find the charge flown in loop if loop is rotated about axis AB by 90degree 180degree and 360degreeks

To determine the charge that flows in a loop when it is rotated about an axis, we need to consider the principles of electromagnetic induction, particularly Faraday's Law of Induction. This law states that a change in magnetic flux through a loop induces an electromotive force (EMF) in that loop, which can cause a current to flow if the circuit is closed. Let's break this down step by step for the different angles of rotation: 90 degrees, 180 degrees, and 360 degrees.

Understanding Magnetic Flux

Magnetic flux (Φ) is defined as the product of the magnetic field (B) and the area (A) of the loop perpendicular to the field, given by the formula:

Φ = B × A × cos(θ)

Here, θ is the angle between the magnetic field and the normal to the surface of the loop. When the loop is rotated, this angle changes, which affects the magnetic flux through the loop.

Case 1: Rotation by 90 Degrees

When the loop is rotated by 90 degrees, the angle θ changes from 0 degrees (where the magnetic field is perpendicular to the loop) to 90 degrees (where the magnetic field is parallel to the loop). In this case, the magnetic flux changes from:

• Initial flux: Φ_initial = B × A × cos(0°) = B × A
• Final flux: Φ_final = B × A × cos(90°) = 0

The change in magnetic flux (ΔΦ) is:

ΔΦ = Φ_final - Φ_initial = 0 - (B × A) = -B × A

According to Faraday's Law, the induced EMF (ε) is equal to the negative rate of change of magnetic flux:

ε = -d(ΔΦ)/dt

If we assume the rotation happens in a time interval Δt, the induced EMF will lead to a current (I) if the loop is part of a closed circuit:

I = ε/R

where R is the resistance of the circuit. The charge (Q) that flows can be calculated as:

Q = I × Δt

Case 2: Rotation by 180 Degrees

For a 180-degree rotation, the loop rotates from being perpendicular to the magnetic field to being opposite to it. The flux changes as follows:

• Initial flux: Φ_initial = B × A
• Final flux: Φ_final = B × A × cos(180°) = -B × A

The change in magnetic flux is:

ΔΦ = -B × A - B × A = -2B × A

Using Faraday's Law, the induced EMF will be greater than in the 90-degree case, leading to a larger current and thus more charge flowing through the circuit:

Q = (ε/R) × Δt

Case 3: Rotation by 360 Degrees

When the loop is rotated by 360 degrees, it returns to its original position. The magnetic flux does not change from its initial value:

• Initial flux: Φ_initial = B × A
• Final flux: Φ_final = B × A

The change in magnetic flux is:

ΔΦ = B × A - B × A = 0

Since there is no change in flux, there will be no induced EMF, and consequently, no current will flow:

Q = 0

Summary of Charge Flow

To summarize:

• For a 90-degree rotation, charge flows based on the induced EMF from a change in flux.
• For a 180-degree rotation, more charge flows due to a larger change in flux.
• For a 360-degree rotation, no charge flows as there is no change in flux.

This analysis illustrates how the rotation of a loop in a magnetic field can induce current and flow of charge, depending on the angle of rotation and the resulting changes in magnetic flux. Understanding these principles is crucial in fields like electrical engineering and physics.