Question

Two long straight conductors carrying equal current kept parallel at a separation of $d$ metre experience a force per unit length. If the distance of separation is made $2d$, then to maintain the same force per unit length on each conductor, the current in each conductor should be increased to

• A. $2I$
• B. $\sqrt{2}I$
• C. $\dfrac{I}{\sqrt{2}}$
• D. $1.5I$
• E. $3I$

Hint:

Physics Tip: For parallel wires, force $\propto \dfrac{I^2}{r}$. If distance doubles, current must multiply by $\sqrt{2}$ to keep force same.

Answer 1

The Correct Option is B

Concept:
Force per unit length between two long parallel wires: $$\frac{F}{L}=\frac{\mu_0 I_1 I_2}{2\pi r}$$ For equal currents: $$\frac{F}{L}=\frac{\mu_0 I^2}{2\pi r}$$
Step 1: Initial condition.
At separation $d$: $$\frac{F}{L}=\frac{\mu_0 I^2}{2\pi d}$$
Step 2: New condition.
Distance becomes $2d$, new current = $I'$. $$\frac{F}{L}=\frac{\mu_0 {I'}^2}{2\pi (2d)}$$ Force remains same.
Step 3: Equate both expressions.
$$\frac{\mu_0 I^2}{2\pi d}=\frac{\mu_0 {I'}^2}{4\pi d}$$ Cancel common terms: $$2I^2={I'}^2$$ $$I'=\sqrt{2}I$$
Hence required current is Option (B). :contentReference[oaicite:1]{index=1}