Question

# 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?

A
B
C
D

Solution

## 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).

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