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 $\displaystyle \int \lambda .dl$
GIven, $\lambda =\frac{\lambda _{0} x}{L}$
$Q_{t o t a l}=\displaystyle \int _{0}^{L}\frac{\lambda _{0} x}{L}dx=\left[\frac{\lambda _{0} x^{2}}{2 L}\right]_{0}^{L}=\frac{\lambda _{0} L}{2}$