Question

The magnetic field at any point in the xy plane is given by B vector = A r vector times K, where r vector is the position vector of the point, A is a constant, and K is a unit vector in the +z direction. The net current through a circle of radius R, in the xy plane and centered at the origin is given by:

A) π AR^2/μ0
B) 2 π AR/μ0
C) 4 π AR^3/3μ0
D) 2 π AR^2/μ0
E) π AR^2/2μ0

Answer 1

The net current through a circle in the xy plane subjected to a magnetic field defined by B vector = A r vector times K is given by 2 π AR/μ0, which corresponds to option B.

Step-by-step explanation:

The net current through a circle in the xy plane with a magnetic field B vector given by A r vector times K can be found using Ampère's law. Given that r vector is the position vector, A is a constant, and K is a unit vector in the +z direction, we use the symmetry of the problem and the fact that the field must be tangent to and constant in magnitude along any circular path centered on the origin to calculate the enclosed current.

According to Ampère's law, the line integral of the magnetic field B vector around the loop is equal to μ0 (the permeability of free space) times the net current I enclosed by the path. The integral around a circle of radius R is simply the magnitude of the magnetic field times the circumference of the circle, which gives us B(2πR). As B vector = A times the radial distance R times K, we find that the line integral is 2πAR, and the net current through the circle equals (2πAR)/μ0, which corresponds to option B) 2 π AR/μ0.