31-33] The Potential 27 



the distribution of electricity on the conductors in the field is not affected 

 by it. Then the work done in bringing the charge e to a point is pro- 

 portional to e, and may be taken to be Fe. The amount of work done will 

 of course depend on the position from which the charged particle started. 

 It is convenient, in measuring Fe, to suppose that the particle started at a 

 point outside the field altogether, i.e. from a point so far removed from all 

 the charges of the field that their effect at this point is inappreciable for 

 brevity, we may say the point at infinity. We now define F to be the 

 potential at the point 0. Thus 



The potential at any point in the field is the work per unit charge which 

 has to be done on a charged particle to bring it to that point, the charge on the 

 particle being supposed so small that the distribution of electricity on the 

 conductors in the field is not affected by its presence. 



In moving the small charge e from x, y, z to x + dx, y + dy, z + dz, we 

 shall have to perform an amount of work 



-(Xdx + Ydy + Zdz)e, 



so that in bringing the charge e into position at x, y, z from outside the field 

 altogether, we do an amount of work 



7dy+Zde) t 



where the integral is taken along the path followed by e. 



Denoting the work done on the charge e in bringing it to any point 



x, y y z in the electric field by Fe, we clearly have 



/ 



(Xdx + Ydy + Zdz) (6), 



giving a mathematical expression for the potential at the point x, y, z. 



The same result can be put in a different form. If ds is any element of 

 the path, and if the intensity R at the extremity of this element makes an 

 angle 6 with ds, then the component of the force acting on e when moving 

 along ds, resolved in the direction of motion of e, is Re cos 6. The work 

 done in moving e along the element ds is accordingly 



Re cos 6ds, 

 so that the whole work in bringing e from infinity to x, y, z is 



-eT '''RcosQds, 



J oo 



and since this is equal, by definition, to Fe, we must have 



; 



Bcosffds (7). 



