A thick uniformly charged hollow cylinder of | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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

Vedclass · Accepted Answer

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

Vedclass · Answer

(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.

Similar Questions

Write the equation for the magnetic field at an inside point of a very long solenoid.

$A$ long solenoid has $200$ turns per $cm$ and carries a current of $2.5 \ A$. The magnetic field at its centre is $(\mu_0 = 4\pi \times 10^{-7} \ \text{Wb/A} \cdot \text{m})$.

How can the magnetic field of a solenoid be increased for a given current?

An infinitely long hollow conducting cylinder with radius $R$ carries a uniform current along its surface. Choose the correct representation of magnetic field $(B)$ as a function of radial distance $(r)$ from the axis of the cylinder.

$A$ solenoid has a turn density of $5000 \, \text{turns/m}$ and a cross-sectional area of $10 \, \text{cm}^2$. If a current of $1 \, \text{A}$ flows through it and the core material has a relative permeability of $1000$, find the energy per unit length of the solenoid in $\text{J/m}$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module