het316 l10 magnetmc fields ii

HET316 Electromagnetic Waves: Magnetic Fields 10.1

Magnetic Forces and Fields


Magnetic Flux

As with electric fields it is possible to visualise the magneticfield in terms of field lines which are everywhere tangential to the magnetic field B.

Magnetic field lines for a current carrying loop


Magnetic field lines have properties very similar to those of electric field lines. The properties are:

1. The field lines form closed loops (or extend from - to +). 2. Field lines do not branch or cross (except maybe where the fieldfalls to zero). 3. The area density of the field lines is proportional to the magnetic field.

One difference is that the field lines do not start and end on magnetic charges. This is an experimental result: magnetic monopoles are predicted by some grand unified theories but no true elementary magnetic charges have ever been discovered.


Notwithstanding the differences it is still true that the natural definition of the Magnetic Flux through a surface, S, has the same form as for electric flux:


Gauss Law for B

Since magnetic field lines form closed loops, the interpretation of flux in terms of number of field lines crossing a surfacesuggests that the total magnetic flux through any closed surface is zero.

This is, in fact, true, and forms the third of Maxwells equations:

For any closed surface S, (10)


This plays a different role for B fields than the corresponding law for E (or D) fields, because it tells us nothing about how the field is generated, only that the field is constrained to obey equation (10). In this it is like the conservation condition for E. It is any easy exercise to derive the differential form of the law (just repeat the arguments that led to Gauss law for E, with no charge density). The result is


Amperes Law

The Biot-Savart law for a static magnetic field can be recast in a more sophisticated form, called Amperes Law. The reason for doing this is that this form of the law generalises more easily to time-varying fields, in addition to allowing the calculation of fields for some special current distributions that would be hard using the Biot-Savart law directly.

Amperes law , together with Gauss laws for electric and magnetic fields and the conservation condition for static electric fields, form the complete set of Maxwells equations for static fields.

Amperes law gives the circulation of the magnetic field, just as the conservation condition gives the circulation of the electricfield. Consider the case of a long straight wire.


Density


A solenoid is a long cylindrical coil. (The word apparently derives from the Greek solen meaning channel or pipe.) Its main use is to provide a region of almost uniform magnetic field.

It can be thought of as a stack of current loops and, as for single loops, the off axis field can be hard to calculate, For an infinitely long solenoid with small diameter windings, however, symmetry allows us to use Amperes law.

het316 l10 magnetmc fields ii

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