Electromagnetism II

Cristina Lazzeronic.lazzeroni@bham.ac.uk

Lecture 2

Lecture 2:

- Maxwell’s only contribution to the laws of EM- Need for displacement current- Maxwell’s equations in free space in differential form- Equation of continuity (conservation of charge)

Maxwell’s equations, integral form (from lecture 1)

Ampere’s Law is incomplete

For current distributions involving a high degree of symmetry,instead of the Biot-Savat law, Ampere’s law can be used tocalculate the magnetic field

Maxwell knew that a change in B-field produces an E-field:

Can a changing E-field produce a B-field ?

M4

M3

Induced magnetic fields

Direct experiment shows thatB field is generated by changing E field both inside the plates and outside

Capacitor is being charged

Ampere’s law to a loop inside the plates: conduction current inside the gap is zero !!!Maxwell realized that something called displacement current flows in the gap

Maxwell argued as follows:

Ampere’s Law will work at all times if we add this term to thecurrent density

Displacementcurrent

Maxwell’s Law of induction(induced magnetic field)His sole contribution to the laws of E,B fields but a crucial one

Ampere-Maxwell Law

B field can be set up either by a conduction or a displacementcurrent

Maxwell’s equations in free space, differential form

Gauss’ law in differential form:

M1

Similarly:

Faraday’s law in differential form:

Differentiate at a fixed place wrt time so time derivative can go inside integral

Similarly, Ampere-Maxwell’s law in differential form:

Lorentz equation:

Equation of continuity:

Derivation of equation of continuity:

Conservation of charge: any variation in the total chargewithin a closed surface must be due to charges that flowacross the surface

For any net charge that leaves Sthere must be an equivalent reduction in Q

Charge leaving Surface S =change in the amount ofcharge inside the volume bounded by S

Differential form of thelaw of conservationof electric charge

Can also be derived from M1 and M4, taking divergence of M4

Can also start from steady state equationUse continuity equation to justify displacement current:

Take the divergence: (A)

But from

(A)must be modified by adding to the right hand sidea quantity that will make the divergence everywhere zero.Start from Gauss’ law and make ρ change with time:

Therefore the quantity to add is

Indeed the displacement current

Example:

A parallel plate capacitor with circular plats is being charged

Derive expressions forthe induced magneticfield at various radii r

Example:

A parallel plate capacitor with circular plats is being charged

Derive expressions forthe induced magneticfield at various radii r

Summary

Next Lecture:Electrostatic solutions to Maxwell’s equations

Recommended readings:Grant+Phillips: 1.4.3, 1.4.4, 4.2.2, 4.5, 4.5.1, 4.5.2 6.14

Before next lecture please : Redo the examples in lectures 1,2 by yourself Revise vector calculusMemorize Maxwell’s equations in free space

Digression: vector calculus

See also Maths notes from lectures 22,23,24,25 of Term 1