Question

In the given figure, the magnetic field at ‘O’.

• A. \frac{3}{4} \frac{\mu_o I}{r} + \frac{\mu_o I}{4 \pi r}
• B. \frac{3}{10} \frac{\mu_o I}{r} - \frac{\mu_o I}{4 \pi r}
• C. \frac{3}{8} \frac{\mu_o I}{r} + \frac{\mu_o I}{4 \pi r}
• D. \frac{3}{8} \frac{\mu_o I}{r} - \frac{\mu_o I}{4 \pi r}

Answer 1

The Correct Option is C

In the given problem, the magnetic field at point O is due to the two different segments of the current-carrying wire. We apply the Biot-Savart law to calculate the magnetic field contributions.
1. For the straight segment of the wire (along the horizontal direction): The magnetic field at point O due to this segment is given by the formula: \[ B_{\text{straight}} = \frac{\mu_0 I}{4 \pi r} \]
2. For the curved segment of the wire (in the circular loop): The magnetic field at point O due to this loop is given by the formula: \[ B_{\text{loop}} = \frac{3 \mu_0 I}{8 r} \] Thus, the total magnetic field at point O is the sum of the two contributions: \[ B_{\text{total}} = B_{\text{straight}} + B_{\text{loop}} = \frac{3 \mu_0 I}{8 r} + \frac{3 \mu_0 I}{4 \pi} \] Therefore, the correct answer is (C).

Answer 2

To calculate the magnetic field at point \( O \), we use the Biot-Savart law, which gives the magnetic field due to a current-carrying wire. 1. The magnetic field at the center of a current-carrying loop is given by the formula: \[ B_{\text{loop}} = \frac{\mu_0 I}{2r} \] Where: - \( I \) is the current, - \( r \) is the radius of the loop, - \( \mu_0 \) is the permeability of free space. 2. The magnetic field at a point along the axis of the loop is given by the formula: \[ B_{\text{axis}} = \frac{\mu_0 I}{4\pi r^2} \left( \frac{3}{2} \right) \] This field is along the axis of the loop, which is perpendicular to the plane of the circular current loop. Now, combining these two components, we calculate the resultant magnetic field at point \( O \): \[ B_{\text{total}} = B_{\text{loop}} + B_{\text{axis}} = \frac{3 \mu_0 I}{8r} + \frac{\mu_0 I}{4\pi} \] Thus, the magnetic field at point \( O \) is \( \frac{3 \mu_0 I}{8r} + \frac{\mu_0 I}{4\pi} \), which corresponds to option (C).