The electric charges +2q, +2q, -2q and -2q ar | vedclass

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Question

(B) The electric potential $V$ at a point due to a charge $q$ at a distance $r$ is given by $V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$.Point $A$ is at

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$\frac{q}{\pi \epsilon_0 L}\left[1-\frac{1}{\sqrt{5}}\right]$

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(B) The electric potential $V$ at a point due to a charge $q$ at a distance $r$ is given by $V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$.Point $A$ is at the midpoint of the side with charges $+2q$ and $+2q$. The distance of $A$ from each of these charges is $L$.The distance of $A$ from each of the charges $-2q$ at the opposite corners can be found using the Pythagorean theorem. The horizontal distance is $2L$ and the vertical distance is $L$, so the distance $r = \sqrt{(2L)^2 + L^2} = \sqrt{5L^2} = L\sqrt{5}$.The total potential at $A$ is the sum of potentials due to all four charges:$V_A = \frac{1}{4\pi\epsilon_0} \left[ \frac{2q}{L} + \frac{2q}{L} + \frac{-2q}{L\sqrt{5}} + \frac{-2q}{L\sqrt{5}} \right]$$V_A = \frac{1}{4\pi\epsilon_0} \left[ \frac{4q}{L} - \frac{4q}{L\sqrt{5}} \right]$$V_A = \frac{4q}{4\pi\epsilon_0 L} \left[ 1 - \frac{1}{\sqrt{5}} \right] = \frac{q}{\pi\epsilon_0 L} \left[ 1 - \frac{1}{\sqrt{5}} \right]$.

The electric charges $+2q$, $+2q$, $-2q$ and $-2q$ are placed at the corners of a square of side $2L$ as shown in the figure. The electric potential at point $A$, midway between the two charges $+2q$ and $+2q$, is $(\epsilon_0 = \text{permittivity of free space})$

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