(A): When number of turns in a coil doubled, coefficient of self inductance of the coil becomes four times.
(R): Coefficient of self inductance is proportional to the square of number of turns.

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 1: Understanding the Assertion
The assertion states that when the number of turns in a coil is doubled, the coefficient of self-inductance of the coil becomes four times.

### Step 2: Understanding the Formula for Self-Inductance
The self-inductance \( L \) of a coil is given by the formula:
\[ L = \mu_0 \frac{N^2 A}{l} \]
where:
- \( L \) = self-inductance
- \( \mu_0 \) = permeability of free space
- \( N \) = number of turns in the coil
- \( A \) = cross-sectional area of the coil
- \( l \) = length of the coil

From this formula, we can see that self-inductance \( L \) is proportional to the square of the number of turns \( N \).

### Step 3: Doubling the Number of Turns
If the number of turns \( N \) is doubled, then:
\[ N' = 2N \]
Substituting this into the formula for self-inductance:
\[ L' = \mu_0 \frac{(2N)^2 A}{l} = \mu_0 \frac{4N^2 A}{l} = 4L \]
This shows that when the number of turns is doubled, the self-inductance becomes four times the original value.

### Step 4: Evaluating the Reason
The reason states that the coefficient of self-inductance is proportional to the square of the number of turns. From our analysis, we have confirmed that this is indeed true, as we derived from the formula for self-inductance.

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

### Final Answer
Both statements are true, and the reason correctly explains the assertion.

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