This page needs to be proofread.

CHAPTER XI.

ON ENERGY AND STRESS IN THE ELECTROMAGNETIC FIELD.

Electrostatic Energy.

630.] The energy of the system may be divided into the Potential Energy and the Kinetic Energy.

The potential energy due to electrification has been already considered in Art. 85. It may be written

W={\frac {1}{2}}\sum (e\Psi )

,

(1)

where $e$ is the charge of electricity at a place where the electric

potential is $\Psi$ , and the summation is to be extended to every place where there is electrification.

If $f$ , $g$ , $h$ are the components of the electric displacement, the quantity of electricity in the element of volume $dx\,dy\,dz$ is

e=\left({\frac {df}{dx}}+{\frac {dg}{dy}}+{\frac {dh}{dz}}\right)\,dx\,dy\,dz

,

(2)

and

W={\frac {1}{2}}\iiint \left({\frac {df}{dx}}+{\frac {dg}{dy}}+{\frac {dh}{dz}}\right)\,\Psi \,dx\,dy\,dz

,

(3)

where the integration is to be extended throughout all space.

631.] Integrating this expression by parts, and remembering that when the distance, $r$ , from a given point of a finite electrified system becomes infinite, the potential $\Psi$ becomes an infinitely small

quantity of the order $r^{-1}$ , and that $f$ , $g$ , $h$ become infinitely small quantities of the order $r^{-2}$ , the expression is reduced to

W=-{\frac {1}{2}}\iiint \left(f{\frac {d\Psi }{dx}}+g{\frac {d\Psi }{dy}}+h{\frac {d\Psi }{dz}}\right)\,dx\,dy\,dz

,

(4)

where the integration is to be extended throughout all space. If we now write $P$ , $Q$ , $R$ for the components of the electromotive force, instead of $-{\frac {d\Psi }{dx}}$ , $-{\frac {d\Psi }{dy}}$ , and $-{\frac {d\Psi }{dz}}$ , we find