Question

Self inductance of solenoid is

• A. directly proportional to current flowing through the coil.
• B. directly proportional to the length.
• C. directly proportional to its area of cross-section.
• D. inversely proportional to the area of cross-section.

Hint:

Logic Tip: Inductance depends on the geometry of the coil; increasing the cross-sectional area allows more magnetic flux to pass through, thereby increasing self-inductance.

Answer 1

The Correct Option is C

Concept:
The self-inductance of a long solenoid depends on its number of turns, length, permeability of core material, and cross-sectional area. For an air-core solenoid: \[ L=\frac{\mu_0 N^2 A}{l} \] where:

• $L$ = self-inductance
• $\mu_0$ = permeability of free space
• $N$ = total number of turns
• $A$ = area of cross-section
• $l$ = length of solenoid

Step 1: Observe dependence on area
From the formula: \[ L \propto A \] This means if cross-sectional area increases, self-inductance also increases in the same ratio.
Step 2: Reason physically
A larger cross-sectional area allows more magnetic flux to pass through each turn of the solenoid. More magnetic flux linkage per unit current means greater self-inductance.
Step 3: If area changes
If area becomes $kA$, then: \[ L'=\frac{\mu_0 N^2 (kA)}{l}=kL \] So inductance changes directly with area. Examples:

• If area doubles $\Rightarrow L$ doubles
• If area becomes half $\Rightarrow L$ becomes half

Step 4: Final Answer
The self-inductance of a solenoid is directly proportional to its area of cross-section. \[ \boxed{L \propto A} \] Quick Tip:
A wider solenoid stores more magnetic flux, so its inductance becomes larger.