64 Electrostatics Field of Force [OH. n 



8. The potentials at the four corners of a small tetrahedron J, B, C, D are F 1? F 2> 

 F 3 , F 4 respectively. G is the centre of gravity of masses J/ x at A, J/ 2 at B, M 3 at C, 

 J/4 at /). Shew that the potential at G is 



9. Charges 3e, e, -e are placed at A, B, C respectively, where B is the middle 

 point of AC. Draw a rough diagram of the lines of force ; shew that a line of force which 

 starts from A making an angle a with A B > cos" 1 ( - ) will not reach B or (7, and shew 

 that the asymptote of the line of force for which a=cos~ 1 ( ) is at right angles to AC. 



10. If there are three electrified points A, B, C in a straight line, such that AC=f, 

 BC = -j , and the charges are e, ^-' and Va respectively, shew that there is always a 

 spherical equipotential surface, and discuss the position of the points of equilibrium on 



the line ABC when V=e ~ .- and when V=e ~ - ^- 



(/-a) 2 (/+) 2 



11. A and C are spherical conductors with charges e + e' and e respectively. Shew 

 that there is either a point or a line of equilibrium, depending on the relative size and 

 positions of the spheres, and on e'/e. Draw a diagram for each case giving the lines of 

 force and the sections of the equipotentials by a plane through the centres. 



12. An electrified body is placed in the vicinity of a conductor in the form of a 

 surface of anticlastic curvature. Shew that at that point of any line of force passing from 

 the body to the conductor, at which the force is a minimum, the principal curvatures of 

 the equipotential surface are equal and opposite. 



13. Shew that it is not possible for every family of non-intersecting surfaces in free 

 space to be a family of equipotentials, and that the condition that the family of surfaces 



/(A, *,y, z) = 

 shall be capable of being equipotentials is that 



8X\ 2 /8X\ 2 /8X 



shall be a function of X only. 



14. In the last question, if the condition is satisfied find the potential. 



15. Shew that the confocal ellipsoids 



can form a system of equipotentials, and express the potential as a function of X. 



16. If two charged concentric shells be connected by a wire, the inner one is wholly 

 discharged. If the law of force were - , prove that there would be a charge B on the 



inner shell such that if A were the charge on the outer shell, and /, g the sum and diS'er- 

 ence of the radii, 



2gB=-Ap {( 

 approximately. 



