Magnetic Field Produced by a Current-carrying Wire

When a wire carries a current, it produces a magnetic field around it. The magnitude of this magnetic field depends on the current and the distance from the wire. The direction of the magnetic field is given by the right-hand rule, where if you point your thumb in the direction of the current, your fingers curl in the direction of the magnetic field.

Magnetic Field Produced by Two Current-carrying Wires

When two wires carry currents in the same direction, the magnetic fields produced by each wire at a given point add up to create a stronger magnetic field. The magnitude of the magnetic field produced by each wire is given by the equation:

B = (μ0 * I) / (2π * r)

Where B is the magnetic field, μ0 is the permeability of free space (a constant), I is the current, and r is the distance from the wire.

Magnetic Field at the Mid-point

When two wires carrying currents i and 2i in the same direction are placed parallel to each other, the magnetic field at the mid-point is the sum of the magnetic fields produced by each wire. Let's call the magnetic field produced by the wire carrying current i as B1 and the magnetic field produced by the wire carrying current 2i as B2.

Initially, when both currents are present, the magnetic field at the mid-point is given by:

B = B1 + B2 = (μ0 * i) / (2π * r) + (μ0 * 2i) / (2π * r)

B = (μ0 * 3i) / (2π * r)

Switching Off the Current 2i

When the current 2i is switched off, we only have the current i flowing through one wire. Therefore, the magnetic field at the mid-point will only be due to the current i. The magnetic field can be calculated using the same equation:

B' = (μ0 * i) / (2π * r)

Comparing the Magnetic Fields

To determine the change in the magnetic field, we can compare the magnetic field when both currents are present (B) and when only the current i is present (B').

B / B' = [(μ0 * 3i) / (2π * r)] / [(μ0 * i) / (2π * r)]

B / B' = (3i) / i

B / B' = 3

Therefore, the magnetic field when the current 2i is switched off is three times smaller than the magnetic field when both currents are present. In other words, B' = B/3.

Hence, the correct answer is option c) B.