For a long, tightly wound coil (solenoid-like) with magnetic core, how does its inductance vary with basic physical parameters, all else held constant?

Introduction / Context:
Inductance quantifies a coil’s ability to store magnetic energy for a given current. Designers adjust turns, core materials, and geometry to achieve target inductance values for filters, chokes, and resonant circuits. Knowing the correct proportionalities prevents costly trial-and-error.

Given Data / Assumptions:

• Long, uniformly wound coil with a linear magnetic core (no saturation).
• Fringing and leakage are neglected for first-order analysis.
• We compare qualitative dependences (increase/decrease), not exact numbers.

Concept / Approach:
A widely used approximate relation is L ∝ μ * N^2 * A / l, where μ is permeability, N is turns count, A is cross-sectional area, and l is magnetic path length. Thus, inductance grows with permeability, with area, and with the square of the turns, and it decreases with longer magnetic paths.

Step-by-Step Solution:
Start from the proportionality: L ∝ μ * N^2 * A / l.Analyze parameter effects: increasing N strongly raises L because of the N^2 factor.Increasing length l lowers L; increasing area A raises L; higher μ raises L.Therefore, the statement “directly proportional to the number of turns squared” is the most accurate among the options.

Verification / Alternative check:
Doubling turns approximately quadruples inductance if μ, A, and l remain constant. This square-law behavior is well established in inductor design references and verified in practical winding experiments.

Why Other Options Are Wrong:
Direct proportionality to length is incorrect; L falls as length increases.

Inverse proportionality to area and to permeability are both wrong; L increases with A and μ.

Independence from core material is false; μ of the core is a dominant factor.

Common Pitfalls:
Assuming L increases linearly with turns; the correct dependence is N^2 for tightly coupled coils.

Final Answer:
directly proportional to the number of turns squared