Question

Two wires carrying equal current $I$ in opposite directions are placed a distance $a$ apart. The wires experience a force $F$ due to the current. The wires are moved to a distance $\frac{a}{2}$.
If the current in wire one remains the same, which of the following values of $I_2$ would result in the same force $F$ between the wires?

• A. $\frac{1}{4}I$
• B. $\frac{1}{2}I$
• C. $I$
• D. $2I$
• E. $4I$

Answer 1

Answer: (B)

$\frac{1}{2}I$

The force acting on the wires initially $F=\frac{{\mu }_o{I}_1{I}_2}{2\pi a}$
where $I_1=I_2=I$

$⟹$ $F=\frac{{\mu }_o{I}^2}{2\pi a}$

New distance between the wires is $a^{\prime }$ with current in the wires $I_1^{\prime }$ & $I_2^{\prime }$.

Given : $a^{\prime }=\frac{a}{2}$ and $I_1^{\prime }=I$

Thus new force $F^{\prime }=\frac{{\mu }_{o}I{I}_2^{\prime }}{2\pi \left(a/2\right)}=\frac{{\mu }_{o}I{I}_2^{\prime }}{\pi a}$

But $F^{\prime }=F$

$⟹I_2^{\prime }=\frac{I}{2}$