## Related questions with answers

You are working for a company that creates special magnetic environments. Your new supervisor has come from the financial side of the organization rather than the technical side. He has promised a client that the company can provide a device that will create a magnetic field inside a cylindrical chamber that is directed along the cylinder axis at all points in the chamber and increases in the axial direction as the square of the value of y, where y is the in the axial direction and y = 0 is at the bottom end of the cylinder. Prepare a calculation to show that the field requested by your supervisor and promised to a client is impossible.

Solution

Verified$\vec B \cdot \vec dA =0$

, on the side. Now, since $\nabla \cdot \vec B=0$ flux trough closed surface must be $0$ so flux trough top and bottom surface must be the same magnitude and opposite sign, or both should be $0$. Since field is along the axis, its flux is not $0$ infact it is:

$|\phi|=B\cdot A$

both on top and on bottom, and since the field grows, on the top part flux is big and positive, while at the bottom, flux is small, and thus we cannot satisfy $\nabla \cdot \vec B=0$.

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