Examples 293 



89. Verify that, if r, s be real positive constants, z = x+iy, a = pe il *, - = - + -, the 



C T S 



field of force outside the conductors x 2 +y 2 + 2sx = 0, x 2 +y 2 -2rx=Q due to a doublet at 

 the point z=a, outside both the circles, of strength p. and inclination a to the axis, is 

 given by putting 



U+iV^ la* (~W cot c* (---}- e--* cot c, (1 - 1 

 P 2 I \z aJ V ao 



where s= is the inverse point to z a with regard to either of the circles. 



90. A very thin indefinitely great conducting plane is bounded by a straight edge of 

 indefinite length, and is connected with the earth. A unit charge is placed at a point P. 

 Prove that the potential at any point Q due to the charge at P and the electricity induced 

 on the conducting plane is 



where P' is the image of P in the plane, the cylindrical coordinates of. Q and P are 

 (r, (p, z), (/, <', z'\ the straight edge is the axis of 2, the angles 0, <' lie between and 27r, 

 <p = on the conductor, 



4r/ J ' 



and those values of the inverse functions are taken which lie between TT and TT. 



91. A semi-infinite conducting plane is at zero potential under the influence of an 

 electric charge q at a point Q outside it. Shew that the potential at any point P is 

 given by 



J 



7T 



q F, . i /coshin + cosi(0-0i) 



* _ {cosh 77 - cos (0 - 00} Han- 1 . / , f ' 4r7i ^ 



V2rr! L V cosh .7 - cos J (0 - 00 



where r, 6, z are the cylindrical coordinates of the point P, (r lt l5 0) of the point Q, = 

 is the equation of the conducting plane, and 



2rri cosh 77 = r 2 + r^+^s 2 . 



Hence obtain the potential at any point due to a spherical bowl at constant potential, 

 and shew that the capacity of the bowl is 



sma) 



where a is the radius of the aperture, and a is the angle subtended by this radius at the 

 centre of the sphere of which the bowl is a part. 



92. A thin circular conducting disc is connected to earth and is under the influence 

 of a charge q of electricity at an external point P. The position of any point Q is denoted 

 by the peri-polar coordinates p, 0, <, where p is the logarithm of the ratio of the distances 

 from Q to the two points R, S in which a plane QRS through the axis of the disc cuts its 

 rim, is the angle RQS, and is the angle the plane QRS makes with a fixed plane 

 through the axis of the disc, the coordinate having values between - TT and + TT, and 

 changing from + TT to - TT in passing through the disc. Prove or verify that the potential 

 of the charge induced on the disc at any point Q (p, 0, <) is 



= sin" 1 (cos |(0 - ) sech a} - -^L |- + - sin" 1 { - cos (0 + ^o) sech \ 



A 7T _J tyf 1 |_2 IT 



