Three charged particles are placed at the corners of an equilateral triangle that has edge length 2.0 cm. One particle has charge +3.0 nC and a second has charge +6.0 nC. What is the third charge if the electric potential energy of the three charged particles is zero?

Verified step by step guidance

1. Understand the problem: The electric potential energy of a system of three charges is given by the sum of the potential energy contributions from each pair of charges. The goal is to find the value of the third charge such that the total electric potential energy of the system is zero.
2. Write the formula for the electric potential energy between two charges: \( U = \frac{k \cdot q_1 \cdot q_2}{r} \), where \( k \) is Coulomb's constant (\( 8.99 \times 10^9 \ \text{N·m}^2/\text{C}^2 \)), \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between them.
3. Set up the total electric potential energy equation for the system: \( U_{total} = U_{12} + U_{13} + U_{23} \), where \( U_{12} \), \( U_{13} \), and \( U_{23} \) are the potential energies between the respective pairs of charges. Substitute \( U_{12} = \frac{k \cdot (3.0 \times 10^{-9}) \cdot (6.0 \times 10^{-9})}{0.02} \), \( U_{13} = \frac{k \cdot (3.0 \times 10^{-9}) \cdot q_3}{0.02} \), and \( U_{23} = \frac{k \cdot (6.0 \times 10^{-9}) \cdot q_3}{0.02} \).
4. Set \( U_{total} = 0 \) and solve for \( q_3 \): Combine the terms into a single equation: \( \frac{k \cdot (3.0 \times 10^{-9}) \cdot (6.0 \times 10^{-9})}{0.02} + \frac{k \cdot (3.0 \times 10^{-9}) \cdot q_3}{0.02} + \frac{k \cdot (6.0 \times 10^{-9}) \cdot q_3}{0.02} = 0 \). Factor out common terms and simplify to isolate \( q_3 \).
5. Solve for \( q_3 \): Rearrange the equation to find \( q_3 \) in terms of the known charges and constants. This will give the value of the third charge that makes the total electric potential energy zero.

Key Concepts

Electric Potential Energy

Electric potential energy is the energy a charged particle possesses due to its position in an electric field. It is influenced by the charges involved and the distances between them. In this scenario, the total electric potential energy of the system must equal zero, which means the contributions from all three charges must balance each other out.

Coulomb's Law

Coulomb's Law describes the force between two point charges. 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 law is essential for calculating the interactions between the charged particles in the triangle and determining the conditions for zero potential energy.

Equilateral Triangle Geometry

An equilateral triangle has all sides of equal length and all angles measuring 60 degrees. This symmetry simplifies calculations involving the distances between the charges. In this problem, knowing the geometry allows us to apply the principles of electric potential energy and Coulomb's Law effectively, as the distances between the charges are uniform.