SATHEE: Magnetics 4 Question 2

Magnetics 4 Question 2

3. An insulating thin rod of length $l$ has a linear charge density $ρ (x) = ρ_{0} \frac{x}{l}$ 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

(2019 Main, 10 Jan I)

(a) $n ρ l^{3}$

(b) $π n ρ l^{3}$

(d) $\frac{π}{4} n ρ l^{3}$

Show Answer

Answer:

Correct Answer: 3. (d)

Solution:

$\begin{aligned} (R_{2} = radius of coil \\ having N loops) \end{aligned}$

Key Idea A rotating charge constitutes a current. Hence, a rotating charged rod behaves like a current carrying coil. If charge q rotates with a frequency $n$ , then equivalent current is $I = q n$ and magnetic moment associated with this current is $M = I A$

where, $A$ =area of coil or area swept by rotating rod. Let $d q$ be the charge on $d x$ length of rod at a distance $x$ from origin as shown in the figure below.

The magnetic moment $d m$ of this portion $d x$ is given as

$\begin{aligned} d m & = (d I) A \\ d m & = n d q A [∵ I = q n, ∴ d I = n \cdot d q] \\ = n ρ d x A \end{aligned}$

where, $ρ =$ charge density of $rod = ρ_{0} \frac{x}{l}$ .

So, $d m = \frac{n ρ_{0} x d x π x^{2}}{l} = \frac{π n ρ_{0}}{l} \cdot x^{3} \cdot d x$

Total magnetic moment associated with rotating rod is sum of all the magnetic moments of such differentiable elements of rod.

So, magnetic moment associated with complete rod is

$\begin{aligned} M & = \int_{x = 0}^{x = l} d m \\ = \int_{0}^{l} \frac{π n ρ_{0}}{l} \cdot x^{3} d x = \frac{π n ρ_{0}}{l} \cdot \int_{0}^{l} x^{3} d x \\ = \frac{π n ρ_{0}}{l} \frac{x^{4}}{4} = \frac{π n ρ_{0} l^{3}}{4} \end{aligned}$

$\begin{aligned} at & x & = l \\ ρ & = ρ_{0} \\ ∴ & M & = \frac{π}{4} n ρ l^{3} \end{aligned}$

Magnetics 4 Question 2

3. An insulating thin rod of length $l$ has a linear charge density $ρ (x) = ρ_{0} \frac{x}{l}$ 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

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Magnetics 4 Question 2

3. An insulating thin rod of length l has a linear charge density ρ(x)=ρ0xl 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

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu