Question

The linear charge density of a thin rod of length $l$ varies with the distance $x$ from the left-most end as $\lambda ={\lambda }_0{x}^2(0\le x\le l)$. The total charge on the rod is

• A. $\frac{{\lambda }_0{l}^3}{3}$
• B. $\frac{{\lambda }_0{l}^2}{3}$
• C. $\frac{2{\lambda }_0{l}^2}{3}$
• D. $\frac{{\lambda }_0{l}^3}{4}$

Answer 1

The correct option is A $\frac{{\lambda }_0{l}^3}{3}$


Let us choose an element of length $dx$ at a distance $x$ from the one end as shown in the figure.

Infinitesimally small charge $dq$ on the element :
$dq=\lambda dx$

The total charge on the rod,

$q=\int dq=\underset{x=0}{\overset{x=l}{\int }}{\lambda }_0{x}^{2}dx$

$\Rightarrow q={\lambda }_0{\left[\frac{x^3}{3}\right]}_{x=0}^{x=l}$

$\therefore q=\frac{{\lambda }_0{l}^3}{3}$

Hence, option (a) is the correct answer.