EMF induced in a rotating conducting rod

When a J-shaped conducting rod is rotating in its own plane with a constant angular velocity (ω) about one of its ends (point P) in a uniform magnetic field (B) directed normally into the plane of the paper, an electromotive force (EMF) is induced across the rod.

Explanation:

To understand the magnitude of the induced EMF, we can consider the following points:

1. Magnetic field:
The uniform magnetic field (B) directed normally into the plane of the paper creates a magnetic flux through the area enclosed by the rotating rod.

2. Change in magnetic flux:
As the conducting rod rotates, the area enclosed by the rod changes, resulting in a change in the magnetic flux through the rod.

3. Faraday's Law of Electromagnetic Induction:
According to Faraday's law, the magnitude of the induced EMF is directly proportional to the rate of change of magnetic flux through the circuit.

4. Calculation of EMF:
The rate of change of magnetic flux can be calculated as the product of the magnetic field strength (B) and the rate of change of the enclosed area (dA/dt).

5. Calculation of the enclosed area:
The enclosed area can be approximated as a rectangular loop formed by the J-shaped rod. The length of the rod (L) is perpendicular to the magnetic field, and the width (w) is the distance between the two ends of the rod.

6. Rate of change of the enclosed area:
Since the rod is rotating with a constant angular velocity (ω), the rate of change of the enclosed area is given by dA/dt = w * (dL/dt), where dL/dt is the linear velocity of the rotating rod.

7. Final expression for EMF:
Combining the above equations, the magnitude of the induced EMF (ε) is given by ε = B * w * (dL/dt).

8. Simplification:
Since the linear velocity (v) of a point on the rotating rod is given by v = ω * r, where r is the distance from the rotation axis (point P), we can express dL/dt as ω * r. Therefore, the expression for EMF simplifies to ε = B * w * ω * r.

9. Conclusion:
The magnitude of the induced EMF across the rotating J-shaped conducting rod is given by ε = B * w * ω * r, where B is the magnetic field strength, w is the width of the rod, ω is the angular velocity, and r is the distance from the rotation axis.