282 Methods for the Solution of Special Problems [CH. vni 



On Conjugate Functions and Conformal Kepresentation : 

 MAXWELL. Electricity and Magnetism. Chap. xn. 



LAMB. Hydrodynamics. (Camb. Univ. Press, 1895 and 1906.) Chap. iv. 

 J. J. THOMSON. Recent Researches in Electricity and Magnetism. (Clarendon 



Press, 1893.) Chap. in. 

 WEBSTER. Electricity and Magnetism. Introduction, Chap. iv. 



EXAMPLES. 



1. An infinite conducting plane at zero potential is under the influence of a charge of 

 electricity at a point 0. Shew that the charge on any area of the plane is proportional to 

 the angle it subtends at 0. 



2. A charged particle is placed in the space between two uninsulated planes which 

 intersect at right angles. Sketch the sections of the equipotentials made by an imaginary 

 plane through the charged particle, at right angles to the planes. 



3. In question 2, let the particle have a charge e, and be equidistant from the planes. 

 Shew that the total charge on a strip, of which one edge is the line of intersection of the 

 planes, and of which the width is equal to the distance of the particle from this line of 

 intersection, is -\e. 



4. In question 3, the strip is insulated from the remainder of the planes, these being 

 still to earth, and the particle is removed. Find the potential at the point formerly 

 occupied by the particle, produced by raising the strip to potential V. 



5. If two infinite plane uninsulated conductors meet at an angle of 60, and there is a 

 charge e at a point equidistant from each, and distant r from the line of intersection, find 

 the electrification at any point of the planes. Shew that at a point in a principal plane 

 through the charged point at a distance r^/3 from the line of intersection, the surface 

 density is 



6. Two small pith balls, each of mass m, are connected by a light insulating rod. 

 The rod is supported by parallel threads, and hangs in a horizontal position in front of an 

 infinite vertical plane at potential zero. If the balls when charged with e units of 

 electricity are at a distance a from the plate, equal to half the length of the rod, shew 

 that the inclination 6 of the strings to the vertical is given by 



tan0 = 



7. What is the least positive charge that must be given to a spherical conductor, 

 insulated and influenced by an external point-charge e at distance r from its centre, in 

 order that the surface density may be everywhere positive ? 



8. An uninsulated conducting sphere is under the influence of an external electric 

 charge; find the ratio in which the induced charge is divided between the part of its 

 surface in direct view of the external charge and the remaining part. 



9. A point-charge e is brought near to a spherical conductor of radius a having a 

 charge E. Shew that the particle will be repelled by the sphere, unless its distance from 



the nearest point of its surface is less than \a \J -p^ approximately. 



