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(D) Using Ampere's circuital law,$\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{	ext{enclosed}}$.Case-$I$: For $r < R/2$,the enclosed current $I_{	ext{

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(D) Using Ampere's circuital law,$\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{	ext{enclosed}}$.Case-$I$: For $r Case-$II$: For $R/2 \leq r \leq R$,the enclosed current is $I_{	ext{enclosed}} = J \cdot \pi(r^2 - (R/2)^2)$.Applying Ampere's law: $|\vec{B}|(2\pi r) = \mu_0 J \pi(r^2 - R^2/4)$.Thus,$|\vec{B}| = \frac{\mu_0 J}{2r}(r^2 - R^2/4)$. This shows that $|\vec{B}|$ increases from $0$ at $r = R/2$ to a maximum at $r = R$.Case-$III$: For $r > R$,the enclosed current is constant: $I_{	ext{enclosed}} = J \cdot \pi(R^2 - (R/2)^2) = J \cdot \pi(3R^2/4)$.Applying Ampere's law: $|\vec{B}|(2\pi r) = \mu_0 J \pi(3R^2/4)$.Thus,$|\vec{B}| = \frac{3\mu_0 J R^2}{8r}$. This shows that $|\vec{B}|$ decreases as $1/r$ for $r > R$.The graph that shows $|\vec{B}| = 0$ for $r  R$ is represented by option $D$.

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

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