Assertion : An emf `vec(E)` is induced in a closed loop where magnetic flux is varied. The induced `vec(E)` is not a conservative field.
Reason : The line intergral `vec(E).vec(dl)` around the closed loop is non-zero.

Options

A. a.If both Assertion & Reason are true and the reason is the correct explanation of the assertion.
B. b.if both assertion and reason are true but reason is not the correct expalanation of the assertion
C. c.if assertion is true , but reason is false
D. d.both assertion and reason are false statements.

Answer

To solve the given question, we need to analyze both the assertion and the reason provided.

### Step-by-Step Solution:

1. **Understanding the Assertion**:
The assertion states that an electromotive force (emf) \(\vec{E}\) is induced in a closed loop when the magnetic flux through the loop is varied. According to Faraday's law of electromagnetic induction, a changing magnetic flux induces an emf in a closed loop.

**Hint**: Recall Faraday's law, which relates changing magnetic flux to induced emf.

2. **Induced Field Characteristics**:
The assertion further claims that this induced emf \(\vec{E}\) is not a conservative field. A conservative electric field is one where the line integral around any closed loop is zero. For an induced electric field due to a changing magnetic flux, this is not the case.

**Hint**: Consider the definition of conservative fields and how they relate to line integrals.

3. **Understanding the Reason**:
The reason states that the line integral \(\oint \vec{E} \cdot d\vec{l}\) around the closed loop is non-zero. This aligns with the assertion because if the induced emf is not conservative, then the line integral around a closed loop must indeed be non-zero.

**Hint**: Think about how the line integral relates to the work done in a conservative field.

4. **Connecting Assertion and Reason**:
Since the assertion is true (the induced emf is not conservative), and the reason provided correctly explains why the assertion is true (the line integral is non-zero), we can conclude that both the assertion and reason are true.

**Hint**: Verify the logical connection between the assertion and the reason.

5. **Final Conclusion**:
Therefore, both the assertion and the reason are true, and the reason is the correct explanation for the assertion.

**Hint**: Summarize your findings to confirm the relationship between the assertion and the reason.

### Final Answer:
Both the assertion and the reason are true, and the reason is the correct explanation for the assertion.