Question

The three charges $\frac{q}{2},q$ and $\frac{q}{2}$ are placed at the corners $A,B\text{and}C$ of a square of side 'a' as shown in figure. The magnitude of electric field $\left(E\right)$ at the corner $D$ of the square is

• A. $\frac{q}{4\pi {ϵ}_0{a}^2}(1-\frac{1}{\sqrt{2}})$
• B. $\frac{q}{4\pi {ϵ}_0{a}^2}(\frac{1}{\sqrt{2}}+\frac{1}{2})$
• C. $\frac{q}{4\pi {ϵ}_0{a}^2}(\frac{1}{\sqrt{2}}-\frac{1}{2})$
• D. $\frac{q}{4\pi {ϵ}_0{a}^2}(1+\frac{1}{\sqrt{2}})$

Answer 1

The correct option is B $\frac{q}{4\pi {ϵ}_0{a}^2}(\frac{1}{\sqrt{2}}+\frac{1}{2})$

Overall $EF$ can be written as:
$|E_0|=\frac{kq/2}{a^2}\sqrt{2}+\frac{kq}{(a\sqrt{2}{)}^2}$
$=\frac{kq}{\sqrt{2}{a}^2}+\frac{kq}{(a\sqrt{2}{)}^2}$
$=\frac{kq}{\sqrt{2}{a}^2}+\frac{kq}{2a^2}$
$=\frac{kq}{a^2}(\frac{1}{\sqrt{2}}+\frac{1}{2})(\text{Here}k=\frac{1}{4\pi {\epsilon }_0})$
$\Rightarrow$ Option $A$ is correct