Question

Derive an expression for the force between two parallel conductors carrying currents. Hence define ampere.

Answer 1

Let the distance between the two conductors carrying current $I_1$ and $I_2$ be $d$.

Magnetic field produced by conductor 1 at a distance $d$, $B_1=\frac{{\mu }_o}{4\pi }\frac{2I_1}{d}$

Force acting the conductor 2 placed in this magnetic field $F=B_1{I}_{2}L$

Thus we get $F=\frac{{\mu }_o}{4\pi }\frac{2I_1{I}_2}{d}L$

Force er unit length of conductor 1 $f=\frac{{\mu }_o}{4\pi }\frac{2I_1{I}_2}{d}$

Now put $d=1m$, $I_1=I_2=1A$ and $L=1m$

We get $F=\frac{{\mu }_2}{2\pi }=2\times {10}^{-7}$ $N$

Thus we define ampere as the current flowing in each conductor separated by a unit distance so that one conductor applies a force of $2\times {10}^{-7}$ N on a unit length of another parallel conductor.