97] Examples 85 



2. A sheet of metal of thickness t is introduced between the two plates of a parallel 

 plate condenser which are at a distance d apart, and is placed so as to be parallel to the 

 plates. Shew that the capacity of the condenser is increased by an amount 



t 



4-rrd(d-t} 



per unit area. Examine the case in which t is very nearly equal to d. 



3. A high-pressure main consists first of a central conductor, which is a copper tube 

 of inner and outer diameters of T 9 F and }f inches. The outer conductor is a second copper 

 tube coaxal with the first, from which it is separated by insulating material, and of 

 diameters Iff and l^f inches. Outside this is more insulating material, and enclosing 

 the whole is an iron tube of internal diameter 2 T ^ inches. The capacity of the conductor 

 is found to be -367 microfarad per mile : calculate the inductive capacity of the insulating 

 material. 



4. An infinite plane is charged to surface density or, and P is a point distant half an 

 inch from the plane. Shew that of the total intensity STTO- at P, half is due to the charges 

 at points which are within one inch of P, and half to the charges beyond. 



5. A disc of vulcanite (non-conducting) of radius 5 inches, is charged to a uniform 

 surface density a- by friction. Find the electric intensities at points on the axis of the 

 disc distant respectively 1, 3, 5, 7 inches from the surface. 



6. A condenser consists of a sphere of radius a surrounded by a concentric spherical 

 shell of radius b. The inner sphere is put to earth, and the outer shell is insulated. 



Shew that the capacity of the condenser so formed is ^ - 



7. Four equal large conducting plates A, B, C, D are fixed parallel to one another. 

 A and D are connected to earth, B has a charge E per unit area, and C a charge E' per 

 unit area. The distance between A and B is a, between B and C is 6, and between C and 

 D is c. Find the potentials of B and C. 



8. A circular gold-leaf of radius b is laid on the surface of a charged conducting 

 sphere of radius a, a being large compared to b. Prove that the loss of electrical energy 

 in removing the leaf from the conductor assuming that it carries away its whole charge 

 is approximately 1 b 2 E 2 /a?, where E is the charge of the conductor, and the capacity of the 

 leaf is comparable to 6. 



9. Two condensers of capacities Ci and C 2 , and possessing initially charges & and Q 2 

 are connected in parallel. Shew that there is a loss of energy of amount 



10. Two Leyden Jars A, B have capacities Ci, C 2 respectively. A is charged and a 

 spark taken: it is then charged as before and a spark passed between the knobs of 

 A and B. A and B are then separated and are each discharged by a spark. Shew that 

 the energies of the four sparks are in the ratio 



C 2 . 



11. Assuming an adequate number of condensers of equal capacity , shew how a 

 compound condenser can be formed of equivalent capacity 0C, where 6 is any rational 

 number. 



