Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/400

308.] We shall now apply the same method to find the correction which must be applied to the length of a cylindrical conductor of radius $a$ when its extremity is placed in metallic contact with a massive electrode, which we may suppose of a different metal.

For the lower limit of the resistance we shall suppose that an infinitely thin disk of perfectly conducting matter is placed between the end of the cylinder and the massive electrode, so as to bring the end of the cylinder to one and the same potential throughout. The potential within the cylinder will then be a function of its length only, and if we suppose the surface of the electrode where the cylinder meets it to be approximately plane, and all its dimensions to be large compared with the diameter of the cylinder, the distribution of potential will be that due to a conductor in the form of a disk placed in an infinite medium. See Arts. 152, 177.

If $E$ is the difference of the potential of the disk from that of the distant parts of the electrode, $C$ the current issuing from the surface of the disk into the electrode, and $\rho '$ the specific resistance of the electrode,

$\rho 'C=4aE.$

(18)

Hence, if the length of the wire from a given point to the electrode is $L,$ and its specific resistance $\rho ,$ the resistance from that point to any point of the electrode not near the junction is

$R=\rho {\frac {L}{\pi a^{2}}}+{\frac {\rho '}{4a}},$

and this may be written

$R={\frac {\rho }{\pi a^{2}}}\left(L+{\frac {\rho '}{\rho }}{\frac {\pi a}{4}}\right),$

(19)

where the second term within brackets is a quantity which must be added to the length of the cylinder or wire in calculating its resistance, and this is certainly too small a correction.

To understand the nature of the outstanding error we may observe, that whereas we have supposed the flow in the wire up to the disk to be uniform throughout the section, the flow from the disk to the electrode is not uniform, but is at any point inversely proportional to the minimum chord through that point. In the actual case the flow through the disk will not be uniform, but it will not vary so much from point to point as in this supposed case. The potential of the disk in the actual case will not be uniform, but will diminish from the middle to the edge.

309.] We shall next determine a quantity greater than the true resistance by constraining the flow through the disk to be uniform