To find the electromotive force (emf) induced in the metallic rod when it touches the near end of the circular ring, we can apply Faraday's law of electromagnetic induction. This law states that the induced emf in a closed loop is equal to the negative rate of change of magnetic flux through that loop. Let's break this down step by step.

Understanding the Setup

We have a circular ring of wire with a radius \( R \) placed in a time-varying magnetic field given by \( B = B_0 t \). The magnetic field is only present within the area bounded by the ring. A metallic rod of length \( 2R \) is fixed at one end and can rotate about that end. The distance from the center of the ring to the point where the rod touches the ring is \( 1.732R \).

Magnetic Flux Calculation

The magnetic flux \( \Phi \) through the ring can be calculated using the formula:

• \( \Phi = B \cdot A \)

where \( A \) is the area of the ring. The area \( A \) of the circular ring is given by:

• \( A = \pi R^2 \)

Substituting the expression for the magnetic field, we have:

• \( \Phi = (B_0 t) \cdot (\pi R^2) = \pi B_0 R^2 t \)

Rate of Change of Magnetic Flux

Next, we need to find the rate of change of magnetic flux with respect to time:

• \( \frac{d\Phi}{dt} = \frac{d}{dt}(\pi B_0 R^2 t) = \pi B_0 R^2 \)

Induced EMF Calculation

According to Faraday's law, the induced emf \( \mathcal{E} \) in the rod when it touches the ring is given by:

• \( \mathcal{E} = -\frac{d\Phi}{dt} \)

Substituting our expression for the rate of change of magnetic flux, we find:

• \( \mathcal{E} = -\pi B_0 R^2 \)

Since we are interested in the magnitude of the induced emf, we can drop the negative sign:

• \( \mathcal{E} = \pi B_0 R^2 \)

Conclusion

Thus, the induced emf in the metallic rod when it touches the near end of the ring is \( \mathcal{E} = \pi B_0 R^2 \). This result illustrates how a changing magnetic field can induce an emf in a conductor, a fundamental principle of electromagnetism that has numerous applications in technology, such as in electric generators and transformers.