Introduction:
When a semicircular loop of radius R is rotated with an angular velocity w perpendicular to the plane of a magnetic field B, an electromotive force (emf) is induced in the loop. This phenomenon is known as electromagnetic induction and is governed by Faraday's law of electromagnetic induction.

Faraday's Law of Electromagnetic Induction:
Faraday's law states that the magnitude of the induced emf in a circuit is directly proportional to the rate of change of magnetic flux through the circuit. Mathematically, it can be expressed as:
emf = -d(Φ)/dt
where emf is the induced electromotive force, d(Φ) is the change in magnetic flux, and dt is the change in time.

Explanation:
When a magnetic field B is applied perpendicular to the plane of the semicircular loop, the magnetic flux through the loop changes as it rotates. The magnetic flux through the loop is given by:
Φ = B * A
where B is the magnetic field strength and A is the area of the loop.

As the loop rotates, the area A changes due to the changing orientation of the loop with respect to the magnetic field. The rate of change of the area can be expressed as:
dA/dt = R * dw/dt
where R is the radius of the loop and dw/dt is the rate of change of angular velocity.

Applying Faraday's Law:
Substituting the values of dA/dt and Φ in Faraday's law, we get:
emf = -d(Φ)/dt = -dB * A/dt - B * dA/dt

Since the magnetic field B is constant, the first term becomes zero. Thus, the induced emf can be simplified as:
emf = - B * dA/dt = - B * R * dw/dt

Conclusion:
The induced electromotive force (emf) in the semicircular loop of radius R, rotating with an angular velocity w perpendicular to the plane of a magnetic field B, is given by emf = - B * R * dw/dt. The negative sign indicates that the direction of the induced current opposes the change in magnetic flux.