Question

A straight conductor carrying current $i$ splits into two parts as shown in the figure. The radius of the circular loop is $R$. The total magnetic field at the centre $P$ at the loop is :

• A. Zero
• B. 3$\mu_o$i/32 R, outward
• C. 3$\mu_o$i/32R, inward
• D. $\frac{\mu_{0}i}{2R}$,inward

Answer 1

The Correct Option is A

 To determine the total magnetic field at the center \( P \) of the loop, we must consider the contributions from each part of the conductor carrying current \( i \). Let's analyze the situation step-by-step:

1. Given the geometry of a circular loop, the magnetic field due to a complete loop of radius \( R \) carrying a current \( i \) is given by the Biot-Savart Law: \(B_{\text{loop}} = \frac{\mu_0 i}{2R}\). However, in this setup, the wire divides into two paths resulting in a semicircular loop.
2. The magnetic field at the center of a semicircular loop is half of that of a complete loop since only half of the loop contributes: \(B_{\text{semi-loop}} = \frac{\mu_0 i}{4R}\).
3. A straight wire section extending radially outward contributes no net magnetic field at the center \( P \) because a straight current segment contributes magnetic fields in directions that result in cancellation at the loop's center.
4. The segments of wire that are not part of the loop include paths that go radially outward from the center and do not contribute to the center's field.
5. Thus, the total magnetic field at the center \( P \) is the sum of the contributions from the semicircular loop and the straight segments:
   • Contribution from the semicircular loop: \(B_{\text{semi-loop}} = \frac{\mu_0 i}{4R}\). However, considering both parts cancel each other when aggregated symmetrically about the center, total net effect results in zero.
   • Contributions from the straight wires cancel out as they produce opposing fields at center \( P \).
6. Therefore, the net magnetic field at the center \( P \) is zero: \(B_{\text{net}} = 0\).

After analyzing the effect of each segment of the conductor, we conclude that the total magnetic field at the center \( P \) of this setup is zero. Hence, the correct answer is Zero.