Four identical point charges, each of charge $q$, are placed at the corners of a square of side $a$. What is the magnitude of the net electric force experienced by one of the charges due to the other three?

• A. $2$$\frac{1}{4\pi \epsilon 0_{}}$$\frac{q^2}{a^2}$
• B. $3$$\frac{1}{4\pi \epsilon 0_{}}$$\frac{q^2}{a^2}$
• C. $2+\sqrt{2}$$\frac{1}{4\pi \epsilon 0_{}}$$\frac{q^2}{a^2}$
• D. $\sqrt{2}$$\frac{1}{4\pi \epsilon 0_{}}$$\frac{q^2}{a^2}$

Answer: (C)

Verified step by step guidance

1. Identify the charges and their positions: Four identical point charges, each with charge \(q\), are placed at the corners of a square with side length \(a\). We want to find the net force on one charge due to the other three.
2. Calculate the forces due to the two adjacent charges: Each adjacent charge is a distance \(a\) away. Use Coulomb's law for the magnitude of the force between two point charges: \[F = \frac{1}{4\pi\epsilon_0} \frac{q^2}{a^2}\] Since there are two adjacent charges, calculate the forces from each and note their directions (along the sides of the square).
3. Calculate the force due to the diagonally opposite charge: The diagonal distance between opposite corners of the square is \(a\sqrt{2}\). Use Coulomb's law again: \[F_{diag} = \frac{1}{4\pi\epsilon_0} \frac{q^2}{(a\sqrt{2})^2} = \frac{1}{4\pi\epsilon_0} \frac{q^2}{2a^2}\] This force acts along the diagonal direction.
4. Resolve the forces into components: Choose a coordinate system with the charge experiencing the force at the origin. The two adjacent forces act along the x and y axes, while the diagonal force acts at a 45° angle. Break the diagonal force into x and y components using trigonometry: \[F_{diag,x} = F_{diag} \cos 45^\circ, \quad F_{diag,y} = F_{diag} \sin 45^\circ\]
5. Sum the components of all forces: Add the x-components of the two adjacent forces and the diagonal force to get the total x-component of the net force. Do the same for the y-components. Then, find the magnitude of the net force using the Pythagorean theorem: \[F_{net} = \sqrt{(F_x)^2 + (F_y)^2}\]