Introduction:
When a conducting circular loop is exposed to a uniform magnetic field perpendicular to its plane, an electromotive force (emf) is induced in the loop. This phenomenon is known as electromagnetic induction and is the basis for many practical applications such as electric generators and transformers.

Explanation:
To understand the emf induced in the conducting circular loop, let's break down the process step by step:

1. Magnetic Field: The loop is placed in a uniform magnetic field B, which is perpendicular to the plane of the loop. This means that the magnetic field lines are parallel to the loop.

2. Flux: As the magnetic field passes through the loop, it creates a magnetic flux. The magnetic flux is the product of the magnetic field strength and the area of the loop perpendicular to the field. In this case, the area of the loop is given by A = πR^2, where R is the radius of the loop.

3. Change in Flux: If the magnetic field or the area of the loop changes, the magnetic flux passing through the loop will also change. In this scenario, the radius of the loop changes with time, given by R = R₀ + vt, where R₀ is the initial radius, v is the rate of change of radius, and t is the time.

4. Faraday's Law: According to Faraday's law of electromagnetic induction, the emf induced in a loop is proportional to the rate of change of magnetic flux passing through the loop. Mathematically, this can be expressed as emf = -dΦ/dt, where dΦ/dt represents the rate of change of magnetic flux.

5. Calculating the emf: To find the emf induced in the loop, we need to calculate the rate of change of magnetic flux. The magnetic flux passing through the loop is given by Φ = B * A. Substituting the expression for A and differentiating with respect to time, we get dΦ/dt = B * dA/dt = 2πBRv.

6. Final Expression: Finally, substituting the value of dΦ/dt in Faraday's law, we obtain the expression for the emf induced in the loop as emf = -dΦ/dt = -2πBRv.

Conclusion:
In conclusion, when a conducting circular loop of radius R is exposed to a uniform magnetic field B perpendicular to its plane, the emf induced in the loop is given by emf = -2πBRv, where R is the radius of the loop, v is the rate of change of radius, and B is the magnetic field strength. This phenomenon of electromagnetic induction plays a crucial role in various electrical devices and technologies.