Question

An infinite line of charge with uniform line charge density of $1C/m$ is palced along the $y$ -axis. A charge of $1C$ is placed on the $x$ -axis at a distance of $d=3m$ from the origin. At what distance $r$ from the origin on the $x$ -axis, the total electric field is zero. (Assume, $0<r<d$ )

• A. 1 m
• B. 2 m
• C. 2.5 m
• D. 1.75 m

Answer 1

Answer: (B)

An infinite line of uniform charge density is placed along $y$ -axis and a charge $(+1C)$ is placed at $x=+3m$, as shown in the figure. For electric field to be zero at $x$, there should be
$E_{\text{line charge }}=E_{\text{point charge }}$ ...(i)
As we know, for line charge
$E_{\text{line charge }}=\frac{2k\lambda }{x}=\frac{2k}{x}$ ...(ii)
Similarly, for point charge,
$E_{\text{point charge }}=\frac{kq}{(3-x{)}^2}=\frac{k}{(3-x{)}^2}$ ...(iii)
From Eqs. (i), (ii) and (iii), we get
$\frac{2k}{x}=\frac{k}{(3-x{)}^2}$
$\Rightarrow \frac{2}{x}=\frac{1}{(3-x{)}^2}$
$\Rightarrow 2x^2-13x+18=0$
This is a quadratic equation, which have solution
$x_1,x_2=\frac{13\pm \sqrt{13^2-144}}{4}$
So, $x_1=4.5m$ (not possible, as $0<x<3$ ) and $x_2=2m$