A conducting rod is rotated about one end in a plane perpendicular to a uniform magnetic field with constant angular velocity. The correct graph between the induced emf (e) across the rod and time (t) is

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a conducting rod that is rotated about one end in a plane perpendicular to a uniform magnetic field.
- The magnetic field is uniform and directed into the plane (indicated by a cross sign).

2. **Identifying the Motion**:
- As the rod rotates, it sweeps out an area in the magnetic field. The angular velocity of the rod is constant, denoted as \( \omega \).

3. **Calculating the Induced EMF**:
- The induced EMF (\( e \)) in the rod can be calculated using Faraday's law of electromagnetic induction, which states that:
\[ e = -\frac{d\Phi}{dt} \]
- Here, \( \Phi \) is the magnetic flux through the area swept by the rod.

4. **Finding the Magnetic Flux**:
- The magnetic flux \( \Phi \) is given by:
\[ \Phi = B \cdot A \]
- Where \( B \) is the magnetic field strength, and \( A \) is the area swept by the rod.

5. **Calculating the Area Swept**:
- As the rod rotates, the area \( A \) swept out in a small time \( dt \) can be expressed as:
\[ dA = \frac{1}{2} L^2 d\theta \]
- Where \( L \) is the length of the rod and \( d\theta \) is the small angle swept in time \( dt \).

6. **Substituting into the Flux Equation**:
- The magnetic flux can now be expressed as:
\[ \Phi = B \cdot \frac{1}{2} L^2 \theta \]
- Taking the derivative with respect to time \( t \):
\[ \frac{d\Phi}{dt} = B \cdot \frac{1}{2} L^2 \frac{d\theta}{dt} \]

7. **Using the Constant Angular Velocity**:
- Since \( \frac{d\theta}{dt} = \omega \) (constant angular velocity), we have:
\[ \frac{d\Phi}{dt} = B \cdot \frac{1}{2} L^2 \omega \]

8. **Calculating the Induced EMF**:
- Substituting back into the EMF equation:
\[ e = -\frac{d\Phi}{dt} = -B \cdot \frac{1}{2} L^2 \omega \]
- Since \( B \), \( L \), and \( \omega \) are constants, the induced EMF \( e \) is also constant.

9. **Graphing the Induced EMF vs. Time**:
- Since the induced EMF \( e \) is constant over time, the graph of \( e \) versus \( t \) will be a horizontal line.

### Conclusion:
The correct graph between the induced EMF (\( e \)) across the rod and time (\( t \)) is a horizontal line, indicating that \( e \) is constant over time.