Four electric charges $+\mathrm{q},+\mathrm{q},-\mathrm{q}$ and $-\mathrm{q}$ are placed in order at the…

Four electric charges $+\mathrm{q},+\mathrm{q},-\mathrm{q}$ and $-\mathrm{q}$ are placed in order at the corners of a square of side $2 \mathrm{~L}$. The electric potential at point midway between the two positive charges is
  1. $\frac{1}{4 \pi \epsilon_0} \frac{2 \mathrm{q}}{\mathrm{L}}(1-\sqrt{5})$
  2. zero
  3. $\frac{1}{4 \pi \epsilon_0} \frac{2 \mathrm{q}}{\mathrm{L}}\left(1+\frac{1}{\sqrt{5}}\right)$
  4. $\frac{1}{4 \pi \in_0} \frac{2 \mathrm{q}}{\mathrm{L}}\left(1-\frac{1}{\sqrt{5}}\right)$

Solution

$\begin{aligned} & \mathrm{DP}=\sqrt{5} \mathrm{~L}=\mathrm{CP} \\ & \mathrm{AP}=\mathrm{BP}=\mathrm{L} \end{aligned}$ Potential at point $\mathrm{P}$ due to the four charges $\begin{aligned} & \mathrm{V}=\frac{1}{4 \pi \varepsilon_0}\left(\frac{\mathrm{q}}{\mathrm{L}}+\frac{\mathrm{q}}{\mathrm{L}}-\frac{\mathrm{q}}{\sqrt{5} \mathrm{~L}}-\frac{\mathrm{q}}{\sqrt{5} \mathrm{~L}}\right) \\ & =\frac{1}{4 \pi \varepsilon_0}\left(\frac{2 \mathrm{q}}{\mathrm{L}}-\frac{2 \mathrm{q}}{\sqrt{5} \mathrm{~L}}\right) \\ & =\frac{1}{4 \pi \varepsilon_0} \cdot \frac{2 \mathrm{q}}{\mathrm{L}}\left(1-\frac{1}{\sqrt{5}}\right) \end{aligned}$ :

Asked in: MHT CET 2021 (24 Sep Shift 2)

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