71004

Linear magnetostatic case

A not entirely parallel examination of the work done, perhaps by mechanical or perhaps

by electrical means, to create a magnetic field leads to the result Um =

1

2

Z

vH Hdv (7.4)

for the total energy stored. Of course in the case of a linear medium this result has several alternative forms, one of which is Um =

1

2

Z

vH- B dv. (7.5)

To see which of these results appears to be the closer to physical processes, we consider

the case where the field results from the creation in a toroidal inductor of a current by

means of a suitably variable voltage generator connected to the inductor terminals.

In

any portion of the field creation process, we find the power being delivered to the Circuit

is the product of the total current i (which is related to the total magnetic field H) and the currently induced voltage v (which is related to the time ratę of change of the flux density B).

Thus in an incremental change of flux taking place over time Ót, we will find the component 6Um = vi6t of energy change is related to the product of H, óB/ót and 6t i.e. to the product of H and 6B. Thus as far as the incremental component of energy supplied to bring about this field change is concerned, it seems that the most directly appropriate formula is dUm =

Z

vH- dB dv.