GATE 2024 EE – Question 51
In the $(x,y,z)$ coordinate system, three point-charges $Q$, $Q$, and $\alpha Q$ are located in free space at $(-1,0,0)$, $(1,0,0)$, and $(0,-1,0)$, respectively. The value of $\alpha$ for the electric field to be zero at $(0,0.5,0)$ is ______________ (rounded off to 1 decimal place).
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Correct answer: -1.61 to -1.59
Explanation
At $(0,0.5,0)$ the $x$-components of the fields of the two $Q$ charges cancel. Each $Q$ is at distance $\sqrt{1.25}$ and contributes a $y$-component $\dfrac{kQ}{1.25}\cdot\dfrac{0.5}{\sqrt{1.25}}=0.358\,kQ$, so together $0.716\,kQ$ (along $+y$, away from the charges). The charge $\alpha Q$ is 1.5 m below, contributing $\dfrac{k\alpha Q}{1.5^2}=0.444\,k\alpha Q$ along $+y$ if positive. Zero field needs $0.716+0.444\alpha=0$, so $\alpha=-1.6$.