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 given by:
\[ L = \frac{\mu_0 N^2 A}{l} \] where:
\( \mu_0 \) is the permeability of free space,
\( N \) is the number of turns,
\( A \) is the cross-sectional area,
\( l \) is the length of the solenoid.
Step 2: Analyzing how \( L \) changes with \( l \) and \( A \).
From the formula, it is clear that:
Increasing \( A \) increases \( L \) because \( A \) is in the numerator.
Decreasing \( l \) increases \( L \) because \( l \) is in the denominator.
Step 3: Evaluating the options.
Increasing both \( l \) and \( A \) would have conflicting effects on \( L \), making the net effect less predictable.
Decreasing \( l \) and increasing \( A \) simultaneously provides the most direct increase in \( L \), as both changes contribute positively to increasing \( L \).
Increasing \( l \) and decreasing \( A \) would lead to a decrease in \( L \).
Decreasing both \( l \) and \( A \) would also have conflicting effects, but the decrease in \( A \) would be more detrimental.