in a dielectric

3.7 Interfaces between dielectrics

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3.8 Inductance & relative permeability

3.9 Magnetization & magnetic susceptibility

3.10 Diamagnetism and paramagnetism

3.11 Magnetostatics in a magnet

3.12 Interfaces between magnets

3.13 Ferromagnetism

3.14 Ideal ferromagnetism

3.15 Electromagnets and hysteresis

3. EM in simple materials

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Week 5 Part ii)

- Ohm’s law

- Relaxation time

- Method of images

- Force on a particle near a conductor

Conductivity

σ = conductivity [Ω−1m-1]

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dl

dS

Example : a metal

Metal with one free electron per atom

- RMS velocity : <v2>1/2

- Mean free path : L

- Time between collisions : τ- Drift velocity : vdrift

NB insulators have σ ∼ 10-15 Ω−1m-1

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Relaxation time

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8 x 10-19 s metal

8 x 103 s insulator

“Characteristic time for electrons to rearrange themselves”

Point charge near conductor

Physical picture :

Conduction electrons are attracted to the surface of the

conductor creating a surface charge density, σs.(NB tR = 0!)

Mathematical picture :

Solve Poisson’s equation for Φ subject to the boundary

condition that Φ(x,y,0)=0. NB this will be unique as

shown in Week 1

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Physical picture

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Mathematical problem

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Comparison of field lines

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Methods of images

Image charge