Question

A neophyte magnet designer tells you that he can produce a magnetic field B in vacuum that points everywhere in the x -direction and that increases in magnitude with increasing x . That is, B =B₀(x/a)ı^, where B₀ and a are constants with units of teslas and meters, respectively. Use Gauss's law for magnetic fields to show that this claim is impossible. (Hint: Use a Gaussian surface in the shape of a rectangular box, with edges parallel to the x ; y . and z -axes.)

Answer 1

Using Gauss's law for magnetic fields, the claim of a magnetic field increasing in magnitude with increasing x in a vacuum is shown to be impossible, as it would result in an unequal amount of magnetic flux through a closed surface, which violates Gauss's law.

Step-by-step explanation:

The claim that a magnetic field B in a vacuum can point everywhere in the x-direction and increase in magnitude with increasing x is impossible according to Gauss's law for magnetic fields. Gauss's law states that the total magnetic flux out of any closed surface is zero. If we take a Gaussian surface in the shape of a rectangular box with edges parallel to the x, y, and z-axes encompassing such a field, the sides of the box normal to the y and z axes would have no magnetic flux passing through them because B is in the x-direction. However, the sides parallel to the yz-plane would then have an unequal amount of flux entering and leaving the box if B were to increase with x, which would contradict Gauss's law. Since the net magnetic flux through the surface must be zero, the magnetic field cannot increase with x as claimed by the neophyte magnet designer.