(A) : The magnetic field at the centre of a circular coil carrying current could be calculated using Amperes law.
(R) : Biot savart law could be derived from Amperes law.

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 false

Answer

To solve the question, we need to evaluate the two statements:

(A): The magnetic field at the center of a circular coil carrying current could be calculated using Ampere's law.
(R): Biot-Savart law could be derived from Ampere's law.

### Step-by-Step Solution:

1. **Understanding Ampere's Law**:
- Ampere's law states that the line integral of the magnetic field (B) around a closed loop is equal to μ₀ times the total current (I) passing through the loop. Mathematically, it is expressed as:
\[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I \]

2. **Magnetic Field at the Center of a Circular Coil**:
- For a circular coil of radius \( R \) carrying a current \( I \), the magnetic field at the center can be derived using Ampere's law. The magnetic field \( B \) at the center of the coil is given by:
\[ B = \frac{\mu_0 I}{2R} \]
- Thus, the assertion (A) is true.

3. **Understanding Biot-Savart Law**:
- The Biot-Savart law describes the magnetic field generated by a current-carrying element. It states that the magnetic field \( dB \) at a point due to a small segment of current \( Idl \) is given by:
\[ dB = \frac{\mu_0 I}{4\pi} \frac{dl \sin \theta}{r^2} \]
- This law can indeed be derived from Ampere's law under certain conditions.

4. **Evaluating the Reason (R)**:
- The reason (R) states that the Biot-Savart law could be derived from Ampere's law. This statement is also true. The derivation involves considering the contributions of infinitesimal current elements and integrating over the entire current distribution.

5. **Conclusion**:
- Both the assertion (A) and reason (R) are true, and the reason (R) correctly explains the assertion (A). Therefore, the answer is that both statements are true, and (R) is the correct explanation of (A).

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

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