Question

In the Ampere's Law derivation for a solenoid, what is the contribution of the perpendicular sides \(bc\) (inside to outside) and \(da\) (outside to inside) to \(\oint \vec{B} \cdot d\vec{l}\)?

Options

• Zero for both, because \(\vec{B}\) is perpendicular to \(d\vec{l}\)

• \(Bl\) for both, because they have the same length as \(ab\)

• \(\mu_0 n I\) for both, from the enclosed current

• \(Bl/2\) for each, since the field varies linearly

Answer 1

For the perpendicular sides \(bc\) and \(da\), the magnetic field \(\vec{B}\) is perpendicular to \(d\vec{l}\), so \(\vec{B} \cdot d\vec{l} = 0\) at every point along these sides. Hence their total contribution to \(\oint \vec{B} \cdot d\vec{l}\) is zero.