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

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154
SIMPLE CASES.
[126.

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

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

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


Two Infinite Coaxal Cylindric Surfaces.

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

whence

Since when , and when ,

If are the surface-densities on the inner and outer surfaces,