Discussion
An infinitely long cylinder of radius \(R\) carries a frozen-in magnetization \(\vec{M} = ke^{-s}\hat{z}\), where \(k\) is a constant and \(s\) is the distance from the axis of cylinder. The magnetic permeability of free space is \(\mu_{0}\). There is no free current present anywhere. The magnetic flux density \((\vec{B})\) inside the cylinder is
Answer: C
The flux density inside is \(\mu_{0}ke^{-s}\hat{z}\).
With no free current, ∮ H · dl = 0. Take a rectangular loop with one side inside the cylinder parallel to the axis and the other far outside. For an infinitely long cylinder with axial magnetization, H outside is zero and H inside is parallel to the axis, so H must be zero inside as well. Then
\[ \begin{aligned} \vec{B} &= \mu_{0}(\vec{H} + \vec{M}) \\ &= \mu_{0}\vec{M} = \mu_{0}ke^{-s}\hat{z} \end{aligned} \]
Zero would ignore the magnetization entirely. μ0ke-R uses the value of M at the surface everywhere, and the extra factor R/s has no source in this geometry: B simply follows M point by point.
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