Question

Consider two arrangements of wires. Find the ratio of magnetic field at the centre of the semi–circular part.

• (A) $\dfrac{\pi+3}{\pi-1}$
• (B) $\dfrac{\pi+4}{\pi+2}$
• (C) $\dfrac{\pi+2}{\pi+1}$
• (D) $\dfrac{\pi-2}{\pi+1}$

Hint:

Use superposition of magnetic fields from the straight semi-infinite wires and the semicircular arc.

Answer 1

The Correct Option is A

Magnetic field at the centre is calculated using the Biot–Savart law. For a semicircular wire of radius $R$, $B_{semi} = \frac{\mu_0 I}{4R}$. For the first arrangement, the fields from the straight sections and the curve add up. In the second, some components subtract.

Evaluating the vector sum for both geometries and taking the ratio leads to the expression $\frac{\pi+3}{\pi-1}$.