Question

A wire carrying current I has the shape as shown in adjoining figure. Linear parts of the wire are very long and parallel to . X-axis while semicircular portion of radius R is lying in Y-Z plane. Magnetic field at point O is

• A. $\overrightarrow{B}=-\frac{\mu_0}{4 \pi}\frac{I}{R}\Big(\pi \hat{i}+2 \hat{k}\Big)$
• B. $\overrightarrow{B}=\frac{\mu_0}{4 \pi}\frac{I}{R}\Big(\pi \hat{i}-2 \hat{k}\Big)$
• C. $\overrightarrow{B}=\frac{\mu_0}{4 \pi}\frac{I}{R}\Big(\pi \hat{i}+2 \hat{k}\Big)$
• D. $\overrightarrow{B}=-\frac{\mu_0}{4 \pi}\frac{I}{R}\Big(\pi \hat{i}-2 \hat{k}\Big)$

Answer 1

The Correct Option is A

To determine the magnetic field at point O, we will analyze the contribution to the magnetic field from each section of the wire using the Biot-Savart Law. The wire consists of very long linear parts parallel to the X-axis and a semicircular arc in the Y-Z plane.

1. The Biot-Savart Law states that the magnetic field d\overrightarrow{B} due to an element of wire d\overrightarrow{l} carrying a current I is given by: d\overrightarrow{B} = \frac{\mu_0}{4 \pi} \cdot \frac{I \, d\overrightarrow{l} \times \hat{r}}{r^2}, where \hat{r} is the unit vector from the current element to the point where the field is being calculated and r is the distance between the element and the point.
2. For the infinite straight sections parallel to the X-axis, their contributions to the magnetic field at point O will cancel each other out due to symmetry. This is because the magnetic field due to a long straight wire is concentric circles around the wire, which results in equal and opposite contributions to point O from the two linear sections.
3. The magnetic field due to the semicircular arc is next computed. For a semicircular arc of radius R: B_{\text{arc}} = \frac{\mu_0}{4 \pi} \cdot \frac{\pi I}{R^2} \cdot R = \frac{\mu_0 I}{4R}
4. The direction of the magnetic field due to the semicircular arc can be found using the right-hand rule. Since the current flows in the semicircle within the Y-Z plane, the magnetic field at point O will be directed along the negative X-axis.

Thus, the total magnetic field at point O is due to the semicircular portion only and given by:

\overrightarrow{B} = -\frac{\mu_0}{4 \pi}\frac{I}{R} \Big(\pi \hat{i} + 2 \hat{k}\Big).

The correct answer is: \overrightarrow{B} = -\frac{\mu_0}{4 \pi}\frac{I}{R} \Big(\pi \hat{i} + 2 \hat{k}\Big), matching option (A).