Sunday 18 September 2016

electromagnetism - Flux of magnetic field not zero


Consider the magnetic part of a single electromagnetic wave in empty space, propagating along the $x$-axis of some reference frame. If we take as surface a cylinder with the axis along $x$ and height $L < \lambda /2$ (where $\lambda$ is the wavelenght) placed in a way that it is only crossed by upward (or downward) pointing $\textbf{B}$ vectors, and then we calculate the flux of B through this surface, how do we possibly get zero? The magnetic field is outgoing from the surface, so the flux should be positive.




My solution


I thought of a way to solve this apparent paradox but I am not sure about it, so I ask. My solution is that, since B is contained in a plane, and the intersection between this plane and the surface of the cylinder is a line, the surface integral to calculate the flux is zero because B is different from zero only in a region of negligible measure, namely a line with respect to a surface (I am thinking of Lebesgue integration).


Is my view correct?




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