Examples 87 



19. Outside a spherical charged conductor there is a concentric insulated but un- 

 charged conducting spherical shell, which consists of two segments. Prove that the two 

 segments will not separate if the distance of the separating plane from the centre is less 

 than 



ab 



where a, b are the internal and external radii of the shell. 



20. A soap-bubble of radius a is formed by a film of tension T, the external 

 atmospheric pressure being II. The bubble is touched by a wire from a large conductor 

 at potential F, and the film is an electrical conductor. Prove that its radius increases to 

 r, given by 



FV 



n(r3-a3) + 27> 2 -a2)=-^. 



O7T 



21. If the radius and tension of a spherical soap-bubble be a and T respectively, 

 shew that the charge of electricity required to expand the bubble to twice its linear 

 dimensions, would be 



4 Vn-tt^er+Tna), 

 n being the atmospheric pressure. 



22. A thin spherical conducting envelope, of tension T for all magnitudes of its 

 radius, and with no air inside or outside, is insulated and charged with a quantity Q of 

 electricity. Prove that the total gain in mechanical energy involved in bringing a charge 

 q from an infinite distance and placing it on the envelope, which both initially and finally 

 is in mechanical equilibrium, is 



f (2^)* {($ + ?)*-#}. 



23. A spherical soap-bubble is blown inside another concentric with it, and the 

 former has a charge E of electricity, the latter being originally uncharged. The latter 

 now has a small charge given to it. Shew that if a and 2a were the original radii, the 

 new radii will be approximately a + x, 2a+y, where 



where n is the atmospheric pressure, and T is the surface-tension of each bubble. 



24. Shew that the electric capacity of a conductor is less than that of any other 

 conductor which can completely surround it. 



25. If the inner sphere of a concentric spherical condenser is moved slightly out of 

 position, so that the two spheres are no longer concentric, shew that the capacity is 

 increased. 



