Question

Find the total charge on the given wire of length $L$ if its linear charge density varies with $x$ as $\lambda =\frac{{\lambda }_{0}x}{L}$ .

• A. ${\lambda }_{0}L$
• B. $\frac{{\lambda }_{0}L}{2}$
• C. $\frac{{\lambda }_{0}L}{4}$
• D. $2{\lambda }_{0}L$

Answer 1

Answer: (B)

Total charge of wire will be equal to $\int \lambda .dl$
GIven, $\lambda =\frac{{\lambda }_{0}x}{L}$
$Q_{total}={\int }_0^L\frac{{\lambda }_{0}x}{L}dx={\left[\frac{{\lambda }_0{x}^2}{2L}\right]}_0^L=\frac{{\lambda }_{0}L}{2}$