Question

Express the total charge $Q$ on the rod in terms of $\alpha $ and $L$.

(A) $\dfrac{{\alpha L}}{2}$
(B) $\dfrac{{\alpha {L^2}}}{2}$
(C) $\dfrac{{\alpha {L^3}}}{2}$
(D) $\dfrac{{\alpha {L^2}}}{3}$

Answer 1

Hint: Linear charge density varies continuously at a constant rate throughout the rod when the charge is kept in line with the axis of the rod. Which means that at every point on the rod, the linear charge density is different.

Complete step by step answer:
Linear charge density is the ratio of the total charge present on the rod to total length. Linear charge density on a small segment of rod can be given by,
$\lambda = \dfrac{{dq}}{{dx}}$
Where, $dq = $small charge present on small segment $dx$. So,
$dq = \lambda dx$
This can also be written as,
$dq = \alpha xdx$
Now integrating the above equation,
$\Rightarrow \int\limits_0^Q {dq} = \int\limits_0^L {\alpha xdx} $
$\Rightarrow Q = \alpha \left[ {\dfrac{{{x^2}}}{2}} \right]_0^L$
$\Rightarrow Q = \dfrac{{\alpha {L^2}}}{2}$

Hence option B is the correct answer.

Note: Surface Charge Density is the ratio of total charge on a plane sheet to total area of sheet. And volume charge density is the ratio of total charge present on a body to total volume of the body. All the charge densities are born from one theorem and Gauss' theorem.