Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/324

=292 If $$\rho$$ denotes the specific resistance of the substance per unit of volume, the electromotive force at any point is $$\rho w$$, and this may be expressed in terms of the electric potential and the vector potential $$H$$ by equations (B), Art. 598, Rh

or

Comparing the coefficients of like powers of $$r$$ in equations (3) and (5),

Hence we may write Rh Rh

690.] To find the total current $$C$$, we must integrate $$w$$ over the section of the wire whose radius is $$a$$, Rh

Substituting the value of $$\pi w$$ from equation (3), we obtain Rh

The value of $$H$$ at any point outside the wire depends only on the total current $$C$$, and not on the mode in which it is distributed within the wire. Hence we may assume that the value of $$H$$ at the surface of the wire is $$A C$$, where $$A$$ is a constant to be determined by calculation from the general form of the circuit. Putting $$H=AC$$ when $$r=a$$, we obtain Rh

If we now write $$\frac{\pi a^2}{\rho} = \alpha$$, $$\alpha$$ is the value of the conductivity of unit of length of the wire, and we have