Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/196

154 If a spherical soap bubble is electrified to a potential $$A$$, then, if its radius is $$a$$, the charge will be $$Aa$$, and the surface-density will be

The resultant electrical force just outside the surface will be $$4\pi\sigma$$, and inside the bubble it is zero, so that by Art. 79 the electrical force on unit of area of the surface will be $$2\pi\sigma^{2}$$, acting outwards. Hence the electrification will diminish the pressure of the air within the bubble by $$2\pi\sigma^{2}$$, or

But it may be shewn that if $$T$$ is the tension which the liquid film exerts across a line of unit length, then the pressure from $$T$$ within required to keep the bubble from collapsing is $$2\tfrac{T}{a}$$. If the electrical force is just sufficient to keep the bubble in equilibrium when the air within and without is at the same pressure

126.] Let the radius of the outer surface of a conducting cylinder be $$a$$, and let the radius of the inner surface of a hollow cylinder, having the same axis with the first, be $$b$$. Let their potentials be $$A$$ and $$B$$ respectively. Then, since the potential $$V$$ is in this case a function of $$r$$, the distance from the axis, Laplace’s equation becomes

whence

Since $$V=A$$ when $$r=a$$, and $$V=B$$ when $$r=b$$,

If $$\sigma_{1},\ \sigma_{2}$$ are the surface-densities on the inner and outer surfaces,