Page:A Dynamical Theory of the Electromagnetic Field.pdf/28

quantities, namely,

For Electromagnetic Momentum	F	G	H
„ Magnetic Intensity	$\alpha$	$\beta$	$\gamma$
„ Electromotive Force	P	Q	R
„ Current due to true conduction	p	q	r
„ Electric Displacement	f	g	h
„ Total Current (including variation of displacement)	p'	q'	r'
„ Quantity of free Electricity	e
„ Electric Potential	$\Psi$

Between these twenty quantities we have found twenty equations, viz.

Three equations of	Magnetic Force	(B)
„	Electric Currents	(C)
„	Electromotive Force	(D)
„	Electric Elasticity	(E)
„	Electric Resistance	(F)
„	Total Currents	(A)
One equation of	Free Electricity	(G)
„	Continuity	(H)

These equations are therefore sufficient to determine all the quantities which occur in them, provided we know the conditions of the problem. In many questions, however, only a few of the equations are required.

Intrinsic Energy of the Electromagnetic Field.

(71) We have seen (33) that the intrinsic energy of any system of currents is found by multiplying half the current in each circuit into its electromagnetic momentum. This is equivalent to finding the integral

$E={\frac {1}{2}}\sum (Fp'+Gq'+Hr')dV$

(37)

over all the space occupied by currents, where $p,q,r$ are the components of currents, and F, G, H the components of electromagnetic momentum.

Substituting the values of $p',q',r'$ from the equations of Currents (C), this becomes

${\frac {1}{8\pi }}\sum \left\{F\left({\frac {d\gamma }{dy}}-{\frac {d\beta }{dz}}\right)+G\left({\frac {d\alpha }{dz}}-{\frac {d\gamma }{dx}}\right)+H\left({\frac {d\beta }{dx}}-{\frac {d\alpha }{dy}}\right)\right\}dV$

Integrating by parts, and remembering that $\alpha ,\beta ,\gamma$ vanish at an infinite distance, the expression becomes

${\frac {1}{8\pi }}\sum \left\{\alpha \left({\frac {dH}{dy}}-{\frac {dG}{dz}}\right)+\beta \left({\frac {dF}{dz}}-{\frac {dH}{dx}}\right)+\gamma \left({\frac {dG}{dx}}-{\frac {dF}{dy}}\right)\right\}dV$

where the integration is to be extended over all space. Referring to the equations of Magnetic Force (B), p. 482, this becomes

$E={\frac {1}{8\pi }}\sum \{\alpha \cdot \mu \alpha +\beta \cdot \mu \beta +\gamma \cdot \mu \gamma \}dV$

(38)