(A) : A magnetic field is produced either by a steady current or by a time varying electric field (R) : According to Ampere’s law `ointbarB.bardl = mu_(0)i_(0)+mu_(0)epsilon_(0)(doint_(E ))/(dt)`

Options

A. Both 'A' and 'R' are true and 'R' is the correct explanation of 'A'.
B. Both 'A' and ’R' are true and 'R' is not the correct explanation of 'A
C. A' is true and 'R' is false
D. 'A' is false and R' is true

Answer

To solve the question, we need to analyze both the assertion (A) and the reason (R) provided.

### Step-by-Step Solution:

1. **Understanding the Assertion (A)**:
- The assertion states: "A magnetic field is produced either by a steady current or by a time varying electric field."
- This statement is partially true. A magnetic field can indeed be produced by a steady current. However, it is also true that a time-varying electric field can produce a magnetic field, but this is specifically through the concept of displacement current.

2. **Understanding the Reason (R)**:
- The reason states: "According to Ampere’s law `∮B·dl = μ₀i₀ + μ₀ε₀(d/dt)∮E·dA`."
- This equation is known as Ampere-Maxwell Law, which expands Ampere's Law to include the displacement current term. It shows that a magnetic field can be produced not only by a steady current (i₀) but also by a changing electric field (represented by the displacement current term).

3. **Analyzing the Relationship between A and R**:
- The assertion (A) claims that a magnetic field can be produced by both steady current and time-varying electric fields.
- The reason (R) provides the mathematical foundation for this assertion, confirming that both steady currents and changing electric fields contribute to the generation of magnetic fields.

4. **Conclusion**:
- Since the assertion is true in the context that both steady currents and changing electric fields can produce magnetic fields, and the reason correctly explains this phenomenon through Ampere's law, both A and R are true.
- However, the assertion does not fully encompass the role of displacement current as described in the reason. Thus, while A is true, R is a correct explanation of A.

### Final Answer:
- Both A and R are true, and R is the correct explanation of A.

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