Question

The force acting per unit length on one conductor due to the other for two long parallel current-carrying wires is

• A. $\frac{{\mu }_0}{4\pi }\frac{I_1{I}_2}{r}$
• B. $\frac{{\mu }_0}{4\pi }\frac{I_1{I}_2}{2r}$
• C. $\frac{{\mu }_0}{4\pi }\frac{2I_1{I}_2}{r}$
• D. $\frac{{\mu }_0}{4\pi }\frac{I_1{I}_2}{4r}$

Answer 1

Answer: (C)

Force between two parallel current carrying wires: $\mathrm{AB}$ and $\mathrm{CD}$ are two infinitely long conductors, placed parallel to each other and separated by distance $r$. They carry currents ${\mathrm{I}}_1$ and ${\mathrm{I}}_2$ in the same direction.

Magnetic field produced by current ${\mathrm{I}}_1$ at any point of $\mathrm{CD}$ is
${\mathrm{B}}_1=\frac{{\mu }_0{\mathrm{I}}_1}{2\pi \mathrm{r}}$
This field acts perpendicular to $\mathrm{CD}$ and into the plane of paper. It exerts a force on wire $\mathrm{CD}$ carrying current ${\mathrm{I}}_2$. Force exerted on unit length of $\mathrm{CD}$ is
$F=B_1{I}_{2}l=\frac{{\mu }_0{I}_1}{2\pi r}\times I_2\times 1$
$F=\frac{{\mu }_0{I}_1{I}_2}{2\pi r}\text{ or }\frac{{\mu }_0}{4\pi }\cdot \frac{2I_1{I}_2}{r}$
By Fleming's left hand rule, this force acts on CD towards AB. Similarly, conductor $\mathrm{CD}$ also exerts an equal force $\mathrm{AB}$ towards itself. Hence the two wires get attracted towards each other, if currents are in same direction.