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(D) Consider a thin cylindrical shell of radius $r$ and thickness $dr$ within the thick cylinder.The charge on this thin shell is $dQ = \rho (2\pi r d

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Graph of magnetic field on the axis of cylinder with distance from point $P$ is as shown in the image.

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(D) Consider a thin cylindrical shell of radius $r$ and thickness $dr$ within the thick cylinder.The charge on this thin shell is $dQ = \rho (2\pi r dr) L$.The current $dI$ produced by this rotating shell is $dI = \frac{dQ}{T} = \frac{dQ \cdot \omega}{2\pi} = \rho \omega r L dr$.This acts as a solenoid with $n = \frac{1}{L}$ turns per unit length (effectively). The magnetic field produced by this shell at the center $P$ is $dB = \mu_0 n dI = \mu_0 \left(\frac{1}{L}\right) (\rho \omega r L dr) = \mu_0 \omega \rho r dr$.Integrating from $r = a$ to $r = b$,the total magnetic field at $P$ is $B_P = \int_a^b \mu_0 \omega \rho r dr = \frac{\mu_0 \omega \rho (b^2 - a^2)}{2}$.For a long solenoid,the magnetic field at the ends ($A$ or $B$) is exactly half of the field at the center: $B_A = B_B = \frac{B_P}{2} = \frac{\mu_0 \omega \rho (b^2 - a^2)}{4}$.The direction of the magnetic field is determined by the right-hand rule and is the same along the axis. The graph of the magnetic field along the axis shows a constant value in the middle and a decrease towards the ends,which is consistent with the properties of a long solenoid.

$A$ thick uniformly charged hollow cylinder of inner radius $a$ and outer radius $b$ rotates with constant angular speed $\omega$ about its axis $APB$,with charge density $\rho$. Given that $L \gg a$ and $L \gg b$,and $P$ is the midpoint of $AB$. Choose the incorrect option.

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