Examples 351 



EXAMPLES. 



1. The ends of a rectangular conducting lamina of breadth c, length a, and uniform 

 thickness T, are maintained at different potentials. If f(x, y} be the specific resistance p 

 at a point whose distances from an end and a side are #, y, prove that the resistance of 



the lamina cannot be less than 



dx 





or greater than 



2. Two large vessels filled with mercury are connected by a capillary tube of uniform 

 bore. Find superior and inferior limits to the conductivity. 



3. A cylindrical cable consists of a conducting core of copper surrounded by a thin 

 insulating sheath of material of given specific resistance. Shew that if the sectional 

 areas of the core and sheath are given, the resistance to lateral leakage is greatest when 

 the surfaces of the two materials are coaxal right circular cylinders. 



4. Prove that the product of the resistance to leakage per unit length between two 

 practically infinitely long parallel wires insulated by a uniform dielectric and at different 

 potentials, and the capacity per unit length, is Kp/^Tr, where K is the inductive capacity 

 and p the specific resistance of the dielectric. Prove also that the time that elapses before 

 the potential difference sinks to a given fraction of its original value is independent of the 

 sectional dimensions and relative positions of the wires. 



5. If the right sections of the wires in the last question are semicircles described on 

 opposite sides of a square as diameters, and outside the square, while the cylindrical space 

 whose section is the semicircles similarly described on the other two sides of the square is 

 filled up with a dielectric of infinite specific resistance, and all the neighbouring space is 

 filled up with a dielectric of resistance p, prove that the leakage per unit length in unit 

 time is 2 F/p, where V is the potential difference. 



6. If ^ + ^=/(^+*y), and the curves for which <p = cons. be closed curves, shew that 

 the insulation resistance between lengths I of the surfaces <p = (p > 0=<pi, is 



p (<Pi ~ <Po) 



where [\^] is the increment of ^ on passing once round a <p-curve, and p is the specific 

 resistance of the dielectric. 



7. Current enters and leaves a uniform circular disc through two circular wires of 

 small radius e whose central lines pass through the edge of the disc at the extremities of 

 a chord of length d. Shew that the total resistance of the sheet is 



(2cr/ir) log (die). 



8. Using the transformation 



prove that the resistance of an infinite strip of uniform breadth TT between two electrodes 

 distant 2a apart, situated on the middle line of the strip and having equal radii 5, is 



log ( TT tanh a ) . 

 \fl / 



