28 Electrostatics Field of Force [OH. n 



We see at once that the two expressions (6) and (7) just obtained for V 

 are identical, on noticing that 6 is the angle between two lines of which the 

 direction cosines are respectively 



X Y Z . dx dy dz 

 R' R' R '' ds' fa' di' 



We therefore have C os0 = > 



R ds R ds R ds 



so that R cos 6 ds = Xdx + Ydy + Zdz, 



and the identity of the two expressions becomes obvious. 



If the Theorem of the Conservation of Energy is true in the Electro- 

 static Field, the work done in bringing a small charge 6 from infinity to any 

 point P must be the same whatever path to P we choose. For if the 

 amounts of work were different on two different paths, let these amounts 

 be V P 6 and Fp'e, and let the former be the greater. Then by taking the 

 charge from P to infinity by the former path and bringing it back by the 

 latter, we should gain an amount of work ( V P V P ') e, which would be 

 contrary to the Conservation of Energy. Thus V P and V P ' must be equal, 

 and the potential at P is the same, no matter by what path we reach P. 

 The potential at P will accordingly depend only on the coordinates #, y, z 

 of P. 



As soon as we introduce the special law of the inverse square, we shall 

 find that the potential must be a single-valued function of x, y, z, as a 

 consequence of this law (39), and hence shall be able to prove that the 

 Theorem of Conservation of Energy is true in an Electrostatic field. For 

 the moment, however, we assume this. 



34. Let us denote by W the work done in moving a charge e from P 

 to Q. In bringing the charge from infinity to P, we do an amount of work 



FIG. 5. 



which by definition is equal to Vpe where V P denotes the value of V at the 

 point P. Hence in taking it from infinity to Q, we do a total amount of 

 work Vpe 4- W. This, however, is also equal by definition to F Q e. Hence 

 we have 



or TT=(F Q - F P )e .............................. (8). 



