A current is flowing through a thin cylindrical shell of radius R. If energy density in the medium, due to magnetic field, at a distance r from the axis of the shell is equal to U, then which of the following graphs is correct?
The correct option is B
Using Ampere's Circuital law over the cross section of the thin shell, ∮B.dl=μ0ienclosed
Case 1: For r<R
ienclosed=0 ⇒∮B.dl=0 ∴Binside=0
Case 2: For r=R ∮B.dl=μ0ienclosed ⇒B×2πR=μ0I ∴Bsurface=μ0I2πR
Case 2: For r>R
∮B.dl=μ0ienclosed ⇒B×2πr=μ0I ∴Bsurface=μ0I2πr
Hence, the variation of B w.r.t r is represented by the following graph.
Energy density due to a magnetic field B is given by U=B22μ0 where μ0 is the magnetic permeability of free space.
i.e U∝B2 So, the graph representing variation of U with r is (b).