Question

What is the magnetic field at O due to current in the infinite wire forming a loop as shown in the following figure?

• A. $\frac{\mu_0 I}{4 \pi d} (sec\phi_1 + sec\phi_2)$
• B. $\frac{\mu_0}{4 \pi} \times \frac{2I}{d}$

Answer 1

Answer: (A)

(a)

Answer 2

The problem asks for the magnetic field at point O due to the current in the given wire configuration. The configuration consists of an infinite wire that forms a rectangular loop.

Given the options, it is highly probable that the question intends for the angles $\phi_1$ and $\phi_2$ in option (a) to be the angles $\theta_1$ and $\theta_2$ as defined in the diagram (angles at O from the vertical to the diagonal lines). And it also implies that only the vertical segments contribute to the field. This is a common simplification in some problems, or a mistake.

Magnetic field due to left vertical wire at O: $B_1 = \frac{\mu_0 I}{4 \pi d} \sec\theta_1$.

Magnetic field due to right vertical wire at O: $B_2 = \frac{\mu_0 I}{4 \pi d} \sec\theta_2$.

Both fields are directed out of the page.

Total field (under this assumption) $B = B_1 + B_2 = \frac{\mu_0 I}{4 \pi d} (\sec\theta_1 + \sec\theta_2)$.

Replacing $\theta_1, \theta_2$ with $\phi_1, \phi_2$ as per the option format, we get option (a).