Question

Consider a solenoid of length \( l \) and area of cross-section \( A \) with a fixed number of turns. The self-inductance of the solenoid will increase if:

• A. both \( l \) and \( A \) are increased
• B. \( l \) is decreased and \( A \) is increased
• C. \( l \) is increased and \( A \) is decreased
• D. both \( l \) and \( A \) are decreased

Hint:

When evaluating the effect of physical changes on the inductance of a solenoid, remember that increasing the area of cross-section and decreasing the length will enhance the inductance, benefiting from the direct proportionality to \( A \) and inverse proportionality to \( l \).

Answer 1

The Correct Option is B

Step 1: Understanding self-inductance.
The self-inductance \( L \) of a solenoid is defined by:
\[ L = \frac{\mu_0 N^2 A}{l} \]
where:
\( \mu_0 \) represents the permeability of free space,
\( N \) denotes the number of turns,
\( A \) is the cross-sectional area,
\( l \) is the solenoid's length.
Step 2: Analyzing the impact of \( l \) and \( A \) on \( L \).
The formula indicates that:
An increase in \( A \) directly increases \( L \) due to its position in the numerator.
A decrease in \( l \) directly increases \( L \) due to its position in the denominator.
Step 3: Evaluating potential modifications.
Simultaneously increasing both \( l \) and \( A \) would result in opposing effects on \( L \), making the overall change uncertain.
The most effective method to increase \( L \) is to decrease \( l \) while increasing \( A \), as both actions independently promote an increase in \( L \).
Increasing \( l \) and decreasing \( A \) would cause \( L \) to decrease.
Decreasing both \( l \) and \( A \) would also yield conflicting outcomes, with the reduction in \( A \) having a more significant negative impact.