Question

In the Amperian loop derivation for a solenoid, what is the contribution of side \(ab\) (inside, along the axis) to the line integral \(\oint \vec{B} \cdot d\vec{l}\)?

Options

• \(0\), because \(\vec{B}\) is perpendicular to \(d\vec{l}\)

• \(Bl\), because \(\vec{B}\) is parallel to \(d\vec{l}\)

• \(\mu_0 n I\), directly from Ampere's Law

• \(B l / 2\), because only half the turns contribute

Answer 1

For side \(ab\) inside the solenoid along the axis, \(\vec{B}\) is parallel to \(d\vec{l}\) (\(\vec{B} \parallel d\vec{l}\)), so \(\vec{B} \cdot d\vec{l} = B \, dl\). Integrating over the length \(l\) gives a contribution of \(Bl\). This is the only side contributing to the line integral.