MHTCET 11 May 2024 Shift 2 Paper

Concept:The net magnetic field at the centre of the loop becomes zero when the field due to the loop and the field due to the straight wire cancel each other in magnitude.Explanation:The magnetic field at the centre of a circular loop of radius $R$ carrying current $I_c$ is:$$B_{\text{loop}} = \frac{\mu_0 I_c}{2R}$$where $\mu_0$ is the permeability of free space.The magnetic field produced by a long straight wire at a perpendicular distance $d$ is:$$B_{\text{wire}} = \frac{\mu_0 I_w}{2\pi d}$$For the resultant field at the centre to be zero, these two field magnitudes must be equal:$$B_{\text{loop}} = B_{\text{wire}}$$$$\frac{\mu_0 I_c}{2R} = \frac{\mu_0 I_w}{2\pi d}$$Cancelling $\mu_0$ from both sides and solving for $d$:$$\frac{I_c}{2R} = \frac{I_w}{2\pi d}$$$$d = \frac{R I_w}{\pi I_c}$$Answer:Hence, the required separation is $d = \frac{R I_w}{\pi I_c}$, which corresponds to Option A.