Question

A charge of magnitude Q is placed at the origin. A second charge of magnitude 4Q 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. $x=-d$
• B. $x=-\frac{d}{2}$
• C. $x=\frac{d}{3}$
• D. $x=\frac{3d}{2}$
• E. At no position other than infinitely far away will the positive test charge experience zero net force

Answer 1

The correct option is C $x=\frac{d}{3}$

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 $4Q$ which is at a distance $d$ from $Q$ at origin

$E_2=4KQ/(d-x{)}^2$

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 $KQ/x^2=4KQ/(d-x{)}^2$

or $4x^2=(d-x{)}^2$

or $x=d/3$ or $-d$

but $x=-d$ , it means point $P$ is not in between the two given charges of $Q$ and $4Q$ so it is not possible as both of the charges will repel it in same direction, so the net force will not be zero.

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

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