Solenoid

Printed on the AP Physics C: Electricity and Magnetism sheet and derived there from Ampere's law at topic 12.4. It is not on the AP Physics 2 sheet, and AP Physics 2 sets no solenoid problems.

$n$ is turns per unit length, not the total number of turns. That single substitution accounts for most wrong answers on this equation. If a problem gives you $N$ turns over a length $\ell$, form $n = N/\ell$ first and write it down. A 500-turn coil 0.25 m long has $n = 2000$ turns per metre, so substituting 500 understates the field by a factor of four.

The CED sets the idealisation explicitly. Unless a question says otherwise, solenoids are assumed very long, with a uniform field inside and a negligible field outside. That is what makes the Amperian loop argument work: three of its four sides contribute nothing, so $\oint \vec{B} \cdot d\vec{\ell}$ collapses to $B\ell$.

Notice that the interior field carries no radius and no distance term. Move off the axis, stay well inside, and the field is the same. Notice too that it is the field the coil produces; $\vec{F}_B$ is a separate question.

The inductance of the same coil, $L_{\text{sol}} = \mu_{\text{core}} N^2 A / \ell$, does use the total turn count $N$, and squares it. Two equations for one object, two different turn quantities.