Three electrons form an equilateral triangle 1.0 nm on each side. A proton is at the center of the triangle. What is the potential energy of this group of charges?

Verified step by step guidance

1. Identify the charges involved: Each electron has a charge of \(-e = -1.6 \times 10^{-19} \; C\), and the proton has a charge of \(+e = 1.6 \times 10^{-19} \; C\). The system consists of three electrons and one proton.
2. Determine the distances between charges: The electrons form an equilateral triangle with side length \(1.0 \; \text{nm} = 1.0 \times 10^{-9} \; \text{m}\). The proton is located at the center of the triangle. The distance from the center to each vertex (electron) is the circumradius of the triangle, given by \(r = \frac{a}{\sqrt{3}}\), where \(a\) is the side length of the triangle.
3. Calculate the potential energy contributions: The total potential energy of the system is the sum of the potential energy between all pairs of charges. Use the formula for potential energy between two charges: \(U = \frac{k q_1 q_2}{r}\), where \(k = 8.99 \times 10^9 \; \text{N·m}^2/\text{C}^2\) is Coulomb's constant, \(q_1\) and \(q_2\) are the charges, and \(r\) is the distance between them.
4. Break the total potential energy into components: (1) The potential energy between each pair of electrons (three pairs in total), and (2) the potential energy between the proton and each electron (three interactions). For the electron-electron pairs, the distance is \(1.0 \; \text{nm}\). For the proton-electron interactions, the distance is \(\frac{1.0 \; \text{nm}}{\sqrt{3}}\).
5. Sum all contributions: Add the potential energy from the three electron-electron pairs and the three proton-electron interactions to find the total potential energy of the system. Be mindful of the signs of the charges (electron-electron interactions are repulsive, while proton-electron interactions are attractive).

Key Concepts

Coulomb's Law

Coulomb's Law describes the electrostatic force between two charged particles. It states that the 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 principle is essential for calculating the interactions between the electrons and the proton in the given configuration.

Electric Potential Energy

Electric potential energy is the energy a charged particle possesses due to its position in an electric field. It is calculated based on the configuration of charges and their distances from one another. In this scenario, the potential energy of the system can be determined by summing the potential energies between each pair of charges, considering both the electrons and the proton.

Equilateral Triangle Geometry

The geometry of an equilateral triangle is crucial for determining the distances between the charges. In this case, all sides are equal, and the angles are each 60 degrees. Understanding this geometry allows for accurate calculations of the distances between the electrons and the proton, which is necessary for applying Coulomb's Law and calculating the total potential energy of the system.