A long straight solid cylinder of radius R carries a steady current along its length. The current density J across the cross-section of the cylinder varies with the radial distance r from the axis as J = kr , where k is a positive constant. The magnetic field B inside the cylinder ( r < R ) varies with r as:

Step-by-step solution

To find the magnetic field inside the cylinder at a distance r ( r First, calculate the current enclosed by an Amperian loop of radius r . Since the current density J is non-uniform, we must integrate over the cross-sectional area: I_ enclosed = J dA = ₀^ r (kr) (2 r') dr' I_ enclosed = 2 k ₀^ r r'^2 dr' = 2 k ( r^3 3 ) Now, apply Ampere's law: B(2 r) = ₀ ( 2 k r^3 3 ) B = ₀ k r^2 3 Therefore, the magnetic field inside the cylinder varies as B r^2 . Option 1 is incorrect as it assumes a uniform current density. Opt