Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/135

$\Phi =-{\tfrac {1}{2}}\Sigma _{r}\Sigma _{s}\Sigma _{t}(E_{s}V_{t}q_{rt}{\frac {dp_{rs}}{d\phi }})$.

(9)

Now the coefficients of potential are connected with those of induction by n equations of the form

$\Sigma _{r}(p_{ar}q_{ar})=1\,\!$,

(10)

and ${\tfrac {1}{2}}n(n-1)$ of the form

$\Sigma _{r}(p_{ar}q_{ar})=0\,\!$.

(11)

Differentiating with respect to $\phi$ we get ${\tfrac {1}{2}}n(n+1)$ equations of the form

$\Sigma _{r}(p_{ar}{\frac {dq_{br}}{d\phi }})+\Sigma _{r}(q_{br}{\frac {dp_{ar}}{d\phi }})=0$,

(12)

where $a$ and $b$ may be the same or different.

Hence, putting $a$ and $b$ equal to $r$ and $s$,

$\Phi ={\tfrac {1}{2}}\Sigma _{r}\Sigma _{s}\Sigma _{t}(E_{s}V_{t}p_{rs}{\frac {dq_{rt}}{d\phi }})$,

(13)

but \Sigma_s(E_s p_{rs})=V_r, so that we may write

$\Phi ={\tfrac {1}{2}}\Sigma _{r}\Sigma _{t}(V_{r}V_{t}{\frac {dq_{rt}}{d\phi }})$,

(14)

where $r$ and $t$ may have each every value in succession from 1 to $n$. This expression gives the resultant force in terms of the potentials.

If each conductor is connected with a battery or other contrivance by which its potential is maintained constant during the displacement, then this expression is simply

$\Phi ={\frac {dQ}{d\phi }}\,\!$,

(15)

under the condition that all the potentials are constant.

The work done in this case during the displacement $\delta \phi$ is $\Phi \delta \phi$, and the electrical energy of the system of conductors is increased by $\delta Q$; hence the energy spent by the batteries during the displacement is

$\Phi \delta \phi +\delta Q=2\Phi \delta \phi =2\delta Q\,\!$.

(16)

It appears from Art. 92, that the resultant force $\Phi$ is equal to $-{\tfrac {dQ}{d\phi }}$, under the condition that the charges of the conductors are constant. It is also, by Art. 93, equal to ${\tfrac {dQ}{d\phi }}$, under the condition that the potentials of the conductors are constant. If the conductors are insulated, they tend to move so that their energy is diminished, and the work done by the electrical forces during the displacement is equal to the diminution of energy.

If the conductors are connected with batteries, so that their