A point charges Q is moving in a circular orbit of radius R in the x-y plane with an angular velocity `omega`. This can be considered as equivalent to a loop carrying a steady current `(Q omega)/(2 pi)`. S uniform magnetic field along the positive z-axis is now switched on, which increases at a constant rate from 0 to B in one second. Assume that the radius of hte orbit remains constant. The application of hte magnetic field induces an emf in the orbit. The induced emf is defined as the work done by an induced electric field in moving a unit positive charge around a closed loop. It si known that, for an orbiting charge, the magnetic dipole moment is proportional to the angular momentum with a porportionality constant `lambda`.
The magnitude of the induced electric field in the orbit at any instant of time during the time interval of the mangnetic field change is

Options

A. `(BR)/(4)`
B. `(BR)/(2)`
C. `BR`
D. `2BR`

Explanation

To find the magnitude of the induced electric field in the orbit due to the changing magnetic field, we can follow these steps:

### Step 1: Understand the Problem
We have a point charge \( Q \) moving in a circular orbit of radius \( R \) with angular velocity \( \omega \). When a uniform magnetic field \( B \) is switched on and increases at a constant rate from \( 0 \) to \( B \) in one second, it induces an electromotive force (emf) in the orbit.

### Step 2: Calculate the Rate of Change of Magnetic Field
The rate of change of the magnetic field \( \frac{dB}{dt} \) can be calculated as follows:
\[ \frac{dB}{dt} = \frac{B - 0}{1 \text{ s}} = B \text{ (since it increases from 0 to B in 1 second)} \]

### Step 3: Determine the Area of the Circular Orbit
The area \( A \) of the circular orbit is given by:
\[ A = \pi R^2 \]

### Step 4: Relate Induced EMF to the Change in Magnetic Flux
The induced emf \( \mathcal{E} \) is related to the rate of change of magnetic flux \( \Phi \) through the area \( A \):
\[ \mathcal{E} = -\frac{d\Phi}{dt} \]
Where the magnetic flux \( \Phi \) is given by:
\[ \Phi = B \cdot A = B \cdot \pi R^2 \]
Thus, the rate of change of magnetic flux is:
\[ \frac{d\Phi}{dt} = A \cdot \frac{dB}{dt} = \pi R^2 \cdot B \]

### Step 5: Calculate the Induced EMF
Taking the magnitude of the induced emf:
\[ |\mathcal{E}| = \frac{d\Phi}{dt} = \pi R^2 \cdot B \]

### Step 6: Relate Induced EMF to Electric Field
The induced emf can also be expressed in terms of the electric field \( E \) around the loop:
\[ \mathcal{E} = E \cdot (2\pi R) \]
Where \( 2\pi R \) is the circumference of the circular orbit.

### Step 7: Solve for the Electric Field
Setting the two expressions for induced emf equal to each other:
\[ E \cdot (2\pi R) = \pi R^2 \cdot B \]
Dividing both sides by \( 2\pi R \):
\[ E = \frac{\pi R^2 \cdot B}{2\pi R} = \frac{R B}{2} \]

### Conclusion
The magnitude of the induced electric field in the orbit at any instant of time during the time interval of the magnetic field change is:
\[ E = \frac{R B}{2} \]