Question

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?

Accepted Answer

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. 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 ($$ 8.99 	imes 10^9 \ 	ext{N·m}^2/	ext{C}^2 $$), $$ q_1 $$ and $$ q_2 $$ are the charges, and $$ r  is $$the distance between them. 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 $$ U_{12} = \frac{k \cdot (3.0 	imes 10^{-9}) \cdot (6.0 	imes 10^{-9})}{0.02} $$, $$ U_{13} = \frac{k \cdot (3.0 	imes 10^{-9}) \cdot q_3}{0.02} $$, and $$ U_{23} = \frac{k \cdot (6.0 	imes 10^{-9}) \cdot q_3}{0.02} . $$Set $$ U_{total} = 0 $$ and solve for $$ q_3 $$: Combine the terms into a single equation: $$ \frac{k \cdot (3.0 	imes 10^{-9}) \cdot (6.0 	imes 10^{-9})}{0.02} + \frac{k \cdot (3.0 	imes 10^{-9}) \cdot q_3}{0.02} + \frac{k \cdot (6.0 	imes 10^{-9}) \cdot q_3}{0.02} = 0 . $$Factor out common terms and simplify to isolate $$ q_3 . $$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.

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?

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Concetti chiave

Electric Potential Energy

Coulomb's Law

Equilateral Triangle Geometry