Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/353

 

On the Proper Thickness of the Wire of a Galvanometer, the External Resistance being given.

716.] Let the form of the channel in which the galvanometer coil is to be wound be given, and let it be required to determine whether it ought to be filled with a long thin wire or with a shorter thick wire.

Let $l$ be the length of the wire, $y$ its radius, $y+b$ the radius of the wire when covered, $\rho$ its specific resistance, $g$ the value of $G$ for unit of length of the wire, and $r$ the part of the resistance which is independent of the galvanometer.

The resistance of the galvanometer wire is

$R={\frac {\rho }{\pi }}{\frac {l}{y^{2}}}$.

The volume of the coil is

$V=4l(y+b)^{2}$

The electromagnetic force is $\gamma G$, where $\gamma$ is the strength of the current and

$G=gl$.

If $E$ is the electromotive force acting in the circuit whose resistance is $R+r$,

$E=y(R+r)$.

The electromagnetic force due to this electromotive force is

$E{\frac {G}{R+r}}$,

which we have to make a maximum by the variation of $y$ and $l$. Inverting the fraction, we find that

${\frac {\rho }{\pi g}}{\frac {1}{y^{2}}}+{\frac {r}{gl}}$

is to be made a minimum. Hence

$2{\frac {\rho }{\pi }}{\frac {dy}{y^{3}}}+{\frac {r\,dl}{l^{2}}}=0$.

If the volume of the coil remains constant

${\frac {dl}{l}}+2{\frac {dy}{y+b}}=0$.

Eliminating $dl$ and $dy$, we obtain

${\frac {\rho }{\pi }}{\frac {y+b}{y^{3}}}={\frac {r}{l}}$,

Hence the thickness of the wire of the galvanometer should be such that the external resistance is to the resistance of the galvanometer coil as the diameter of the covered wire to the diameter of the wire itself.