A long straight wire is oriented in the z direction and carries a current I. A square loop with sides of length w is in the xz-plane with its nearest edge a distance d from the wire. In time of t, the square loop moves a distance of l closer to the wire. What is the emf induced in the loop while its moving?

Question

A long straight wire is oriented in the z direction and carries a current I. A square loop with sides of length w is in the xz-plane with its nearest edge a distance d from the wire. In time of t, the square loop moves a distance of l closer to the wire. What is the emf induced in the loop while its moving?

Answer 1

Write B as a function of distance from the wire, and integrate B over the loop to get BA as a function of the changing B per time.

Answer 2

To find the electromagnetic force (emf) induced in the loop while it is moving, you can use Faraday's law of electromagnetic induction. This law states that the magnitude of the emf induced in a circuit is equal to the rate of change of magnetic flux passing through the circuit.

The magnetic flux through the loop can be calculated by taking the dot product of the magnetic field and the area vector of the loop:

Φ = B · A

In this case, the magnetic field experienced by the loop due to the current in the wire can be calculated using Ampere's law. Ampere's law states that the magnetic field around a long straight wire is inversely proportional to the distance from the wire:

B = (μ₀ * I) / (2π * d)

where μ₀ is the permeability of free space, I is the current in the wire, and d is the distance from the wire to the loop.

The area vector of the loop is perpendicular to the loop and has a magnitude equal to the area of the loop. Since the loop is a square in the xz-plane, the area vector is along the y-axis with magnitude equal to the area of the square:

A = w² * ŷ

where w is the side length of the square loop and ŷ is a unit vector along the y-axis.

Substituting these values into the formula for magnetic flux, we get:

Φ = (B · A) = (B * A * cosθ)

Since the magnetic field and the area vector are perpendicular to each other (θ = 90 degrees), the cosine of the angle between them is zero. Therefore, the magnetic flux simplifies to:

Φ = B * A * cosθ = 0

This means that the magnetic flux passing through the loop is zero, and according to Faraday's law, the induced emf in the loop is also zero. Hence, there is no emf induced in the loop while it is moving.