Problem 24 - Physics

Question

There are four charges, each with a magnitude of \(2.0 \mu \mathrm{C}\). Two are positive and two are negative. The charges are fixed to the corners of a 0.30 -m square, one to a corner, in such a way that the net force on any charge is directed toward the center of the square. Find the magnitude of the net electrostatic force experienced by any charge.

Step-by-Step Solution

Answer

Answer

Step 1: Understanding the Problem

We have a square with sides of length \(0.30 \text{ m}\) and charges of \(2.0 \mu \text{C}\) placed at each corner. Two are positive and two are negative. The task is to find the net electrostatic force on any of the charges, noting that symmetry ensures the force is directed toward the center.

Step 2: Identifying Pair Interactions

Each charge interacts with the three other charges. Use Coulomb's Law, \( F = k \frac{|q_1 q_2|}{r^2}\), to find the force between pairs, where \(k = 8.99 \times 10^9 \text{ Nm}^2/ ext{C}^2\) and \(q_1 = q_2 = 2.0 \mu\text{C} = 2.0 \times 10^{-6}\text{ C} \).

Key Concepts

Coulomb's Law

Coulomb's Law is essential for understanding forces between electric charges. It involves a simple formula that tells us how strong the force is between two charges. The law is expressed as \( F = k \frac{|q_1 q_2|}{r^2} \). Let's break this down:
- \( F \) is the magnitude of the force between the charges.
- \( k \) is the Coulomb's constant, which is approximately \( 8.99 \times 10^9 \text{ Nm}^2/\text{C}^2 \).
- \( q_1 \) and \( q_2 \) are the magnitudes of the charges in coulombs.
- \( r \) is the distance between the two charges.
This law shows that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This implies that as two charges are closer, the force between them increases significantly. Conversely, as the distance increases, the force diminishes. Understanding this proportionality helps in calculating and predicting the behavior of charged particles.

Square Configuration of Charges

Visualizing a square configuration of charges helps simplify complex force interactions. At each corner of a square, a charge is placed. In this scenario, two charges are positive while the other two are negative. This set-up means:
- The distance between adjacent corners is the side length of the square, 0.30 m.
- Diagonal interactions occur over a distance derived from the Pythagorean theorem, resulting in \( 0.30 \sqrt{2} \) m for corners across the square.
This arrangement allows us to systematically examine the forces acting on each charge. Each charge at a corner will experience forces from its three neighboring charges. By calculating these forces using Coulomb's Law, one can derive how these charges influence each other. This particular configuration, due to its symmetry, ensures the net force on each charge points towards the center of the square.

Symmetry in Electrostatics

Symmetry plays a critical role in electrostatics, especially in arrangements like a square configuration of charges. Symmetry simplifies complex calculations by reducing the number of unique interactions that must be considered. For our square, symmetry ensures:
- Each charge experiences equal magnitude forces from opposite direction pairs.
- The net force on each charge is directed inward towards the center of the square due to the symmetry of opposing forces.
In this configuration, the forces from opposite charges cancel out some components and augment others, directing all forces towards the center. This means you don't have to individually solve for each force in three dimensions. Instead, you can rely on symmetry to simplify calculations and predict force directions. Symmetry also helps in visualizing the electrostatic landscape, making it easier to understand these interactions conceptually.