Step 1: Understanding the Concept:
Electrostatic potential energy of a configuration of point charges represents the total external work performed to bring these charges from an infinite separation to their specific coordinates in space.
Because the electrostatic force is a conservative force, the energy stored in the system is independent of the path taken to assemble it.
Potential energy is a scalar quantity, meaning we sum the values algebraically rather than vectorially.
Crucially, when assembling a system of \(n\) charges, the total energy is the sum of the potential energies of every unique pair of charges.
For a system of three charges, there are exactly three unique pair interactions to account for: pair (1,2), pair (2,3), and pair (3,1).
The sign of each interaction energy depends on the signs of the participating charges; like charges produce positive energy (repulsion), while unlike charges produce negative energy (attraction).
Step 2: Key Formula or Approach:
The potential energy between any two point charges \(q_i\) and \(q_j\) separated by a distance \(r\) is defined by the formula:
\[ U_{pair} = \frac{1}{4\pi\epsilon_0} \frac{q_i q_j}{r} \]
For a system of three charges \(q_1, q_2, q_3\), the net potential energy \(U_{total}\) is:
\[ U_{total} = U_{12} + U_{23} + U_{31} \]
Step 3: Detailed Explanation:
Let the three charges at the vertices of the equilateral triangle be defined as follows:
- \(q_1 = +q\)
- \(q_2 = +q\)
- \(q_3 = -q\)
Since the charges are at the corners of an equilateral triangle of side length \(L\), the separation between any two charges is constant: \(r_{12} = r_{23} = r_{31} = L\).
Now, let's calculate the interaction energy for each unique pair systematically:
1. Interaction between \(q_1\) and \(q_2\) (both positive):
\[ U_{12} = \frac{1}{4\pi\epsilon_0} \frac{(+q)(+q)}{L} = \frac{1}{4\pi\epsilon_0} \frac{q^2}{L} \]
2. Interaction between \(q_2\) and \(q_3\) (one positive, one negative):
\[ U_{23} = \frac{1}{4\pi\epsilon_0} \frac{(+q)(-q)}{L} = -\frac{1}{4\pi\epsilon_0} \frac{q^2}{L} \]
3. Interaction between \(q_1\) and \(q_3\) (one positive, one negative):
\[ U_{31} = \frac{1}{4\pi\epsilon_0} \frac{(+q)(-q)}{L} = -\frac{1}{4\pi\epsilon_0} \frac{q^2}{L} \]
To find the net potential energy of the system, we sum these three scalar components:
\[ U_{total} = \left[ \frac{1}{4\pi\epsilon_0} \frac{q^2}{L} \right] + \left[ -\frac{1}{4\pi\epsilon_0} \frac{q^2}{L} \right] + \left[ -\frac{1}{4\pi\epsilon_0} \frac{q^2}{L} \right] \]
Combining the common algebraic terms:
\[ U_{total} = \frac{1}{4\pi\epsilon_0} \frac{q^2}{L} (1 - 1 - 1) \]
\[ U_{total} = \frac{1}{4\pi\epsilon_0} \frac{q^2}{L} (-1) \]
\[ U_{total} = -\frac{1}{4\pi\epsilon_0} \frac{q^2}{L} \]
The resulting negative sign signifies that the system is "bound."
This means work must be done by an external agent to pull these charges apart to infinity.
Physically, the attractive energy between the unlike pairs outweighs the repulsive energy between the like pair in this specific geometric configuration.
Step 4: Final Answer:
The net electrostatic potential energy of the three-charge system is \(-\frac{1}{4\pi\epsilon_0} \frac{q^2}{L}\).