Inductance of a solenoid

The solenoid is the canonical case in which the inductance can be calculated explicitly from the geometry.

For a solenoid with $N$ turns, area $A$ and length $\ell$, the internal magnetic field is $|B| = \mu_0\, i\, N/\ell$. The flux through a single turn is $B\cdot A$, and the total flux, linked with all $N$ turns, is:

$\Phi_B^\text{tot} = N\cdot B\cdot A = \mu_0\,\frac{N^2 A}{\ell}\,i$

Comparing with the definition $\Phi_B^\text{auto} = L\,i$, the inductance can be read off directly:

Inductance of a solenoid

$\ev{L_\text{sol} = \mu_0\,\frac{N^2\,A}{\ell}}$

Two observations about the structure of the formula. First: the inductance grows with the square of the number of turns $N$, not linearly: doubling the windings quadruples $L$ (each extra turn increases both the field and the number of linked turns). Second: $L$ depends only on $\mu_0$, on the geometry ($N$, $A$, $\ell$) and on any core material, never on the current: this confirms that inductance is a characteristic constant of the circuit.

Collegamenti

Argomenti: Electromagnetic induction Concetti: Inductance and self-induction Oggetti: Solenoid

Esercizi collegati: Problem — Energy in the field of a solenoid · Graphs of self-induction · RL transient: time constant