An infinite wire has a circular bend of radius \( a \), and carrying a current \( I \) as shown in the figure. The magnitude of the magnetic field at the origin \( O \) of the arc is given by:
Show Hint
- \( \frac{\mu_0 I}{4 \pi a} \left( \frac{3\pi}{2} + 2 \right) \)
- \( \frac{\mu_0 I}{2 \pi a} \left( \frac{\pi}{2} + 2 \right) \)
- \( \frac{\mu_0 I}{4 \pi a} \left( \frac{3\pi}{2} \right) \)
- \( \frac{\mu_0 I}{2 \pi a} \left( \frac{3\pi}{2} + 1 \right) \)
The Correct Option is C
Solution and Explanation
Step 1: Magnetic Field Due to the Arc
For the arc with radius \( a \) and angle \( \frac{3\pi}{2} \), the magnetic field at the origin is: \[ B_1 = \frac{\mu_0 I}{4\pi a} \]
Step 2: Magnetic Field Due to the Straight Segment
For the straight segment of the wire: \[ B_2 = \frac{\mu_0 I}{4\pi a} \left( \frac{3\pi}{2} \right) \]
Step 3: Magnetic Field Due to Other Segments
Since the magnetic field due to the straight segments at the origin is zero: \[ B_3 = 0 \]
Step 4: Calculate the Total Magnetic Field
Thus, the total magnetic field at the origin is: \[ B = \frac{\mu_0 I}{4\pi a} \left( \frac{3\pi}{2} \right) \]
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