Electromagnetism II

Cristina Lazzeronic.lazzeroni@bham.ac.uk

Lecture 6

Maxwell’s equations for free space

They apply simultaneously at any given point in free space.How do they change in presence of material ?

Lecture 4:

Macroscopic polarization P of a dielectric produces surface and volume charge densities:

Electric displacement:

Lecture 5:

Magnetic materials, macroscopic magnetization M

Lecture 6: Magnetism (II)

Magnetization currents (surface and volume)

New quantity: H-field or magnetic field intensity or magnetic field strength

Relationship between B, M, H

Maxwell’s equations in terms of H

Surface magnetization currents

Consider a cylindrical rod in which magnetization is uniform

Magnetization appears in the form of a surface charge

IS = current per unit length - net effect of microscopic currentsassociated with bound electrons

Surface magnetization currents

Total magnetic moment of a length l of the rod:

Total magnetic moment is also given by magnetizationmultiplied by the volume:

= Unit vector normal to the surface at which IS is determined

Vector form

Volume magnetization currents

Volume magnetization currents occur if magnetization is non-uniform i.e. depends on position.

Then current in loops will vary with the position of the loopsand currents in neighbouring loops may not cancel

net current density in the volume of the material

Consider adjacent loops:

Magnetic moment:

Loop A

Loop B

Current that flows around loop A = magnetic moment / area :

Similarly for loop B :

Net current in y direction at boundary between two loops:

If magnetization has a component Mx that depends on z coordinate, there is a second contribution to Jy :

Hence overall:

Similarly:

Hence the bound magnetic current density is related to M by

The bound currents are the net effect of the microscopic currents associated with magnetic dipoles.They generate currents just as free currents do.

Let’s consider M4:

freepolarization magnetization

Define auxillary field H ormagnetic field intensity

H is a mathematical construction so that we can consider just free currents (like D with polarization)

M4 with dielectric andmagnetic materials

(units = ampere / metre)

Integral form of Ampere-Maxwell law:

If

Ampere’s law in terms of HIf is free current through surface S

Gauss’ magnetic law, and Faraday’s law are unchanged

D and H are auxiliary fields, no simple interpretation

LIH materials: From Lecture 5

Substituting into the definition of H:

Define magnetic relative permeability:

Hence in LIH:

Recall:

Linear: B proportional to HIsotropic: B, H, M in the same directionHomogenous: µr independent of position

Diamagnetics: µr is slightly less than 1 and independent of BParamagnetics: µr is slightly bigger than 1 and independent of B

Ferromagnetics: µr changes as B changes

Above equation can be used but with caution:appropriate value must be used

Example:

B depends on the medium (µr)

Note discontinuityof B on the boundary.

We need to see whathappens to fields on eitherside of boundary betweendifferent media.

Summary :

Recommended readings:Grant+Phillips: 5.3 10.2, 10.3

Next Lecture:

Boundary conditions: what happens to the electric and magnetic fields across the boundary between two different materials ?