Question

Two equal charges \(q\) are placed at a distance of \(2 \mathrm{a}\) and a third charge \(-2 q\) is placed at the midpoint. The potential energy of the system is ....... (A) \(\left[\left(9 \mathrm{q}^{2}\right) /\left(8 \pi \epsilon_{0} \mathrm{a}\right)\right]\) (B) \(\left[q^{2} /\left(8 \pi \epsilon_{0} a\right)\right]\) (C) \(-\left[\left(7 q^{2}\right) /\left(8 \pi \epsilon_{0} a\right)\right]\) (D) \(\left[\left(6 q^{2}\right) /\left(8 \pi \in_{0} a\right)\right]\)

Short answer

The short answer to the question is: (C) \(-\left[\left(7 q^{2}\right) /\left(8 \pi \epsilon_{0} a\right)\right]\)

Step-by-step solution

Identify charges and their positions

Denote the equal charges as \(q_1 = q_2 = q\), and the third charge as \(q_3 = -2q\). Their positions can be represented as follows: - Charge \(q_1\) at position \((-a,0)\) - Charge \(q_2\) at position \((a,0)\) - Charge \(q_3\) at position \((0,0)\)

Calculate distances between charges

We will find the distances between each pair of charges: - Distance between \(q_1\) and \(q_3\): \(r_{13} = a\) - Distance between \(q_2\) and \(q_3\): \(r_{23} = a\) - Distance between \(q_1\) and \(q_2\): \(r_{12} = 2a\)

Calculate potential energy between each pair of charges

Using the formula for potential energy between two point charges, \(U = \frac{k*q_1*q_2}{r}\) (with \(k = \frac{1}{4 \pi \epsilon_0}\)), calculate the potential energy between each pair of charges: 1. Potential energy between \(q_1\) and \(q_3\): \( U_{13} = \frac{k*q_1*q_3}{r_{13}}= \frac{k*q*(-2q)}{a} =-\frac{2kq^2}{a}\) 2. Potential energy between \(q_2\) and \(q_3\): \( U_{23} = \frac{k*q_2*q_3}{r_{23}} = \frac{k*q*(-2q)}{a} =-\frac{2kq^2}{a}\) 3. Potential energy between \(q_1\) and \(q_2\): \(U_{12} = \frac{k*q_1*q_2}{r_{12}} = \frac{k*q*q}{2a} = \frac{kq^2}{2a}\)

Calculate total potential energy of the system

Now, we will sum up the potential energies of each pair of charges to find the total potential energy of the system: \(U_{total} = U_{13} + U_{23} + U_{12} = -\frac{2kq^2}{a} -\frac{2kq^2}{a} + \frac{kq^2}{2a}\) \(U_{total} = -\frac{8kq^2}{2a} + \frac{kq^2}{2a} = -\frac{7kq^2}{2a}\) Recall that \(k = \frac{1}{4 \pi \epsilon_0}\), substituting this value into the expression: \(U_{total} = -\frac{7\left[\frac{1}{4 \pi \epsilon_0}\right]q^2}{2a} = -\frac{7q^2}{8 \pi \epsilon_0 a}\)

Match the answer to the options given

Comparing our result to the given options, we find that the total potential energy of the system matches option (C): \(-\left[\left(7 q^{2}\right) /\left(8 \pi \epsilon_{0} a\right)\right]\)

Key concepts

Potential Energy

In electrostatics, potential energy refers to the energy stored in a system due to the positions of charged particles. It's like a rubber band that's been stretched—the potential energy is stored because of its position and will convert to kinetic energy if released. For point charges, this energy can be calculated using the formula:\[U = \frac{k \cdot q_1 \cdot q_2}{r}\]where \(k\) is Coulomb's constant \(\left( \frac{1}{4 \pi \epsilon_0} \right)\), \(q_1\) and \(q_2\) are the magnitudes of the charges, and \(r\) is the separation between them. The potential energy is positive when charges are of the same sign and negative when they are opposite, indicating attraction. This energy determines how much work would be needed to assemble the charge configuration from infinity.

Point Charges

Point charges are idealized charges that are assumed to be located at a single point in space. This simplification is useful for calculating electric fields and potentials in systems of charges. They are characterized by:

• Having a definite amount of charge (\(q\)).
• Being dimensionless—very small compared to the distances involved.

In our exercise:

• Two equal charges \(q\) are positioned at opposite ends and a third charge \(-2q\) is at the midpoint.
• The distances between these charges are crucial for calculating forces and potentials, as shown in the problem's solution.

Using point charges simplifies the complexities of real physical systems while still providing accurate insights.

Coulomb's Law

Coulomb's Law governs the interactions between point charges, describing the force between them. This fundamental principle is given by:\[F = \frac{k \cdot |q_1 \cdot q_2|}{r^2}\]where \(F\) is the force, \(|q_1\cdot q_2|\) is the product of the magnitudes of the two charges, and \(r\) is the distance between them. The force:

• Is attractive if the charges are opposite.
• Is repulsive if the charges are of the same type.

In our exercise, Coulomb's Law helps determine the potential energy between each pair of charges by first considering these forces. Understanding these interactions scrupulously helps in further analyzing advanced concepts in electrostatics.

Electric Potential

Electric potential is a measure of potential energy per unit charge at a point in an electric field. It helps us understand how much work would be needed to bring a charge from infinity to any given point without changing kinetic energy. Mathematically, it's expressed as:\[V = \frac{k \cdot q}{r}\]where \(V\) is the electric potential, \(k\) is Coulomb's constant, \(q\) is the charge, and \(r\) is the distance from the point charge. In our problem scenario:

• Bringing the charge \(-2q\) into the system required work, and this work correlates directly with the electric potential created by \(q_1\) and \(q_2\).
• Each charge contributes to the overall electric potential of the system.

Electric potential is pivotal for understanding the energy landscape between charges, providing insights into energy efficiencies and distributions within a system of charges.