Question

The linear charge density of a thin metallic rod varies with the distance ${}^{\prime }{x}^{\prime }$ from one end as $\lambda ={\lambda }_0{x}^2\left(0⩽x⩽l\right).$ The total charge on the rod is:

• A. $\frac{{\lambda }_0{l}^3}{3}$
• B. $\frac{{\lambda }_0{l}^4}{3}$
• C. $\frac{2{\lambda }_0{l}^3}{3}$
• D. $\frac{{\lambda }_{0}l}{2}$

Answer 1

The correct option is D. $\frac{{\lambda }_{0}l}{2}$.

Given,

$\lambda =\frac{{\lambda }_{0}x}{L}$

Rearranging and integrating,

$Q={\int }_0^L\lambda dx$

$={\int }_0^L\frac{{\lambda }_{0}x}{L}dx$

$=\frac{{\lambda }_0}{L}[\frac{x^2}{2}{]}_0^L$

$=\frac{{\lambda }_0}{L}\times \frac{L^2}{2}$

$=\frac{{\lambda }_{0}L}{2}$.