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 for the following two arrangements of charges, each having a magnitude equal to the electron charge. a. \(\stackrel{1 \times 10^{-10} \mathrm{~m}}{\longrightarrow(-1) \longleftrightarrow \infty \longrightarrow(+1) \longleftrightarrow 10^{-10} \mathrm{~m}}{\longleftrightarrow} \stackrel{\leftarrow}{\longleftrightarrow}\) b.

Short answer

In summary, the energy of interaction for the two arrangements of charges is approximately the same: - For case A, where the charges (-1) and (+1) are separated by a distance of \(1 \times 10^{-10} m\), the energy of interaction is \(V \approx -2.30 \times 10^{-19} J\). - For case B, where the charges (-1) and (+1) are separated by a distance of \(10^{-10} m\), the energy of interaction is \(V \approx -2.30 \times 10^{-19} J\).

Step-by-step solution

In this case, we are given the charges as -1 and +1, both having a magnitude equal to the electron charge (1e). The distance between the charges is \(r = 1 \times 10^{-10} m = 10^{-10} m\). Substitute the values into the formula above and calculate the energy of interaction: #Step 1: Find Q1 and Q2#

For this case, we have: \(Q_{1} = -1 \times e\) \(Q_{2} = +1 \times e\) where e is the elementary charge \(e = 1.6 \times 10^{-19} C\) #Step 2: Calculate the energy of interaction using the Coulomb's law formula#

Using the Coulomb's law formula given in the problem and substituting the values for \(Q_1, Q_2\), and \(r\): \[ V = \frac{Q_{1} Q_{2}}{4 \pi \epsilon_{0} r} = 2.31 \times 10^{-19} \mathrm{~J\cdot nm} \left(\frac{-e^2}{10^{-10} \mathrm{~m}}\right) \] #Step 3: Solve for the energy of interaction, V#

(Remember to substitute the value of the elementary charge \(e = 1.6 \times 10^{-19} C\)) \[ V = 2.31 \times 10^{-19} \mathrm{~J\cdot nm} \cdot \frac{(-1.6 \times 10^{-19} C)^2}{10^{-10} \mathrm{~m}} \] Calculate the value for V: \(V \approx -2.30 \times 10^{-19} J\) #Case B: (-1), (+1) charges separated by a distance of \(10^{-10}m\)

For this case, the charges arrangement and values are the same as in case A. However, the distance between the charges is now \((10^{-10} m)\). We can follow a similar procedure as we did for case A: #Step 1: Find Q1 and Q2 (same as Case A)#

We have: \(Q_{1} = -1 \times e\) \(Q_{2} = +1 \times e\) where e is the elementary charge \(e = 1.6 \times 10^{-19} C\) #Step 2: Calculate the energy of interaction using the Coulomb's law formula (use new r value)#

Using the Coulomb's law formula given in the problem and substituting the values for \(Q_1, Q_2\), and \(r\): \[ V = \frac{Q_{1} Q_{2}}{4 \pi \epsilon_{0} r} = 2.31 \times 10^{-19} \mathrm{~J\cdot nm} \left(\frac{-e^2}{(10^{-10} \mathrm{~m})}\right) \] #Step 3: Solve for the energy of interaction, V (use new r value)#

(Remember to substitute the value of the elementary charge \(e = 1.6 \times 10^{-19} C\)) \[ V = 2.31 \times 10^{-19} \mathrm{~J\cdot nm} \cdot \frac{(-1.6 \times 10^{-19} C)^2}{(10^{-10} \mathrm{~m})} \] Calculate the value for V: \(V \approx -2.30 \times 10^{-19} J\)

Key concepts

Energy of Interaction

The energy of interaction in an electrostatic field represents the potential energy due to the force between two charges. It is given by Coulomb's law, which quantifies the amount of energy needed to assemble a system of charges at rest, or alternatively, the work done in separating them. This energy is dependent on the magnitudes of the charges and their distance apart.

Understanding the Calculation

The formula used for calculating the energy of interaction is: \[ V = \frac{Q_1 Q_2}{4 \pi \epsilon_0 r} \] Here, \(Q_1\) and \(Q_2\) are the magnitudes of the two interacting charges, \(r\) is the distance between them, and \(\epsilon_0\) is the vacuum permittivity, a constant that mediates the force in free space. By substituting the respective values into this formula, you can calculate the energy for different arrangements of charges. For charges equal in magnitude but opposite in sign, the resulting energy is negative, indicating an attraction between the charges.

When solving problems related to energy of interaction, remember that the principles remain the same, regardless of the charge arrangement. Consistent units are key – typically the system is analyzed using joules for energy and meters for distance. In physics, solving such problems provides insight into the potential energy landscape of charged particles and can be applied to understanding atomic and molecular structures.

Elementary Charge

An elementary charge, denoted by \(e\), is the smallest unit of electric charge that is considered indivisible in the context of electrostatic interactions. It is the magnitude of charge carried by a single proton or the negative of the charge carried by a single electron.

Significance in Calculations

Elementary charge plays a crucial role in calculations dealing with charged particles. In the International System of Units (SI), it is approximately valued at \(1.6 \times 10^{-19} \) coulombs (C). When a problem states that charges are equal to the electron charge, it implies that each charge is either \(+e\) or \(-e\), signifying the respective positive or negative elementary charge.

Understanding the elementary charge is important because it forms the basis of quantifying and dealing with electric charges in various physical scenarios. It's also pivotal in calculations of known forces between subatomic particles which obey Coulomb's law. As elementary charges are fundamental constants in nature, a robust grasping of their values and implications is essential for any study involving electrostatics.

Electrostatic Force

The electrostatic force is the force exerted between charged particles. According to Coulomb's law, this force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.

Dependency and Directionality

Coulomb's law can be expressed as: \[ F = k \frac{|Q_1 Q_2|}{r^2} \] where \(F\) represents the electrostatic force, \(|Q_1 Q_2|\) are the absolute magnitudes of the charges, \(r\) is the separation distance, and \(k\) is Coulomb's constant. The force is attractive when charges are of opposite signs and repulsive when charges are of the same sign.

Understanding electrostatic force is fundamental in physics. It not only governs the behavior of charges at rest and explains phenomena such as electric fields and potentials but also has practical implications in industry and technology, including electrostatic precipitation, photocopiers, and even biological processes within cells. By grasping the essence of electrostatic force, students can unlock a deeper understanding of not only Coulomb's law but also the underpinning principles of electromagnetism and the fundamental interactions between charged particles.