Derive the expression for motional EMF induced in a conductor moving in a uniform magnetic field.

To derive the expression for the motional electromotive force (EMF) induced in a conductor moving in a uniform magnetic field, we can apply the principles of electromagnetic induction.

Let's consider a conductor of length ℓ moving with a velocity v in a direction perpendicular to a uniform magnetic field B. We assume that the conductor is part of a complete circuit.

According to Faraday's law of electromagnetic induction, the induced EMF in a conductor is proportional to the rate of change of magnetic flux through the circuit. Mathematically, we can express this as:

EMF = -dΦ/dt

where EMF is the electromotive force, dΦ is the change in magnetic flux, and dt is the change in time.

The magnetic flux (Φ) through a surface is given by the dot product of the magnetic field (B) and the area vector (A) of the surface. In our case, the conductor moves in a direction perpendicular to the magnetic field, so the area vector is parallel to the magnetic field vector. Therefore, the magnetic flux through the conductor is given by:

Φ = B ⋅ A = B ⋅ (ℓ ⋅ w)

where w is the width of the conductor, and ℓ⋅w is the area of the surface.

Now, we need to consider the rate of change of magnetic flux with respect to time. Since the magnetic field is uniform and the conductor is moving with a constant velocity, the change in magnetic flux is only due to the change in the area of the surface.

The width of the conductor (w) does not change, but the length of the conductor (ℓ) perpendicular to the magnetic field decreases as it moves. The rate of change of the magnetic flux is given by:

dΦ/dt = B ⋅ d(ℓ⋅w)/dt = B ⋅ (dℓ/dt ⋅ w)

Now, we can substitute this expression into the original equation for the EMF:

EMF = -dΦ/dt = -B ⋅ (dℓ/dt ⋅ w)

The term dℓ/dt represents the rate at which the length of the conductor perpendicular to the magnetic field is changing, which is equal to the velocity (v) at which the conductor is moving. Therefore, we can rewrite the expression as:

EMF = -B ⋅ (v ⋅ w)

Finally, we can recognize that v⋅w is the cross product of the velocity vector v and the width vector w, which gives us the area vector A. Therefore, we can rewrite the expression as:

EMF = -B ⋅ A

This is the final expression for the motional EMF induced in a conductor moving in a uniform magnetic field. It states that the induced EMF is equal to the negative of the dot product of the magnetic field and the area vector of the conductor.