Electric Residue in the Leyden Jar. 4^5 



charged everywhere with electricity of the same density, so that 

 the unit of surface contains the quantity e. At a distance a from 

 the plane, and in the normal erected in the centre of the circle, 

 conceive the quantity e' of electricity to be placed ; i-equired the 

 action, i. e. the electromotive force exerted on this point, in the 

 direction of the normal, by the whole electricity on the circular 

 surface. 



An element of surface, which is at the distance r from the 

 centre, and makes an angle ^ with any radius assumed as fixed, 

 can be expressed by rdrd(f) ; and, neglecting the sign, the action 

 of the same on e' in the direction of the normal is 



ee'a?-drd(j) 



Integrating this, first between the limits ^ = and = 27r, and 

 then between the limits r = and ?' = E, it becomes 



^'"'"'''{l-'vww)' 



If we here set R = oo, we obtain the following theorem : — 

 An indefinite plane, charged uniformly with electricity, attracts 

 or repels an electric particle e' with the same force, 27ree', what- 

 ever may be its finite distance from the same. 



If E be not infinite, and a = 7nR, the formula becomes 



v'/rt + l 



In order to see then what influence the distance of the point 

 from the plane of the circle has, we select for m the numerical 

 values 0"1 and O'Ol. Then for the electric action we have the 

 numbers 



2ee'7r . 0-9046 and 2ee'7r . 0-9901. 



We see, therefore, that so long as the distance is small in 

 comparison to the radius, the electric action is almost indepen- 

 dent of the distance ; for instance, when the distance is increased 

 ten times, the action is only one-tenth of what it was before. 



On the other hand, we may conclude that the action exerted 

 on e' is j)rincipally due to the parts of the plane which lie nearest 

 it. For example, if a remains the same, whilst for R we set at 

 one time lOw, and then 100«, we obtain for the second action 

 scarcely one-tenth more than for the first. 



These numbers show, that if the electricities were uniformly 

 distributed over the coatings of a Franklin's plate, and if, whilst 

 their quantity remained the same, the thickness of the glass 

 were increased, the electric moment would also be augmented, 

 because more particles of glass would now be acted upon. But 



