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

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

$\text { Case-I } x<\frac{R}{2} $ $|B|=0 $ $\text { Case-II } \quad \frac{R}{2} \leq x$ $Image$ $\left|\vec{B} . d \vec{\ell}=\mu_0\right| $ $|B| 2 \pi x=\mu_0\left[\pi x^2-\pi\left(\frac{R}{2}\right)^2\right] J $ $|B|=\frac{\mu_0 J}{2 x}\left(x^2-\frac{R^2}{4}\right)$ Case-$III$ $\quad x \geq R$ $\int \vec{B} . d \vec{\ell}=\mu_0$ I $|B| 2 \pi x=\mu_0\left[\pi R^2-\pi\left(\frac{R}{2}\right)^2\right] J$ $|B|=\frac{\mu_0 J}{2 x} \frac{3}{2} R^2$ $|B|=\frac{3 \mu_0 J R^2}{8 x}$ $Image$

Question

IIT 2012, Advanced

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Solution

$\text { Case-I } x<\frac{R}{2} $

$|B|=0 $

$\text { Case-II } \quad \frac{R}{2} \leq x$ $Image$

$\left|\vec{B} . d \vec{\ell}=\mu_0\right| $

$|B| 2 \pi x=\mu_0\left[\pi x^2-\pi\left(\frac{R}{2}\right)^2\right] J $

$|B|=\frac{\mu_0 J}{2 x}\left(x^2-\frac{R^2}{4}\right)$

Case-$III$ $\quad x \geq R$

$\int \vec{B} . d \vec{\ell}=\mu_0$ I

$|B| 2 \pi x=\mu_0\left[\pi R^2-\pi\left(\frac{R}{2}\right)^2\right] J$

$|B|=\frac{\mu_0 J}{2 x} \frac{3}{2} R^2$

$|B|=\frac{3 \mu_0 J R^2}{8 x}$ $Image$

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