Question

Use Coulomb's law, $$V=\frac{Q_{1} Q_{2}}{4 \pi \epsilon_{0} r}=2.31 \times 10^{-19} \mathrm{J} \cdot \mathrm{nm}\left(\frac{Q_{1} Q_{2}}{r}\right)$$ to calculate the energy of interaction, \(V\), for the following two arrangements of charges, each having a magnitude equal to the electron charge.

Short answer

Case 1 (Two charges with the same sign): \[V = 2.31 \times 10^{-19} J \cdot nm \left(\frac{Q_1 Q_2}{r}\right) = 2.31 \times 10^{-19} J \cdot nm \left(\frac{e^2}{r}\right)\] Case 2 (Two charges with opposite signs): \[V = 2.31 \times 10^{-19} J \cdot nm \left(\frac{Q_1 Q_2}{r}\right) = 2.31 \times 10^{-19} J \cdot nm \left(\frac{-e^2}{r}\right)\]

Step-by-step solution

Case 1: Two charges with the same sign

Let's consider two charges with the same sign, both positive or both negative. We can denote the charges as \(Q_1 = e\) and \(Q_2 = e\). To find the energy of interaction, we can use Coulomb's law formula: \[V = \frac{Q_1 Q_2}{4 \pi \epsilon_{0} r}\] Plugging in the values, we get: \[V = \frac{e^2}{4 \pi \epsilon_{0} r}\] Since both charges have the same sign, the energy of interaction is positive.

Case 2: Two charges with opposite signs

Now, let's consider two charges with opposite signs. We can denote the charges as \(Q_1 = e\) and \(Q_2 = -e\). To find the energy of interaction, we can use Coulomb's law formula: \[V = \frac{Q_1 Q_2}{4 \pi \epsilon_{0} r}\] Plugging in the values, we get: \[V = \frac{-e^2}{4 \pi \epsilon_{0} r}\] Since one charge is positive and the other is negative, the energy of interaction is negative. For both arrangements, we can represent the energy of interaction in terms of the given unit factor: \[V = 2.31 \times 10^{-19} J \cdot nm \left(\frac{Q_1 Q_2}{r}\right)\] By substituting the expression for each case, we can find the energy of interaction in terms of \(J \cdot nm\).

Key concepts

Energy of Interaction

The energy of interaction refers to the potential energy that exists between two charges. This energy arises due to the electrostatic forces according to Coulomb's law. Coulomb's law provides the foundation for understanding these interactions, mathematically defined as \[V = \frac{Q_{1} Q_{2}}{4 \pi \epsilon_{0} r}\]Here, \(Q_1\) and \(Q_2\) are the charges, \(\epsilon_{0}\) is the permittivity of free space, and \(r\) is the distance between the charges.
This formula explains that the energy of interaction depends on the magnitude of the charges and the distance separating them. A smaller distance between the charges results in a stronger interaction, thus higher interaction energy. Similarly, the nature of the charges (whether they're like or opposite) determines whether this energy shall be positive or negative.
With two like charges (both positive or both negative), this energy is positive, indicating repulsion. For opposite charges, the energy is negative, suggesting attraction.
Understanding the energy of interaction is crucial in fields like physics and chemistry, as it explains why atoms bond or repel each other, influencing the behavior of molecules.

Like Charges

Like charges refer to two charges that have the same sign, either both positive or both negative. According to the principles of electrostatics, like charges repel each other. This repulsion can be quantified using Coulomb's law, which tells us about the energy involved when the charges are at a given distance apart.For like charges, the interaction energy is always positive because of repulsion. Substituting two same-sign electron charges into the formula:\[V = \frac{e^2}{4 \pi \epsilon_{0} r}\]The positive value of \(V\) signifies that external energy would need to be supplied to bring these charges closer together, as they naturally repel.
This is a fundamental concept used to explain various phenomena including why similarly charged objects won't stick together, and it's a key principle in understanding the stability and reactions of ions within chemical compounds.

Opposite Charges

Opposite charges are those with different signs, meaning one charge is positive and the other is negative. In electrostatics, opposite charges attract each other. This attraction results in an energy interaction that is often negative when quantified using Coulomb's law.When opposite charges, such as a positive charge \(Q_1 = e\) and a negative charge \(Q_2 = -e\), are involved, the energy of interaction is given by:\[V = \frac{-e^2}{4 \pi \epsilon_{0} r}\]The negative \(V\) value indicates that the system naturally loses energy when charges come closer, entering a more stable state.
Such interactions are essential in explaining the formation of ionic bonds where positively and negatively charged ions come together to form stable compounds. This principle is foundational in chemistry and is vital for understanding various natural and synthetic chemical processes.

Electron Charge

The electron charge, denoted by \(e\), is a fundamental physical constant representing the basic unit of electric charge. Its value is approximately \(1.602 \times 10^{-19}\) coulombs, and it's the charge of a single electron, or conversely, a single proton (with opposite sign). In electrostatic equations like Coulomb's law, the electron charge is essential for calculating forces and energies between charged particles.
It helps to determine the quantitative interaction energy when using the formula:- \(Q_1 = e\) - \(Q_2 = -e\)Plugging these values into the energy of interaction formula:\[V = \frac{Q_1 Q_2}{4 \pi \epsilon_{0} r} = \frac{-e^2}{4 \pi \epsilon_{0} r} \]Understanding electron charge is crucial in physics and chemistry. It allows us to calculate the interaction forces in molecules and atoms, determine electrical properties, and explore phenomena such as conductivity and magnetism.
The electron charge is a fundamental aspect of electromagnetism and plays a vital role in technological advancements, illustrating the dynamic nature of matter at its core level.