734 



SCIENCE 



[N. S. Vol. XXVIII. No. 725 



SPECIAL ARTICLES 



NOTE ON THE FORMULAS FOR ENERGY STORED IN 

 ELECTRIC AND MAGNETIC FIELDS 



Consider a charged sphere. Let it grow in 

 size. The potential decreases for the same 

 charge as the radius increases. Hence the 

 potential energy also decreases. The tubes of 

 force, everywhere pulling the surface out 

 toward infinity, are losing the potential energy 

 of their stretched condition, and at infinity 

 they have closed up and the potential energy 

 has disappeared from the potential state. 



We may then consider the energy as resi- 

 ding, not in the sphere but in the dielectric 

 outside, and that the amount of energy that 

 disappears from the potential state at each 

 step is entirely in the spherical shell of the 

 dielectric, which makes up the difference in 

 volume between the successive steps in the 

 growth of the sphere. We have then, only to 

 calculate the difference in potential energy for 

 two slightly different radii of the sphere and 

 divide by the volume of the spherical shell, 

 and we shall have the density of the energy 

 in the electric field. It is to be noted that the 

 electric field at any point outside the sphere 

 is unchanged by the growth of the sphere, 

 since the number of tubes of force, and hence 

 the amount of their crowding, depends only 

 on the charge and not on the size of the 

 sphere. 



Let r be the radius of the sphere, v the 

 volume, e the charge, E the electric field, ^ 

 the potential, P the potential energy, c the 

 dielectric constant. 



By definition if/ is the work necessary to 

 carry unit charge from infinity to the sphere. 



f: 



=J'^Edr =J\elr'^)dr = e/r, 



(1) 



which might have been written immediately, 

 since the capacity of a sphere is r. Also by 

 definition 



E = d^p/dr = d/dr ( e/r ) = e/r'. ( 2 ) 



We have also 



P = i<l^e. (3) 



From (1) and (3), 



P = e=/2r. 



Differentiating, we get the change in potential 



energy due to a small change in radius, 

 dP = — e-dr/2r', 



the negative sign meaning a decrease in 

 energy for an increase in radius. The volume 

 of the shell is 4r7rr'dr, and the loss of potential 

 energy per cm' is, by equation (2), 



dP/dv = — eySirr' = — W/8-ir. 



Hence the energy in the dielectric is E'/Stt 

 ergs per cm'. 



If £ =t= 1, the charge for the same ij/ and the 

 same E is greater and we have to write e^ and 

 (eE) instead of i^ and E in equations (1) and 

 (2), to make them hold numerically. This 

 followed through gives, finally, 



E{eE)/8Tr. 



The expression for the energy in a magnetic 

 field follows in exactly the same way; we have 

 only to substitute m for e and H for E in the 

 equations above. We may take a sphere of 

 very great permeability as an isolated pole m. 

 Should it seem clearer, this sphere may be 

 thought of as the pole piece of a long magnet 

 of infinitesimal diameter reaching to infinity, 

 where the other pole piece forms another 

 spherical shell. The tubes of force tend to 

 shorten as in the electrostatic field, closing up 

 when the sphere grows to infinite radius. 



The energy per cm' comes out E^/Stt. 



If all surrounding space is filled with a 

 medium whose permeability is ^ instead of 1, 

 the number of tubes for the same H is fi times 

 as great. So, as before, we must use fiij/ and 

 (fiE) in equations (1) and (2), which, traced 

 through, give jff(mff)/8ir ergs per cm'. 



The first three derivations are rigorous, but 

 awkward questions arise as to what the H and 

 the B, in the last case, represent physically. 

 Yet it can be made satisfactory by supposing 

 that the long thin magnet is divided into two 

 parts, one supplying 11 tubes of force, and the 

 other supplying (;u, — 1) H tubes, the former 

 being rigidly magnetized. 



From the method of this derivation it fol- 

 lows without additional proof that the tension 

 along the lines of force is numerically equal 

 to the energy density. P. G. Agnew 



Washington, D. C. 



