Question

A loop ABCD, carrying current $ I = 12 \, \text{A} $, is placed in a plane, consists of two semi-circular segments of radius $ R_1 = 6\pi \, \text{m} $ and $ R_2 = 4\pi \, \text{m} $. The magnitude of the resultant magnetic field at center O is $ k \times 10^{-7} \, \text{T} $. The value of $ k $ is ______ (Given $ \mu_0 = 4\pi \times 10^{-7} \, \text{T m A}^{-1} $)

Hint:

Use the formula for the magnetic field due to a circular arc at its center. Remember to consider the direction of the magnetic field due to each segment and subtract them.

Answer 1

Correct Answer: 1

$B_0 = |B_{R_1} - B_{R_2}|$

$B_0 = \left| \frac{\mu_0 I}{4R_2} - \frac{\mu_0 I}{4R_1} \right|$

$B_0 = \frac{4\pi \times 10^{-7} \times 12}{4} \left| \frac{1}{4\pi} - \frac{1}{6\pi} \right|$

$B_0 = 12\pi \times 10^{-7} \left| \frac{1}{12\pi} \right|$

$B_0 = 1 \times 10^{-7} T$

$k = 1$