Examples 291 



76. If the cylinder r=a+bco$6 be freely charged, shew that in free space the 

 resultant force varies as 



and makes with the line 0=0, an angle 



"*"-' 



where 2 -& 2 = 2fo. 



77. If <j> + fy=f(x+iy), and the curves for which = constant be closed, shew that 

 the capacity G of a condenser with boundary surfaces < = $!, = <o is 



per unit length, where [^] is the increment of ty on passing once round a 0-curve. 



78. Using the transformation x + iy = c cot ( 7+^F), shew that the capacity per unit 

 length of a condenser formed by two right circular cylinders (radii a, 6), one inside the 

 other, with parallel axes at a distance d apart, is 



/0? + b2-cP\ 



1/2 cosh- 1 --Vr ) 



' \ 26 ) 



79. A plane infinite electric grating is made of equal and equidistant parallel thin 

 metal plates, the distance between their successive central lines being TT, and the breadth 



of each plate 2sin~M =J. Shew that when the grating is electrified to constant 



potential, the potential and charge functions F, U in the surrounding space are given 

 by the equation 



sin ( U+ iV} = K sin (x + iy\ 



Deduce that, when the grating is to earth and is placed in a uniform field of force of unit 

 intensity at right angles to its plane, the charge and potential functions of the portion of 

 the field which penetrates through the grating are expressed by 



and expand the potential in the latter problem in a Fourier Series. 



80. A cylinder whose cross-section is one branch of a rectangular hyperbola is 

 maintained at zero potential under the influence of a line-charge parallel to its axis 

 and on the concave side. Prove that the image consists of three such line charges, and 

 hence find the density of the induced distribution. 



81. A cylindrical space is bounded by two coaxial and confocal parabolic cylinders, 

 whose latera recta are 4a and 46, and a uniformly electrified line which is parallel to the 

 generators of the cylinder intersects the axes which pass through the foci in point distant 

 c from them (>c>6). Shew that the potential throughout the space is 



J Trr* cos - TT 



) COSh r T COS 



.. I *-6* 



Al 8-, ; T 



P*-!-^ 

 *-6* f 



IT (V* sin |+c* -<**-&*' 



V * 



irr cos - 



JCOSh r r + COS ' 



( a i -6 i a^-6* ) 



where r, 6 are polar coordinates of a section, the focus being the pole. Determine A in 

 terms of the electrification per unit length of the line. 



192 



