Four positive point charges $+q$ are kept at the four corners of a square of side $l$. The net electric field at the mid-point of any one side of the square is $ \left( ext { Take, } \frac{1}{4 \pi \varepsilon_0}=k ight) $

Question

Four positive point charges $+q$ are kept at the four corners of a square of side $l$. The net electric field at the mid-point of any one side of the square is $ \left(	ext { Take, } \frac{1}{4 \pi \varepsilon_0}=kight) $

Aicharya · Accepted Answer

The force at midpoint of any side of a square let's say at point $P$ which is midpoint of side $A B$ is due to charges at corners $A, B, C$ and $D$ $ F_P=F_{A P}+F_{B P}+F_{C P}+F_{D P} $

$\begin{aligned} & \mathbf{A P}=\mathbf{D P}=l \\ & \mathbf{B P}=\mathbf{C P}=\sqrt{l^2+(l / 2)^2}=\sqrt{5 l^2 / 4}\end{aligned}$ Electric field due to charges at $A$ and $D$ will be equal in magnitude and opposite in direction, so will cancel each other. Similarly, the vertical components of electric field due to charges at $B$ and $C$ will also cancel each other, but the horizontal components of these will add up to give resultant electric field. $ E=2 \cdot \frac{k q}{\left(\sqrt{5 l^2 / 4}\right)^2} \cdot \cos \theta=2 \times \frac{4 k q}{5 l^2} \times \frac{2}{5}=\frac{16 k q}{5 \sqrt{5} l^2} $

Four positive point charges $+q$ are kept at the four corners of a square of side $l$. The net electric…

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