Induced electromotive force (e.m.f) is the voltage generated in a conductor when the magnetic field through the conductor changes. According to Faraday's law of electromagnetic induction, the magnitude of the induced e.m.f is directly proportional to the rate of change of magnetic flux through the conductor.

When either the inductance or the rate of change of current is doubled, the induced e.m.f doubles as well. This can be explained by examining the factors that affect the magnitude of the induced e.m.f.

1. Faraday's Law: Faraday's law states that the induced e.m.f is directly proportional to the rate of change of magnetic flux through the conductor. Mathematically, it can be expressed as:

ε = -N dΦ/dt

Where ε is the induced e.m.f, N is the number of turns in the coil, and dΦ/dt is the rate of change of magnetic flux.

2. Magnetic Flux: The magnetic flux through a coil is determined by the magnetic field strength and the area of the coil. It can be calculated using the formula:

Φ = B A

Where Φ is the magnetic flux, B is the magnetic field strength, and A is the area of the coil.

Explanation:
- When the inductance is doubled, it means that the coil has more turns. This increases the number of turns (N) in the formula for the induced e.m.f. As a result, the induced e.m.f doubles.
- When the rate of change of current is doubled, it means that the current through the coil is changing at a faster rate. This increases the rate of change of magnetic flux (dΦ/dt) in the formula for the induced e.m.f. Consequently, the induced e.m.f doubles.

Hence, when either the inductance or the rate of change of current is doubled, the induced e.m.f doubles as well. Therefore, the correct answer is option 'C'.