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

Step 1: Identify Magnetic Field Components. The magnetic field at the center of the arc is the sum of fields from the straight wire segments and the semicircular arc.

Step 2: Apply the arc formula. For a semicircle of radius $R$ carrying current $I$, the magnetic field at its center is $B_{arc} = \frac{\mu_{0}I}{4R}$.

Step 3: Analyze Arrangement 1. In the first case, the straight wires contribute in the same direction as the arc. For a specific geometry where straight wires are configured as such, the contributions add up to $B_{1} = \frac{\mu_{0}I}{4R} (\text{constant factor based on geometry})$.

Step 4: Analyze Arrangement 2. In the second case, the direction of current in one or more straight sections is reversed relative to the arc, resulting in a subtraction of field components: $B_{2} = \frac{\mu_{0}I}{4R} (\text{differing factor})$.

Step 5: Form the Ratio. By simplifying the geometric contributions for both cases, the ratio $\frac{B_{1}}{B_{2}}$ evaluates to $\frac{\pi+3}{\pi-1}$.