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Q.
The force acting per unit length on one conductor due to the other for two long parallel current-carrying wires is
NTA AbhyasNTA Abhyas 2020Moving Charges and Magnetism
Solution:
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 $
$F=\frac{\mu_0 I_1 I_2}{2 \pi r} \text { or } \frac{\mu_0}{4 \pi} \cdot \frac{2 I_1 I_2}{r}$
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.
