Question

A line of charge extends along a X-axis whose linear charge density varies directly as x. Imagine a spherical volume with its centre located on X-axis and is moving gradually along it. Which of the graphs shown in figure correspond to the flux with the x coordinate of the centre of the volume

• A. a
• B. b
• C. c
• D. d

Answer 1

The correct option is C c

Given that linar charge density $\lambda$ varies directly to x

$\Rightarrow \lambda \alpha x$

$=\lambda =kx$

k = proportanality constant

electric field due to long straight wire

r = distance of point

$E=\frac{\lambda }{2\pi {\epsilon }_{0}r}$

initially when sphere with its centre at x-axis has flux $=\epsilon .A$

$\varphi =\frac{\lambda }{2\pi {\epsilon }_{0}r}.4\pi r^2=2\lambda r$

as radius is constant

$=\varphi \alpha \lambda$ & $\lambda \alpha x$

$\Rightarrow \varphi \alpha x$

$\Rightarrow$ flux is $\alpha x$