Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/326

 The Guard-ring Accumulator.

$$Bb$$ is a cylindrical vessel of conducting material of which the outer surface of the upper face is accurately plane. This upper surface consists of two parts, a disk $$A$$, and a broad ring $$BB$$ surrounding the disk, separated from it by a very small interval all round, just sufficient to prevent sparks passing. The upper surface of the disk is accurately in the same plane with that of the guard-ring. The disk is supported by pillars of insulating material $$GG$$. $$C$$ is a metal disk, the under surface of which is accurately plane and parallel to $$BB$$. The disk $$C$$ is considerably larger than $$A$$. Its distance from $$A$$ is adjusted and measured by means of a micrometer screw, which is not given in the figure.

This accumulator is used as a measuring instrument as follows:–

Suppose $$C$$ to be at potential zero, and the disk $$A$$ and vessel $$Bb$$ both at potential $$V$$. Then there will be no electrification on the back of the disk because the vessel is nearly closed and is all at the same potential. There will be very little electrification on the edges of the disk because $$BB$$ is at the same potential with the disk. On the face of the disk the electrification will be nearly uniform, and therefore the whole charge on the disk will be almost exactly represented by its area multiplied by the surface-density on a plane, as given at Art. 124.

In fact, we learn from the investigation at Art. 201 that the charge on the disk is

$V\left\{ \frac{R^{2}+R'^{2}}{8A}-\frac{R'{}^{2}-R{}^{2}}{8A}\frac{\alpha}{A+\alpha}\right\} $|undefined

where $$R$$ is the radius of the disk, $$R'$$ that of the hole in the guard-ring, $$A$$ the distance between $$A$$ and $$C$$, and $$\alpha$$ a quantity which cannot exceed $$(R'-R)\tfrac{\log_{e}2}{\pi} $$.

If the interval between the disk and the guard-ring is small compared with the distance between $$A$$ and $$C$$, the second term will be very small, and the charge on the disk will be nearly

$V\frac{R^{2}+R'^{2}}{8A} $|undefined