Question

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

Short answer

The interaction energy for both arrangements of charges, each having a magnitude equal to the electron charge, can be calculated using Coulomb's Law as: \[V = 2.31 \times 10^{-19} \cdot J \cdot nm \left(\frac{2.56 \times 10^{-38}}{r} \right)\] where 'r' is the distance between the charges in each arrangement.

Step-by-step solution

Determine the known variables

For both arrangements, we have the same charges, with both charges having a magnitude equal to the electron charge (\(Q_1 = Q_2 = e\)), where - Electron charge \(e\) : \(1.6 \times 10^{-19} C\) - Vacuum permittivity constant \(\varepsilon_0\) : \(8.85 \times 10^{-12} C^2 / Nm^2\) For each arrangement, we also need the distance between the charges, \(r\). For this exercise, we are not given any specific values for 'r' in the arrangements. Therefore, we will treat 'r' as a variable and calculate the interaction energy in terms of 'r' for both arrangements.

Calculate the energy of interaction using Coulomb's Law

To calculate the energy of interaction, we can plug the known variables into Coulomb's Law formula: \[V = \frac{Q_1 Q_2}{4 \pi \varepsilon_0 r} = 2.31 \times 10^{-19} \cdot J \cdot nm \left(\frac{Q_1 Q_2}{r} \right)\] We can substitute the values of \(Q_1 = Q_2 = e\) and \(\varepsilon_0\): \[V = \frac{e^2}{4 \pi \varepsilon_0 r} = 2.31 \times 10^{-19} \cdot J \cdot nm \left(\frac{e^2}{r} \right)\] Now, let's plug in the value of \(e = 1.6 \times 10^{-19} C\): \[V = \frac{(1.6 \times 10^{-19})^2}{4 \pi (8.85 \times 10^{-12}) r} = 2.31 \times 10^{-19} \cdot J \cdot nm \left(\frac{(1.6 \times 10^{-19})^2}{r} \right)\] Now, we can simplify: \[V = \frac{(2.56 \times 10^{-38})}{4 \pi (8.85 \times 10^{-12}) r} = 2.31 \times 10^{-19} \cdot J \cdot nm \left(\frac{2.56 \times 10^{-38}}{r} \right)\] Therefore, the interaction energy for both arrangements (in terms of 'r') is: \[V = 2.31 \times 10^{-19} \cdot J \cdot nm \left(\frac{2.56 \times 10^{-38}}{r} \right)\]

Key concepts

Energy of Interaction

Understanding the energy of interaction between charged particles is crucial as it defines how they influence each other. In the context of Coulomb's law, this energy is called electrostatic potential energy. It is the work done to bring the charges from infinity to a certain distance 'r' apart in vacuum.

The formula for the energy of interaction derived from Coulomb's law is given by: \[ V = \frac{Q_1 Q_2}{4 \pi \varepsilon_0 r} \]
Where \( V \) is the energy of interaction, \( Q_1 \) and \( Q_2 \) are the charges involved, \( r \) is the separation between the charges, and \( \varepsilon_0 \) is the vacuum permittivity constant.

Electron Charge

The charge of an electron is a fundamental constant and one of the most critical values in physics and chemistry, denoted by \( e \). Its magnitude is essential for calculating the force and energy between charges using Coulomb's law.

The value of an electron's charge is approximately \( e = 1.6 \times 10^{-19} C \) (coulombs). This small but fundamental charge underlies all of electrochemistry and is the basis for our understanding of electric fields and forces.

Vacuum Permittivity

Vacuum permittivity, also called the electric constant, denoted by \( \varepsilon_0 \), is a measure of the resistance encountered when forming an electric field in a vacuum.

The value of vacuum permittivity is approximately \( 8.85 \times 10^{-12} C^2 / Nm^2 \). It appears in Coulomb's law, affecting the force and energy of interaction between two point charges in a vacuum. It's one of the constants that give us insight into the strength of the electromagnetic interaction.

Force Between Charges

Coulomb's law not only allows us to calculate the energy of interaction but also the force between two point charges. The force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

The equation for the force is \[ F = \frac{Q_1 Q_2}{4 \pi \varepsilon_0 r^2} \]
The force calculated here is the electrostatic force, which can be attractive or repulsive depending on the nature of the charges. Understanding this force is essential for exploring electric field concepts, electric potentials, and the behavior of charges in various situations.