Question

A coil is wound on a core of rectangular cross section. If all the linear dimensions of core are increased by a factor 3 and number of turns per unit length of coil remains same, the self inductance increases by a factor

• A. \( \frac{1}{9} \)
• B. 1
• C. 3
• D. 27

Hint:

The self-inductance of a coil depends on the cross-sectional area and the length of the coil. If both dimensions are scaled by a factor, the inductance changes by the square of the scaling factor for area and the scaling factor for length.

Answer 1

The Correct Option is D

Step 1: Formula for self-inductance.
The self-inductance \( L \) of a coil is given by the formula:
\[ L = \frac{\mu_0 N^2 A}{l}, \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( N \) is the number of turns,
- \( A \) is the cross-sectional area of the coil,
- \( l \) is the length of the coil.

Step 2: Effect of increasing linear dimensions.
When the linear dimensions of the core are increased by a factor of 3, the cross-sectional area \( A \) increases by a factor of \( 3^2 = 9 \), and the length \( l \) increases by a factor of 3. Therefore, the new self-inductance \( L' \) is:
\[ L' = \frac{\mu_0 N^2 (3A)}{3l} = 9 \times \frac{\mu_0 N^2 A}{l} = 9L. \]

Step 3: The correct factor for self-inductance.
Thus, the self-inductance increases by a factor of \( 27 \) because the number of turns remains the same.
Final Answer:
The self-inductance increases by a factor of:
\[ \boxed{27}. \]