NCERT · Class 12 · Physics · Moving Charges and MagnetismDerive the expression for the force per unit length between two long parallel current-carrying conductors and hence define one Ampere.

Consider two long parallel conductors $A$ and $B$ separated by a distance $d$ carrying currents $I_1$ and $I_2$ in the same direction. The magnetic field produced by conductor $A$ at the position of conductor $B$ is given by $B_1 = \frac{\mu_0 I_1}{2\pi d}$. According to Lorentz force, the magnetic force acting on a length $L$ of conductor $B$ is $F = I_2 L B_1 \sin(90^\circ)$. Substituting $B_1$, we get $F = \frac{\mu_0 I_1 I_2 L}{2\pi d}$. Therefore, the force per unit length is $\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$. If currents flow in opposite directions, the force is repulsive. Definition of one Ampere: One ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to $2 \times 10^{-7}$ Newtons per meter of length.