Sometimes we show an idealized magnetic field which is uniform in a given region and falls to zero abruptly. One such field is represented in figure. Using Ampere’s law over the path PQRS, show that such a field is not possible.
Half of the given loop is in a region with zero magnetic field, while half of it is in a region with magnetic field B.
Now, we take the Amperian loop PQRS.
Ampere’s circuital law states that the line integral of the magnetic field for a closed surface is μ0 times the current enclosed by the surface.
where
B = magnetic field,
dl = line element,
μ0 = magnetic permeability of vacuum,
I = current enclosed.
Hence, I (current enclosed) is not 0, but is finite (since half of it lies in a region of magnetic field B). Hence, this states the magnetic field inside the loop PQRS is not zero.
However, it is obvious that the magnetic field at any point in the region outside the magnetic field lines = 0.
These are two contradictory statements.
Hence, such a field is not possible. (proved)