352 Steady Currents in continuous Media [OH. x 



9. Shew that the transformation 



x' + iy' = cosh TT (x + iy}la 



enables us to obtain the potential due to any distribution of electrodes upon a thin 

 conductor in the form of the semi-infinite strip bounded by y=0, #=, and #=0. 



If the margin be uninsulated, find the potential and flow due to a source at the point 

 x = c, y = n Shew that if the flows across the three edges are equal, then irc=a cosh" 1 2. 



10. Equal and opposite electrodes are placed at the extremities of the base of an 

 isosceles triangular lamina, the length of one of the equal sides being a, and the vertical 



2?r 

 angle . Shew that the lines of flow and equal potential are given by 



. 2 w 1 + cn u 



2 ~ 1 en u ' 

 where 



and the modulus of en u is sin 75, the origin being at the vertex. 



11. A circular sheet of copper, of specific resistance <TI per unit area, is inserted in a 

 very large sheet of tinfoil (CT O ), and currents flow in the composite sheet, entering and 

 leaving at electrodes. Prove that the current-function in the tinfoil corresponding to an 

 electrode at which a current e enters the tinfoil is the coefficient of i in the imaginary 

 part of 



"o e [~i / \ O-Q O-I i cz 

 - - log (z - c) + -i -- log - , 

 ZTT [_ o-Q + o-i cz - a?j ' 



where a is the radius of the copper sheet, z is a complex variable with its origin at the 

 centre of the sheet, and c is the distance of the electrode from the origin, the real axis 

 passing through the electrode. 



Generalise the expression for any position of the electrode in the copper or in the 

 tinfoil, and investigate the corresponding expressions determining the lines of flow in the 

 copper. 



12. A uniform conducting sheet has the form of the catenary of revolution 



y2 + s*=c 2 cosh 2 - t 



Prove that the potential at any point due to an electrode at # , y > ^05 introducing a 

 current (7, is 



(7(7, / . X Xn 



constant - log I cosh - 



** \ ^ 



