The Induced EMF in a Rotating Copper Rod

When a copper rod of length l is rotated about its hinged end, perpendicular to a uniform magnetic field B, an induced emf (electromotive force) is generated between the middle of the rod and the hinged end. This phenomenon is known as electromagnetic induction and is based on Faraday's Law of electromagnetic induction.

Faraday's Law of Electromagnetic Induction
Faraday's Law states that when a conductor (in this case, the copper rod) is moved in a magnetic field, or when the magnetic field through a conductor changes, an emf is induced in the conductor. The magnitude of the induced emf is given by the equation:

EMF = -N(dΦ/dt)

Where:
- EMF is the induced electromotive force
- N is the number of turns in the coil (in this case, the copper rod is considered as a coil with one turn)
- dΦ/dt is the rate of change of magnetic flux through the coil

Magnetic Flux
Magnetic flux (Φ) is the measure of the magnetic field passing through a given area. It is given by the equation:

Φ = B * A * cos(θ)

Where:
- B is the magnetic field strength
- A is the area perpendicular to the magnetic field
- θ is the angle between the magnetic field and the normal to the area

EMF in the Rotating Copper Rod
In the case of a rotating copper rod, the magnetic field B is perpendicular to the rotational axis. As the rod rotates, the area A perpendicular to the magnetic field changes, resulting in a change in the magnetic flux Φ. This change in flux induces an emf in the rod.

The induced emf can be calculated by taking the derivative of the magnetic flux with respect to time:

dΦ/dt = d(B * A * cos(θ))/dt
= B * d(A * cos(θ))/dt
= B * (dA/dt * cos(θ) - A * sin(θ) * dθ/dt)

Since the rod is rotating, we can assume that the rate of change of area (dA/dt) is negligible compared to the rate of change of the angle (dθ/dt). Therefore, the equation simplifies to:

dΦ/dt ≈ -B * A * sin(θ) * dθ/dt

The negative sign indicates that the emf induces a current in a direction that opposes the change in flux. In this case, it means the induced current tries to resist the rotation of the rod.

Conclusion
In conclusion, when a copper rod is rotated about its hinged end perpendicular to a uniform magnetic field, an induced emf is generated between the middle of the rod and the hinged end. The magnitude of the induced emf can be calculated using Faraday's Law of electromagnetic induction and depends on the rate of change of the magnetic flux through the rod. The induced current flows in a direction that opposes the change in flux, resisting the rotation of the rod.

E=blw as the magnetic field and rod come parallel