You place 4 charges of equal magnitude Q at the corners of a square of side length L, such that two of the charges are negative and two are...

Friday, September 4, 2015

You place 4 charges of equal magnitude Q at the corners of a square of side length L, such that two of the charges are negative and two are...

Hello!

The electric potential of a point charge $\displaystyle{q}$ at some another point is equal to $\displaystyle\frac{{1}}{{{4}\pi\epsilon_{{0}}}}\frac{{q}}{{r}},$ where $\displaystyle{r}$ is the distance from the point to the charge and $\displaystyle\epsilon_{{0}}$ is an absolute constant (the permittivity of vacuum). Note that $\displaystyle{q}$ is a signed value.

Also, it is known that the electric potential of point charges is additive, i.e. the potential of a system of charges is equal to the sum of the point's potentials. So we need to compute $\displaystyle{4}$ potentials and sum them.

The distance $\displaystyle{r}$ is the same for all four charges, and it is $\displaystyle\frac{{L}}{\sqrt{{{2}}}}.$ The magnitudes of the charges are also the same, $\displaystyle{Q}.$ Thus the sum is

$\displaystyle\frac{{1}}{{{4}\pi\epsilon_{{0}}}}\frac{{Q}}{{r}}{\left({1}+{1}-{1}-{1}\right)}.$

$\displaystyle+{1}$ is for positive charges, $\displaystyle-{1}$ is for negative.

We see that the result is zero, and it doesn't depend even on the rearrangement of the charges.

Ninenotes

Friday, September 4, 2015

You place 4 charges of equal magnitude Q at the corners of a square of side length L, such that two of the charges are negative and two are...

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