Given four point charges placed at the vertices of a square of side length aaa, with two adjacent vertices having charge +Q+Q+Q and the other two having charge −Q-Q−Q, the magnitude of the electric field at the center of the square can be found as follows:

Step 1: Distance from each charge to the center

The center of the square is equidistant from each vertex. The distance rrr from a vertex to the center is the half-diagonal of the square:

r=a2r=\frac{a}{\sqrt{2}}r=2​a​

Step 2: Electric field due to a single charge at the center

The magnitude of the electric field at the center due to a single point charge QQQ is given by Coulomb's law:

E=k∣Q∣r2=k∣Q∣(a/2)2=k∣Q∣a2/2=2k∣Q∣a2E=\frac{k|Q|}{r^2}=\frac{k|Q|}{(a/\sqrt{2})^2}=\frac{k|Q|}{a^2/2}=\frac{2k|Q|}{a^2}E=r2k∣Q∣​=(a/2​)2k∣Q∣​=a2/2k∣Q∣​=a22k∣Q∣​

where k=14πϵ0k=\frac{1}{4\pi \epsilon_0}k=4πϵ0​1​ is Coulomb's constant.

Step 3: Direction and superposition

• The electric field due to a positive charge points away from the charge.
• The electric field due to a negative charge points toward the charge.

By symmetry and vector addition, the horizontal components of the electric fields from charges on opposite sides cancel out, while the vertical components add up.

Step 4: Resulting magnitude of the net electric field

Using the vector components and superposition, the net electric field magnitude at the center is:

Enet=4kQa2E_{\text{net}}=\frac{4kQ}{a^2}Enet​=a24kQ​

This result comes from the fact that the contributions from the four charges add vectorially to give a net field of this magnitude at the center of the square when two adjacent corners have +Q+Q+Q and the other two have −Q-Q−Q charges

. Summary:

Parameter| Expression
---|---
Distance from charge to center| r=a2r=\frac{a}{\sqrt{2}}r=2​a​
Electric field from one charge| E=2kQa2E=\frac{2kQ}{a^2}E=a22kQ​
Net electric field magnitude| Enet=4kQa2E_{\text{net}}=\frac{4kQ}{a^2}Enet​=a24kQ​

Thus, the magnitude of the electric field at the center of the square is:

Enet=4kQa2\boxed{E_{\text{net}}=\frac{4kQ}{a^2}}Enet​=a24kQ​​

where k=14πϵ0k=\frac{1}{4\pi \epsilon_0}k=4πϵ0​1​ and aaa is the side length of the square