Assertion (A): The self-inductance of a long solenoid is proportional to the square of the number of turns per unit length.
Reason (R): The magnetic field inside a long solenoid is directly proportional to the current and the number of turns per unit length.

Question

Assertion (A): The self-inductance of a long solenoid is proportional to the square of the number of turns per unit length.

Reason (R): The magnetic field inside a long solenoid is directly proportional to the current and the number of turns per unit length.

TestHub · Accepted Answer

For a long solenoid, the magnetic field inside is $B = μ_{0} n I$ , where $n$ is the number of turns per unit length and $I$ is the current. This confirms Reason (R).

The magnetic flux through each turn is $Φ = B A = (μ_{0} n I) A$ . The total flux linkage for a solenoid of length $l$ with $N$ turns is $N Φ = (n l) (μ_{0} n I A) = μ_{0} n^{2} A l I$ . Since $N Φ = L I$ , the self-inductance $L = μ_{0} n^{2} A l$ .

Therefore, $L$ is proportional to $n^{2}$ , confirming Assertion (A). Reason (R) is a necessary step in deriving Assertion (A).

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