A point charge is placed at each corner of a square with side length a. All charges have magnitude q. Two of the charges are positive and two are negative (Fig. E21.42). What is the direction of the net electric field at the center of the square due to the four charges, and what is its magnitude in terms of q and a?

Verified step by step guidance

1. Identify the configuration of the charges: Place the charges at the corners of the square such that opposite corners have charges of the same sign. This means if the top-left and bottom-right corners have positive charges, the top-right and bottom-left corners will have negative charges.
2. Determine the position of the center of the square: The center of the square is equidistant from all four corners. If the side length of the square is 'a', the distance from the center to any corner is given by \( \frac{a\sqrt{2}}{2} \).
3. Calculate the electric field due to a single charge at the center: The electric field \( E \) due to a point charge \( q \) at a distance \( r \) is given by \( E = \frac{kq}{r^2} \), where \( k \) is Coulomb's constant. Substitute \( r = \frac{a\sqrt{2}}{2} \) to find the electric field due to one charge.
4. Analyze the vector nature of the electric fields: The electric fields due to the positive charges will point away from the charges, while those due to the negative charges will point towards the charges. Use vector addition to determine the net electric field at the center. The fields from opposite charges will cancel each other out in one direction, leaving a net field in the perpendicular direction.
5. Calculate the magnitude of the net electric field: Since the electric fields from opposite charges cancel in one direction, the net electric field is the vector sum of the remaining components. Use symmetry and vector addition to find the magnitude of the net electric field in terms of \( q \) and \( a \).

Key Concepts

Electric Field

The electric field is a vector field around a charged particle that represents the force exerted on other charges. It is defined as the force per unit charge and is directed away from positive charges and towards negative charges. The superposition principle allows us to calculate the net electric field by vectorially adding the fields due to individual charges.

Superposition Principle

The superposition principle states that the net electric field created by multiple charges is the vector sum of the electric fields produced by each charge independently. This principle is crucial for calculating the net field at a point when multiple charges are involved, as it allows us to consider each charge's contribution separately and then combine them.

Symmetry in Electric Fields

Symmetry plays a significant role in simplifying electric field calculations. In this problem, the square's symmetry and the arrangement of charges allow us to predict the direction of the net electric field. The symmetry suggests that certain components of the electric fields from opposite charges will cancel out, simplifying the determination of the net field's direction and magnitude.