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I got the following problem as a part of my assignment in physics:

Equal but opposite currents $J$ flow on the surfaces of two cylinders of radii $R_1$ and $R_2$, respectively, with the axes of the cylinders being located at a distance $d$ from each other. What are the surface current density distributions on these cylinders?

I'm not asking to solve this problem for me, but I want to understand what is going on in this problem or what assumptions I have to make in order to understand the problem from the mathematical standpoint. It is even unclear to me whether I should think in terms of magnetic fields or electric fields. I understand electrostatics and magnetostatics well, but I am confused by this particular problem and would be grateful for any hints that shed some light on this.

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Recall the Maxwell's equations: \begin{equation} \nabla \cdot \vec{E}= \rho/\epsilon_0; \;\nabla \cdot \vec{B}=0; \; \nabla \times \vec{E}=-\frac{\partial \vec{B}}{\partial t}; \; \nabla \times \vec{B} =\mu_0 \left( \vec{J}+\frac{\partial \vec{E}}{\partial t}\right). \end{equation}

Since we are dealing with statics, no time dependence is present in this problem. Furthermore, since $\rho=0$, we can see that we only have magnetic fields (and not electric fields) in this problem. Thus, the remaining tasks are to calculate the magnetic fields using Ampere's law and compute the surface current density distributions using the usual boundary conditions.

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  • $\begingroup$ Thanks a lot! The appropriate boundary condition is that at the surface of each cylinder, the magnetic field has no component perpendicular to the surface, right? Do you have any idea how to solve this mathematically? Should I try to construct some magnetic analogy to the method of image charges in electrostatics? $\endgroup$
    – Thylacine777
    Commented Jun 17, 2021 at 13:27
  • $\begingroup$ The boundary condition that'll be important here will be that the difference in the magnetic field's component parallel to the surface gives the surface current density. For solving the equations, we can apply the superposition principle since the equations are linear. In other words, we can compute the magnetic field contribution from each cylinder and sum two contributions vectorially. $\endgroup$
    – HelloWorld
    Commented Jun 17, 2021 at 13:39
  • $\begingroup$ Thanks,but it eludes me how I can calculate the magnetic field contribution from a cylinder if I don't know the surface current density on its surface $\endgroup$
    – Thylacine777
    Commented Jun 17, 2021 at 21:14
  • $\begingroup$ As I understand, I can calculate the magnetic field distribution in space if I know the distribution of currents, for currents induce a magnetic field as per the Biot Savart law. But in this problem it is the surface current density that I am asked to find, and the magnetic field is not given, so I am at a loss as to where to start... $\endgroup$
    – Thylacine777
    Commented Jun 17, 2021 at 21:18

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