A circular wire loop of radius r is placed in a region of uniform magnetic field B such that the plane of the loop makes an angle `theta` with the direction of B. In which of the following conditions, will no emf be induced in the loop?

Options

A. Change in B with time
B. Change in r with time
C. B being nonuniform in space
D. Change in `theta` with time

Answer

To solve the problem, we need to analyze the conditions under which no electromotive force (emf) is induced in a circular wire loop placed in a magnetic field.

### Step-by-Step Solution:

1. **Understanding Magnetic Flux**:
The magnetic flux (Φ) through the loop is given by the formula:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]
where:
- \( B \) is the magnetic field strength,
- \( A \) is the area of the loop,
- \( \theta \) is the angle between the magnetic field and the normal (perpendicular) to the area of the loop.

2. **Induced EMF and Faraday's Law**:
According to Faraday's law of electromagnetic induction, the induced emf (ε) in the loop is related to the rate of change of magnetic flux:
\[ \epsilon = -\frac{d\Phi}{dt} \]
This means that an emf is induced when there is a change in magnetic flux over time.

3. **Conditions for No Induced EMF**:
For no emf to be induced in the loop, the magnetic flux must remain constant over time. This can happen under the following conditions:
- The magnetic field \( B \) does not change with time.
- The area \( A \) of the loop remains constant (i.e., the radius \( r \) does not change).
- The angle \( \theta \) between the magnetic field and the normal to the loop does not change.

4. **Analyzing the Options**:
- **Change in B with time**: If the magnetic field changes with time, the magnetic flux will change, inducing emf.
- **Change in R with time**: If the radius of the loop changes, the area changes, which will change the magnetic flux and induce emf.
- **Change in theta with time**: If the angle \( \theta \) changes, the cosine factor will change, leading to a change in flux and inducing emf.
- **B is uniform and constant**: If the magnetic field is uniform and does not change with time, and both the area and angle remain constant, then no emf will be induced.

5. **Conclusion**:
Therefore, the condition under which no emf is induced in the loop is when the magnetic field \( B \) is uniform and constant over time, and there are no changes in the radius \( r \) or the angle \( \theta \).

### Final Answer:
No emf will be induced in the loop if the magnetic field \( B \) is uniform and constant over time, and there are no changes in the radius \( r \) or the angle \( \theta \).