A rectangular, a square , a circular and an elliptical loop, all in the `(x-y)` plane, are moving out of a uniform magnetic field with a constant velocity `vec(v)=vhati` . The magnetic field is directed along the negative `z`-axis direction. The induced emf, during the passage of these loops , out of the field region, will not remain constant for :

Options

A. the rectangular, circular and elliptical loops
B. the circular and the elliptical loops
C. only the elliptical loop
D. any of the four loops

Explanation

To solve the problem, we need to analyze the induced electromotive force (emf) in the different shapes of loops as they move out of a uniform magnetic field. The key point here is to understand how the area of the loops that is within the magnetic field changes over time.

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have four types of loops: rectangular, square, circular, and elliptical.
- All loops are in the (x-y) plane and moving out of a uniform magnetic field directed along the negative z-axis with a constant velocity \( \vec{v} = v \hat{i} \).

2. **Induced EMF Formula**:
- The induced emf (\( \mathcal{E} \)) in a loop moving out of a magnetic field can be calculated using Faraday's law of electromagnetic induction:
\[ \mathcal{E} = -\frac{d\Phi_B}{dt} \]
- Where \( \Phi_B \) is the magnetic flux through the loop, given by:
\[ \Phi_B = B \cdot A \]
- Here, \( B \) is the magnetic field strength and \( A \) is the area of the loop within the magnetic field.

3. **Analyze Each Loop**:
- **Rectangular Loop**:
- As it moves out of the magnetic field, the area \( A \) that is within the field decreases uniformly. Therefore, \( \frac{dA}{dt} \) is constant, leading to a constant induced emf.

- **Square Loop**:
- Similar to the rectangular loop, the area decreases uniformly as it exits the magnetic field. Thus, \( \mathcal{E} \) remains constant.

- **Circular Loop**:
- As the circular loop exits the magnetic field, the area that is within the field changes in a non-linear manner. The rate of change of area \( \frac{dA}{dt} \) is not constant throughout the motion, leading to a variable induced emf.

- **Elliptical Loop**:
- The situation is similar to the circular loop. The area changes non-linearly as it exits the magnetic field, resulting in a variable induced emf.

4. **Conclusion**:
- The induced emf will not remain constant for the circular and elliptical loops due to the non-linear change in area as they exit the magnetic field. In contrast, the rectangular and square loops have a constant rate of change of area and thus a constant induced emf.

### Final Answer:
The induced emf will not remain constant for the **circular and elliptical loops**.

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