Question

Two parallel wires of equal lengths are separated by a distance of 3m from each other. The currents flowing through the first and second wire are 3A and 4.5A respectively in opposite directions. The resultant magnetic field at the mid-point of both the wires is \( \mu_0 \) (permeability of free space) is

• A. \( \frac{3 \mu_0}{2 \pi} \)
• B. \( \frac{7 \mu_0}{2 \pi} \)
• C. \( \frac{\mu_0}{2 \pi} \)
• D. \( \frac{5 \mu_0}{2 \pi} \)

Hint:

For magnetic fields produced by two currents, always account for the direction of current flow. Opposite currents produce magnetic fields that oppose each other.

Answer 1

The Correct Option is D

Step 1: Using Ampère’s law.
The magnetic field at a point due to a current-carrying conductor is given by Ampère's law: \[ B = \frac{\mu_0 I}{2 \pi r} \] where \( I \) is the current and \( r \) is the distance from the wire. For two wires with currents in opposite directions, the total magnetic field at the mid-point will be the vector sum of the individual magnetic fields produced by each wire.
Step 2: Calculate the magnetic fields.
The magnetic fields produced by both wires at the mid-point are calculated, considering that the magnetic fields due to each wire oppose each other (because the currents flow in opposite directions). The net magnetic field at the midpoint is given by: \[ B_{net} = B_1 - B_2 = \frac{3 \mu_0}{2 \pi} - \frac{4.5 \mu_0}{2 \pi} = \frac{5 \mu_0}{2 \pi} \] where \( \mu_0 \) is the permeability of free space.
Step 3: Conclusion.
Thus, the correct answer is \( \frac{5 \mu_0}{2 \pi} \).