A: Ampere circuital law is not independent of the Biot- Savart's law.
R: Ampere's Circuital law can be derived from the Blot Savart law.

Options

A. If both Assertion & Reason are true and the reason is the correct explanation of the assertion then mark (1)
B. If both Assertion & Reason are true but the reason is not the correct explanation of the assertion, then mark 2.
C. If assertion is true statement but Reason is false, then mark (3)
D. If both Assertion and Reason are false statements, then mark (4)

Answer

To solve the assertion and reason question regarding Ampere's Circuital Law and Biot-Savart's Law, we will analyze both statements step by step.

### Step-by-Step Solution:

**Step 1: Understand the Assertion**
- The assertion states that "Ampere's Circuital Law is not independent of the Biot-Savart's Law."
- This means that Ampere's Circuital Law relies on or is derived from Biot-Savart's Law.

**Step 2: Understand the Reason**
- The reason states that "Ampere's Circuital Law can be derived from the Biot-Savart's Law."
- This implies that if we start with Biot-Savart's Law, we can mathematically arrive at Ampere's Circuital Law.

**Step 3: Review Biot-Savart's Law**
- Biot-Savart's Law gives the magnetic field \( dB \) due to a small current element \( dl \) as:
\[ dB = \frac{\mu_0 I}{4\pi r^2} \sin \theta \, dl \]
- Here, \( \mu_0 \) is the permeability of free space, \( I \) is the current, \( r \) is the distance from the current element to the point where the magnetic field is being calculated, and \( \theta \) is the angle between \( dl \) and the line connecting the current element to the point.

**Step 4: Derive the Magnetic Field**
- For a closed loop of current, we need to integrate the contributions of \( dB \) around the loop.
- The total magnetic field \( B \) can be calculated by integrating \( dB \):
\[ B = \int dB = \int \frac{\mu_0 I}{4\pi r^2} \sin \theta \, dl \]
- After performing the integration and simplifying, we can express the magnetic field in terms of the total current and the geometry of the loop.

**Step 5: Arrive at Ampere's Circuital Law**
- The result of the integration leads us to the conclusion that:
\[ \oint B \cdot dl = \mu_0 I \]
- This is the statement of Ampere's Circuital Law, which states that the line integral of the magnetic field \( B \) around a closed path is equal to the permeability of free space times the total current \( I \) enclosed by that path.

**Step 6: Conclusion**
- Since we have shown that Ampere's Circuital Law can be derived from Biot-Savart's Law, we conclude that both the assertion and reason are true.
- Therefore, the assertion is true because it is not independent of Biot-Savart's Law, and the reason is true because it correctly explains the assertion.

### Final Answer:
Both the assertion (A) and reason (R) are true, and R correctly explains A.

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