Question

Two long straight parallel conductors carry steady current $I_1$ and $I_2$ separated by a distance d. If the currents are flowing in the same direction, show how the magnetic field set up in one produces an attractive force on the other. Obtain the expression for this force. Hence define one ampere.

Answer 1

Magnetic field due to wire a on b =$\frac{{\mu }_0{I}_a}{2\pi d}$

Force on a current carrying wire placed in magnetic field=$I(\overrightarrow{l}\times \overrightarrow{B})$

Hence force on b is $F_{ba}=I_{b}l\frac{{\mu }_0{I}_a}{2\pi d}$

The direction of force experienced by the wire 'a' is toward the wire 'b' as shown.
Similarly the direction of force experienced by the wire 'b' is toward the wire Thus, the force is attractive.
If two long wires are placed in vacuum at a separation of 1 m, one Ampere would be defined as the current in each wire that would produce a force of per unit length of wire.