Question

As shown in the figure, a long straight conductor with semi-circular are of radius \(\frac{\pi}{10} m\) is carrying current \(I =3 A\). The magnitude of the magnetic field at the center \(O\) of the arc is :(The permeability of the vacuum \(=4 \pi \times 10^{-7} NA ^{-2}\) )

• A. \(6 \mu T\)
• B. \(4 \mu T\)
• C. \(3 \mu T\)
• D. \(1 \mu T\)

Hint:

Remember the formula for the magnetic field due to a circular arc. Also, consider the direction and contribution of the magnetic field due to straight current carrying wires. Visualizing the magnetic field lines can be helpful.

Answer 1

The Correct Option is C

Step 1: Determine the Magnetic Field Due to the Semicircular Arc

The magnetic field at the center of a circular arc of radius R carrying current I is given by:

\( B_c = \frac{\mu_0 I}{4 \pi R} \theta \)

where \( \mu_0 \) is the permeability of free space, and \( \theta \) is the angle subtended by the arc at the center in radians. For a semicircular arc, \( \theta = \pi \).

\( B_c = \frac{\mu_0 I}{4 R} \)

Step 2: Substitute the Given Values

Given \( I = 3 \, \text{A}, \, R = \frac{\pi}{10} \, \text{m}, \, \text{and} \, \mu_0 = 4 \pi \times 10^{-7} \, \text{N A}^{-2} \), we have:

\( B_c = \frac{(4 \pi \times 10^{-7})(3)}{4 \left(\frac{\pi}{10}\right)} \)

\( B_c = \frac{4 \pi \times 10^{-7} \times 3 \times 10}{4 \pi} \)

\( B_c = 3 \times 10^{-6} \, \text{T} = 3 \, \mu \text{T} \)

Step 3: Consider the Magnetic Field Due to the Straight Wires

The straight portions of the wire do not contribute to the magnetic field at point O. This is because the magnetic field lines due to an infinitely long straight wire form concentric circles around the wire. At the center of the semicircle, the straight sections are radial, so the magnetic field produced by them at point O will be perpendicular to the plane of the semicircle and thus won’t affect the field in the plane caused by the semicircular section.

Conclusion: The magnitude of the magnetic field at the center O is \( 3 \, \mu \text{T} \) (Option 3).