Question

A charged wire of length $3\text{m}$ lies along $x-$ axis in such a way that its linear charge density is given by $\lambda =2x^2\text{C/m}$. The total charge on the wire is

• A. $15\text{C}$
• B. $18\text{C}$
• C. $17\text{C}$
• D. $16\text{C}$

Answer 1

The correct option is B $18\text{C}$

We know that, linear charge density,

$\lambda =\frac{dq}{dx}=\frac{q_{total}}{L}$

$\Rightarrow dq=\lambda dx$

integrating both side

$\Rightarrow \underset{0}{\overset{q_{total}}{\int }}dq=\underset{0}{\overset{3}{\int }}\lambda dx$

$\Rightarrow q_{total}=\underset{0}{\overset{3}{\int }}2x^{2}dx=\frac{2}{3}{\left[x^3\right]}_0^3$

$\Rightarrow q_{total}=\frac{2}{3}[3^3-0]$

$\Rightarrow q_{total}=18\text{C}$

Hence, option $\left(b\right)$ is correct.