Problem 8 - University Physics with Modern Physics

Question

Three equal \(1.20-\mu \mathrm{C}\) point charges are placed at the corners of an equilateral triangle whose sides are 0.500 \(\mathrm{m}\) long. What is the potential energy of the system? (Take as zero the potential energy of the three charges when they are infinitely far apart.)

Step-by-Step Solution

Answer

The potential energy of the system is approximately \( 7.77 \times 10^{-2} \ \mathrm{J} \).

Step 1: Understand the concept

The potential energy of a system of point charges is calculated based on the pairwise interactions between them. For each pair of charges, we use the formula for potential energy: \[ U = k \frac{q_1 q_2}{r} \]where \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance between the charges.

Step 2: Write down the knowns

We know the following values:- Each charge, \( q = 1.20 \times 10^{-6} \ \mathrm{C} \)- Distance between each pair, \( r = 0.500 \ \mathrm{m} \)- Coulomb's constant, \( k = 8.99 \times 10^9 \ \mathrm{N} \cdot \mathrm{m}^2/\mathrm{C}^2 \)

Step 3: Calculate the potential energy for one pair

Using the potential energy formula for one pair:\[ U_{pair} = k \frac{(1.20 \times 10^{-6})(1.20 \times 10^{-6})}{0.500} \] Compute: \[ U_{pair} = (8.99 \times 10^9) \frac{1.44 \times 10^{-12}}{0.500} \approx 2.59 \times 10^{-2} \ \mathrm{J} \]

Step 4: Determine the total potential energy of the system

Since the triangle is equilateral, we have three pairs of equal charges. Therefore, the total potential energy \( U \) is three times the potential energy of one pair:\[ U = 3 \times U_{pair} = 3 \times 2.59 \times 10^{-2} \] Calculate:\[ U \approx 7.77 \times 10^{-2} \ \mathrm{J} \]

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in electrostatics that describes the force between two point charges. This law tells us how charges attract or repel each other with a force. The magnitude of this force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.
This can be expressed as:\[ F = k \frac{|q_1 q_2|}{r^2} \]where:

• \( F \) is the magnitude of the force between the charges.
• \( k \) is Coulomb's constant \((8.99 \times 10^9 \ \mathrm{N} \cdot \mathrm{m}^2/\mathrm{C}^2)\).
• \( q_1 \) and \( q_2 \) are the magnitudes of the two charges.
• \( r \) is the distance between the two charges.

Coulomb's Law helps us understand interactions between electrical charges. In the problem, it also forms the cornerstone to calculate potential energy between charges. By knowing how much work it takes to bring charges from infinity to a specific configuration, we can then find potential energy using a modified version of the formula by removing the square from \( r \).

Point Charges

Point charges are idealized charges located at a single point in space. This concept simplifies calculations in electrostatics, as we can ignore any physical size or shape. In real life, charges are often spread out over a volume; however, considering them as point charges lets us focus solely on their effects without complex geometry.
Each of the three charges in our problem is treated as a point charge, with a charge of \(1.20 \mu \mathrm{C} (1.20 \times 10^{-6} \, \mathrm{C})\). By using point charges, we apply Coulomb's Law to evaluate the potential energy of their configuration easily.
Point charges also help us ignore mutual distances. No matter where each point charge is within our equilateral triangle, the interactions are simply calculated using their distances, ensuring the method holds true even when dealing with complex arrangements.

Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are equal in length and all three angles are equal, each measuring 60 degrees. This symmetry is helpful in calculating the potential energy in our setup as it suggests that the interactions between each pair of charges will be identical.
When considering electric potential energy in an equilateral triangle, you can assume that each charge sits at a vertex and that each side, which is the distance between charges, is equal. In our case, each side length is given as 0.500 meters.
This symmetry means that for each pair of charges, the calculation of potential energy using \( U = k \frac{q_1 q_2}{r} \) will yield the same result, simplifying our task to multiplying the potential energy found for one pair by three, since there are three identical pairs of charges in the triangle. This clarity significantly reduces the complexity of our analysis, allowing us to find the total potential energy with simple arithmetic after calculating a single pair's contribution.