An insulating thin rod of length l has a line | vedclass

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

Question

(C) Consider a small element of length $dx$ at a distance $x$ from the origin. The charge on this element is $dq = \rho(x) dx = \rho_0 \frac{x}{l} dx$

Vedclass · Accepted Answer

$\frac{\pi}{4} n \rho_0 l^3$

Vedclass · Answer

(C) Consider a small element of length $dx$ at a distance $x$ from the origin. The charge on this element is $dq = \rho(x) dx = \rho_0 \frac{x}{l} dx$.As the rod rotates with frequency $n$ (rotations per second),the angular velocity is $\omega = 2\pi n$. The current $di$ produced by this rotating charge element is $di = \frac{dq}{T} = dq \cdot n = \rho_0 \frac{x}{l} dx \cdot n$.The magnetic moment $dM$ of this circular current loop of radius $x$ is $dM = di \cdot A = di \cdot (\pi x^2)$.Substituting $di$: $dM = (\rho_0 \frac{x}{l} dx \cdot n) \cdot \pi x^2 = \frac{\pi n \rho_0}{l} x^3 dx$.Integrating from $x = 0$ to $x = l$: $M = \int_0^l \frac{\pi n \rho_0}{l} x^3 dx = \frac{\pi n \rho_0}{l} [\frac{x^4}{4}]_0^l = \frac{\pi n \rho_0}{l} \cdot \frac{l^4}{4} = \frac{\pi}{4} n \rho_0 l^3$.

An insulating thin rod of length $l$ has a linear charge density $\rho(x) = \rho_0 \frac{x}{l}$ on it. The rod is rotated about an axis passing through the origin $(x = 0)$ and perpendicular to the rod. If the rod makes $n$ rotations per second,then the time-averaged magnetic moment of the rod is:

Similar Questions

$A$ current-carrying circular loop is freely suspended by a long thread. The plane of the loop will point in the direction:

$A$ current of $3 \, A$ is flowing in a plane circular coil of radius $4 \, cm$ and number of turns $20$. The coil is placed in a uniform magnetic field of magnetic induction $0.5 \, T$. The magnetic dipole moment of the coil is $... \, A \cdot m^2$.

$A$ loop in the form of four connected semi-circular wires carrying current $I$ lies in the $x-y$ plane as shown in the figure. The unit vector $\hat{k}$ is coming out of the plane of the paper. The magnetic moment of the current loop is

Two thick wires and two thin wires,all of the same material and same length,form a square in three different ways $P, Q,$ and $R$ as shown in the figure with the current connection indicated. The magnetic field at the centre of the square is zero in cases:

Vedclass Test Series

Exam Paper Generator

Online Exam Module