Question

For the two finite line charges shown in figure, find the point on the axial line where net electric field is zero.

• A. $1.7\text{m}$ from $\text{A}$
• B. $2.7\text{m}$ from $\text{A}$
• C. $3.7\text{m}$ from $\text{A}$
• D. $4.7\text{m}$ from $\text{A}$

Answer 1

The correct option is B $2.7\text{m}$ from $\text{A}$

Let the electric field (net) be zero at point P at distance $x$ from A.


Electric field at P due to finite line charge on left
$E=k\lambda [\frac{1}{r}-\frac{1}{r+L}]$
$\Rightarrow E=9\times {10}^9\times 10\times {10}^{-9}\times [\frac{1}{x}-\frac{1}{4+x}]----\left(1\right)$

Electric field at P due to finite line charge on right.
$E=9\times {10}^9\times 20\times {10}^{-9}[\frac{1}{6-x}-\frac{1}{6-x+2}]-----\left(2\right)$

From (1) and (2), for $E_{net}=0$
$(\frac{1}{x}-\frac{1}{4+x})=2(\frac{1}{6-x}-\frac{1}{8-x})$
$\Rightarrow \frac{4+x-x}{x(4+x)}=2\left(\frac{8-x-6+x}{(6-x)(8-x)}\right)$
$\Rightarrow \frac{1}{4x+x^2}=\frac{1}{48-6x-8x+x^2}$
$\Rightarrow x^2-14x+48=4x+x^2$
$\Rightarrow 18x=48$
$\Rightarrow x=2.7\text{m}$ [from A]