Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/293

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This investigation is approximate only when and are large compared with , and when is large compared with . The quantity is a line which may be of any magnitude. It becomes infinite when is indefinitely diminished.

If we suppose there will be no apertures between the wires of the grating, and therefore there will be no induction through it. We ought therefore to have for this case . The formula (11), however, gives in this case

,

which is evidently erroneous, as the induction can never be altered in sign by means of the grating. It is easy, however, to proceed to a higher degree of approximation in the case of a grating of cylindrical wires. I shall merely indicate the steps of this process.


Method of Approximation.

206.] Since the wires are cylindrical, and since the distribution of electricity on each is symmetrical with respect to the diameter parallel to , the proper expansion of the potential is of the form

(14)

where is the distance from the axis of one of the wires, and the angle between and , and, since the wire is a conductor, when is made equal to the radius must be constant, and therefore the coefficient of each of the multiple cosines of must vanish.

For the sake of conciseness let us assume new coordinates , &c. such that

(15)

and let

(16)

Then if we make

(17)

by giving proper values to the coefficients we may express any potential which is a function of and , and does not become infinite except when and .

When the expansion of in terms of and is

(18)

For finite values of the expansion of is

(19)