P19R2017 – Square Configuration of Point Charges

For convenience, we start with the reminder that the diagonal of a square of side length $d$ has length $d\sqrt{2}$. Thus, the center of the square above is located at a distance of $d\sqrt{2}/2$ from each point charge.

The charges $q_1$ and $q_2$ are positive and therefore the fields ${\overrightarrow{E}}_1$ and ${\overrightarrow{E}}_2$ are directed radially outward as shown.

The charges $q_3$ and $q_4$ are negative and therefore the fields ${\overrightarrow{E}}_3$ and ${\overrightarrow{E}}_4$ are directed radially inward as shown.

Note: we drew the fields ${\overrightarrow{E}}_3$ and ${\overrightarrow{E}}_4$ slightly shorter than ${\overrightarrow{E}}_1$ and ${\overrightarrow{E}}_2$ to be able to show all four electric fields in the figure (which will be useful in the next question).

The four charges have the same magnitude $q$ and are located the same distance from the center of the square. Consequently, the four electric fields have the same magnitude, and we write

E_1=E_2=E_3=E_4=\frac{kq}{{\left(d\sqrt{2}/2\right)}^2}=\frac{2kq}{d^2}

In addition, by symmetry, we conclude that the net electric field at the center of the square is vertical, directed downward. Thus, the $x$-component of the net electric field is equal to

\boxed{E_{net\ x}=0}

while the $y$-component of the net electric field is given by

\begin{aligned}
E_{net\ y}&=E_{1y}+E_{2y}+E_{3y}+E_{4y} \\
\\
&=E_1{\mathrm{cos} \left(\theta \right)\ }+E_2{\mathrm{cos} \left(\theta \right)\ }+E_3{\mathrm{cos} \left(\theta \right)\ }+E_4{\mathrm{cos} \left(\theta \right)\ } \\
\\
&=4E_1{\mathrm{cos} \left(\theta \right)\ } \\
\\
&=\frac{8kq}{d^2}{\mathrm{cos} \left(\theta \right)\ }
\end{aligned}

Thus, we conclude that the net electric field at the center of the square is directed downward and has a magnitude equal to

\boxed{E_{net}=\frac{8kq}{d^2}{\mathrm{cos} \left(\theta \right)\ }=\frac{4\sqrt{2}kq}{d^2}}

If the charges $q_3$ and $q_4$ are now positive, with the same magnitude and $q_1$ and $q_2$, then we conclude that the electric fields at the center of the square are directed radially outward with respect to the point charge that created them, as shown in the figure below.

Thus, by symmetry, we conclude that the net electric field at the center of the square is then zero and we write

\boxed{{\overrightarrow{E}}_{net}=\overrightarrow{0}}