Four electric charges $+q,+q,-q$ and $-q$ are placed in order at the corners of a square of side 2 L . The…

Four electric charges $+q,+q,-q$ and $-q$ are placed in order at the corners of a square of side 2 L . The electric potential at point midway between the two positive charges is
  1. $\frac{1}{4 \pi \mathrm{E}_0} \frac{2 \mathrm{q}}{\mathrm{L}}\left(1+\frac{1}{\sqrt{5}}\right)$
  2. $\frac{1}{4 \pi \mathrm{E}_0} \frac{2 \mathrm{q}}{\mathrm{L}}\left(1-\frac{1}{\sqrt{5}}\right)$
  3. $\frac{1}{4 \pi \mathrm{E}_0} \frac{2 \mathrm{q}}{\mathrm{L}}(1-\sqrt{5})$
  4. $\frac{1}{4 \pi \mathrm{E}_0} \frac{2 \mathrm{q}}{\mathrm{L}}(1+\sqrt{5})$

Solution


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

Asked in: MHT CET 2024 (10 May Shift 1)

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