. 



191.] I'll Gratings, fyc., of Conducting Material. 407 



I 

 1 

 the potential due to the grating, and two parallel planes at equal 

 distances, D, on its two sides, each uniformly electrified with half the 

 quantity of electricity of opposite sign to that on the grating. 



6. If now we construct in metal, C, any one complete equipotential 

 surface, V , of this system, and electrify it with the same quantity of 

 electricity as that which we gave originally to the infinite row of 

 infinitely thin bars ; and if we place metal planes, B, B', at the two 

 places of zero-potential (z = +D), we have an insulated conductor 

 at potential YO, between two planes, B, B', at zero potential, and at 

 distance 2D asunder, on each of which the electric density is ^/>. For 

 brevity, I shall denote the insulated conductor by I. 



'ts electrostatic capacity per unit area of its medial plane (the 

 ne of the original infinitely thin bars) is //V - 



'. This conductor, I, is symmetrical on each side of its medial 

 plane, and consists either of an infinite number of isolated parallel 

 bars, each surrounding one of the original infinitely thin bars, or of a 

 late symmetrically corrugated on its two sides, with maximum and 

 inimum thicknesses respectively at the places of the infinitely thin 

 i, and the lines midway between them. For the case of isolated 

 ars, let 2c be the diameter of each, in the medial plane. Then, to 

 nd V , we must put x = +c and z = 0, in (4). Thus we find 



4 sin 2 / 



(5). 



a 



[ence the electrostatic capacity of I in the circumstances is 



rhich is greater or less than l/(27rD), the electrostatic capacity that 

 i would have if reduced to its medial plane, according as c> or < a. 





Mie conductor I, to be a grating, implies c<^a, or sin 2 <1, and 



a 



lerefore requires that 



V >27r/>(D log 4^ = 27r /) (D--22a) (7), 



V 27T J 



When V exceeds this critical value, the conductor I is the con- 

 inuous plate corrugated on each side, which was described in 7. 



le critical value corresponds to an intermediate case of a plate so 

 leeply furrowed on each side as to be just cut through by its two 



irfaces crossing at right angles ; and (7) shows that the electrostatic 



2 E 2 



