# Magnetic field due to circular surface current density

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, $$$$B2\pi r = \mu_0 I_{enc}$$$$ However, I am having problems with the $$I_{enc}$$ part. Since $$$$I_{enc} = \int K \cdot dl_{\perp}$$$$ The length perpendicular to the current flow would be $$l$$. But then we would have, $$$$B = \frac{\mu_0 K_0 cos(\theta)}{2\pi r} \hat{z}$$$$

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 $$0$$.

• 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 Commented Feb 14, 2022 at 12:24
• Current enclosed, is, and should be zero. the current density is perpendicular to your amperian loop and thus the flux integral is 0 Commented Feb 14, 2022 at 12:27