I am trying to find the magnetic field between two coaxial cylindrical shells of radius $a$ and $b$ respectively. They have a surface current density of $K = K_0cos(\theta)\hat{\theta}$ and $K=-K_0cos(\theta)\hat{\theta}$ respectively. My thought process so far is to use an amperian loop, \begin{equation} B2\pi r = \mu_0 I_{enc} \end{equation} However, I am having problems with the $I_{enc}$ part. Since \begin{equation} I_{enc} = \int K \cdot dl_{\perp} \end{equation} The length perpendicular to the current flow would be $l$. But then we would have, \begin{equation} B = \frac{\mu_0 K_0 cos(\theta)}{2\pi r} \hat{z} \end{equation}

But it feels wrong to have the $cos\theta$ in the magnetic field formula. Moreover, I then have to calculate the angular momentum per unit length which would mean I would have to integrate over the $cos\theta$ and that would equal $0$.

  • $\begingroup$ How have you gone from $\int \vec{B} \cdot \vec{dl}$ to $|\vec{B}|2\pi r$? The current has a $\hat\theta$ component meaning the magnetic field does not follow your amperian loop? You're confusing the procedure for solving for a current with a purely $\hat z$ component where the B field is parcelled to dl $\endgroup$ Commented Feb 14, 2022 at 12:24
  • $\begingroup$ Current enclosed, is, and should be zero. the current density is perpendicular to your amperian loop and thus the flux integral is 0 $\endgroup$ Commented Feb 14, 2022 at 12:27


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