Question

A charge of magnitude $Q$ is placed at the origin. A second charge of magnitude $2Q$ is placed at the position $x=d$ along the x-axis.
Other than infinitely far away, at what position on the x-axis will a positive test charge experience a zero net force?

• A. $x=\frac{d}{\sqrt{2}}$
• B. $x=\frac{d}{2\sqrt{2}}$
• C. $x=\frac{d}{3}$
• D. $x=\frac{d}{1+\sqrt{2}}$
• E. At no position other than infinitely far away will the positive test charge experience zero net force.

Answer 1

The correct option is D $x=\frac{d}{1+\sqrt{2}}$

Let the electric field $E$ is zero at a point $P$ between the two given charges , at a distance $x$ from the charge $Q$ which is at the origin .

at point $P$, electric field due to $Q$ at origin

$E_1=KQ/x^2$

at point $P$, electric field due to $2Q$ which is at a distance $d$ from $Q$ at origin

$E_2=2KQ/(d-x{)}^2$

as $E_1$ and $E_2$ are in opposite direction due to two positive charges and electric field is zero at point $P$ , therefore

$E_1=E_2$

or $KQ/x^2=2KQ/(d-x{)}^2$

or $2x^2=(d-x{)}^2$

or $x=d/(1+\sqrt{2})$ or $d/\sqrt{2}-1$

but $x=d/\sqrt{2}-1$ is not possible as point $P$ is in between the two charges .

now if we put a positive test charge q at point $P$ , force on it

$F=qE=q\times 0=0$

option D is correct.