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Moving Charges and Magnetism

Ampere’s Circuital Law

An infinitely long hollow conducting cylinder with inner radiu

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Solving time: 4 mins

An infinitely long hollow conducting cylinder with inner radius $R /2$ and outer radius $R$ carries a uniform current density along its length. The magnitude of the magnetic field, $∣ B ∣$ as a function of the radial distance $r$ from the axis is best represented by

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Book:

Electricity and Magnetism (DC Pandey)

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An infinitely long hollow conducting cylinder with inner radius

R /2

and outer radius

R

carries a uniform current density along its length. The magnitude of the magnetic field,

∣ B ∣

as a function of the radial distance

r

from the axis is best represented by

11 mins ago

Discuss this question LIVE

11 mins ago

Text solutionVerified

r =

distance of a point from center
For

r \leq R /2

Using Ampere's circuital law,

\oint B \cdot d l = μ_{0} i_{net}

Bl B (2 π r) B I_{in} B = μ_{0} (I_{in}) = μ_{0} (I_{in}) = \frac{μ _{0}}{2 π} \frac{I _{in}}{r} = 0 = 0

For

\frac{R}{2} \leq r \leq R

I_{in} = [π r^{2} - π (\frac{R}{2})^{2}] σ

Here,

σ =

current per unit area
Substituting in Eq. (i), we have

B = \frac{μ _{0}}{2 π} \frac{[ π r ^{2} - π \frac{R ^{2}}{4} ] σ}{r}

= \frac{μ _{0} σ}{2 r} (r^{2} - \frac{R ^{2}}{4})

r = \frac{R}{2}, B = 0

r = R, B = \frac{3 μ _{0} σ R}{8}

For

r \geq R I_{in} = I_{Total} = I

(say)
Therefore, substituting in Eq. (i), we have

B = \frac{μ _{0}}{2 π} \cdot \frac{I}{r} or B \propto \frac{1}{r}

∴

The correct graph is (d).

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Question Text	An infinitely long hollow conducting cylinder with inner radius $R /2$ and outer radius $R$ carries a uniform current density along its length. The magnitude of the magnetic field, $∣ B ∣$ as a function of the radial distance $r$ from the axis is best represented by
Topic	Moving Charges and Magnetism
Subject	Physics
Class	Class 12
Answer Type	Text solution:1
Upvotes	108

An infinitely long hollow conducting cylinder with inner radius $R /2$ and outer radius $R$ carries a uniform current density along its length. The magnitude of the magnetic field, $∣ B ∣$ as a function of the radial distance $r$ from the axis is best represented by

Text solutionVerified

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An infinitely long hollow conducting cylinder with inner radius $R /2$ and outer radius $R$ carries a uniform current density along its length. The magnitude of the magnetic field, $∣ B ∣$ as a function of the radial distance $r$ from the axis is best represented by