Page:Elementary Text-book of Physics (Anthony, 1897).djvu/311

§ 261] and improyed by Thomson. The essential portions of the instrument (Fig. 80) are a large flat disk $$B,$$ which can be put in conducting contact with one of the two bodies between which the difference of potential is desired; a similar disk $$C,$$ in the centre of which is cut a circular opening, placed parallel to and a little distance above the former one; a smaller disk $$A$$ with a diameter a little less than that of the opening, which can be placed accurately in the opening and brought plane with the larger disk; and an arrangement, either a balance arm or a spring of known strength, from which the small disk is suspended, and by means of which the force acting on the disk when it is plane with the surface of the larger disk can be measured. The three disks can be conveniently styled the attracting disk, the guard ring, and the attracted disk. The position of the attracted disk when it is in the plane of the guard ring is often called the sighted position. The guard ring is employed in order that the distribution on the attracted disk may be uniform.

To determine the difference of potential between the attracted and attracting disks, we consider them first as forming a flat condenser. If we represent by $$Q$$ the quantity of electricity on the attracted disk, by $$V$$ and $$V_{1}$$ the potentials of the attracted and attracting disks respectively, by $$d$$ the distance between them, and by $$S$$ the area of the attracted disk, then, as has been shown in § 259, the capacity of such a condenser is $$\frac{Q}{V_{1} - V} = \frac{S}{4\pi d}\cdot$$ Now from the nature of the condenser, and in consequence of the regular distribution due to the presence of the guard ring, we have $$\frac{Q}{S} = \sigma,$$ the surface density on either plate, whence $$\sigma = \frac{V_{1} - V}{4\pi d}\cdot$$ The surface density cannot be measured, and must be eliminated by means of an equation obtained by observation of the force with which the two disks are attracted. The plates are never far apart, and the force on a unit charge due to the charge on the lower one may be