A long solenoid has n turns per unit length, length l, and cross-sectional area A. If the number of turns per unit length is doubled while keeping the length and cross-sectional area unchanged, by what factor does its self-inductance change?

Answer and explanation

Correct answer: A. Increases by a factor of 4

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  1. Step 1: The self-inductance of an air-core solenoid is given by: , where is the number of turns per unit length, is the cross-sectional area, and is the length.
  2. Step 2: Let the initial self-inductance be .
  3. Step 3: When the number of turns per unit length is doubled, :
  4. Step 4: Therefore, the self-inductance increases by a factor of 4.
Answer
Increases by a factor of 4
Why not the others: 'Increases by a factor of 2' mistakenly assumes self-inductance depends linearly on turns per unit length () rather than quadratically (); 'Decreases by a factor of 2' assumes an inverse relationship; 'Remains unchanged' ignores the dependence of flux linkage and magnetic field on .

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