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(B) Let the point where the electric field is zero be at a distance $x$ from the origin on the negative $X$-axis. Let this distance be $d = |x|$.The e

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$x = -\sqrt{2} a$

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(B) Let the point where the electric field is zero be at a distance $x$ from the origin on the negative $X$-axis. Let this distance be $d = |x|$.The electric field due to charge $+Q$ at distance $a$ from the origin at point $P$ (distance $d$ from origin) is $E_1 = \frac{kQ}{(a+d)^2}$ directed towards the left.The electric field due to charge $-2Q$ at distance $2a$ from the origin at point $P$ is $E_2 = \frac{k(2Q)}{(2a+d)^2}$ directed towards the right.For the net electric field to be zero,$E_1 = E_2$.$\frac{kQ}{(a+d)^2} = \frac{2kQ}{(2a+d)^2}$$\frac{1}{(a+d)^2} = \frac{2}{(2a+d)^2}$Taking the square root on both sides: $\frac{1}{a+d} = \frac{\sqrt{2}}{2a+d}$$2a + d = \sqrt{2}a + \sqrt{2}d$$d(\sqrt{2} - 1) = a(2 - \sqrt{2})$$d = a \frac{\sqrt{2}(\sqrt{2} - 1)}{\sqrt{2} - 1} = \sqrt{2}a$Since the point is on the negative $X$-axis,the coordinate is $x = -\sqrt{2}a$.

Two point charges $(+Q)$ and $(-2Q)$ are placed at distances $a$ and $2a$ respectively from the origin on the $X$-axis. At what distance from the origin is the electric field zero?

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