Q.37 Which of the following statements is/are CORRECT regarding self-inductance of a long
solenoid having cross sectional area (𝐴𝐴), length (𝑙𝑙) and having 𝑛𝑛 turns per unit length filled
with material of relative permeability 𝜇𝜇𝑟𝑟 ?
(A) It depends on the geometry of solenoid.
(B) It does not depend on geometry of solenoid.
(C) It depends on cross sectional area of solenoid.
(D) It depends on relative permeability of the medium.

Self-inductance of a long solenoid depends directly on its cross-sectional area $A$, length $l$, number of turns per unit length $n$, and the relative permeability μrμr of the core material, as derived from the standard formula L=μ0μrn2AlL=μ0μrn2Al. Options (A), (C), and (D) are correct, while (B) is incorrect.​

Option Analysis

Option (A): It depends on the geometry of solenoid.
Geometry includes $A$ and $l$, both appearing in the formula L=μ0μrn2AlL=μ0μrn2Al, confirming dependence.​

Option (B): It does not depend on geometry of solenoid.
This contradicts the formula, where $A$ and $l$ explicitly factor in, so incorrect.​

Option (C): It depends on cross sectional area of solenoid.
$L\propto A$, directly proportional to cross-sectional area.​

Option (D): It depends on relative permeability of the medium.
L∝μrL∝μr, as the magnetic field B=μ0μrnIB=μ0μrnI incorporates μrμr.​

Derivation

Magnetic field inside: B=μ0μrnIB=μ0μrnI.​
Flux per turn: ϕ=BA=μ0μrnIAϕ=BA=μ0μrnIA.
Total turns $N=nl$, total flux linkage: Φ=Nϕ=μ0μrn2AlIΦ=Nϕ=μ0μrn2AlI.
Self-inductance: L=Φ/I=μ0μrn2AlL=Φ/I=μ0μrn2Al.​

The self-inductance of long solenoid is a key concept in electromagnetic induction, crucial for competitive exams like CSIR NET Physics. For a solenoid with cross sectional area A, length l, n turns per unit length, and core of relative permeability μ_r, the inductance L=μ0μrn2AlL=μ0μrn2Al reveals its dependencies.​

Factors Affecting Inductance

• Geometry Impact: Proportional to $A$ (larger area traps more flux) and $l$ (via $N=nl$), confirming dependence on solenoid shape.​

• Core Material Role: μr>1μr>1 (e.g., iron) boosts $B$ and thus $L$.​

• No Current Dependence: $L$ is independent of current $I$.​

Practical Applications

Solenoids in transformers and inductors optimize $L$ by maximizing $A$, $n$, μrμr, and minimizing $l$.​

This analysis resolves MCQs on self-inductance of long solenoid geometry and μrμr.​