Potential at the Center of a Square

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Electrostatics Intermediate Electric Potential

Source: High school physics (Chinese)

Problem Sets:

Electrostatics

Problem

In air, point charges are placed at the four corners of a square with side length $a$. Their charges are $q$, $2q$, $3q$, and $-4q$, respectively.

Find the electric potential at the center of the square.

[Q1] $U = \frac{2\sqrt{2}kq}{a}$

Electric potential is a scalar quantity. The total potential $U$ at the center is the algebraic sum of the potentials from each corner charge. The distance $r$ from any corner to the center of a square with side $a$ is half the diagonal length.

$$d = \sqrt{a^2 + a^2} = a\sqrt{2}$$ $$r = \frac{d}{2} = \frac{a\sqrt{2}}{2}$$

The total potential $U$ is:

$$U = \sum_{i=1}^{4} \frac{kq_i}{r} = \frac{k}{r}(q_1 + q_2 + q_3 + q_4)$$ $$U = \frac{k}{a\sqrt{2}/2}(q + 2q + 3q - 4q) = \frac{2k}{a\sqrt{2}}(2q) = \frac{4kq}{a\sqrt{2}}$$ $$U = \frac{2\sqrt{2}kq}{a}$$