Question

According to Coulomb's law, which pair of charged particles has the lowest potential energy? a. a particle with a \(1-\) charge separated by \(150 \mathrm{pm}\) from a particle with a \(2+\) charge b. a particle with a \(1+\) charge separated by \(150 \mathrm{pm}\) from a particle with a \(1+\) charge c. a particle with a \(1-\) charge separated by \(100 \mathrm{pm}\) from a particle with a \(3+\) charge

Short answer

Option C has the lowest potential energy because it involves the largest magnitude of opposite charges and the shortest distance between them.

Step-by-step solution

Coulomb's Law formula

Understand that Coulomb's law can be used to calculate the potential energy (U) between two charged particles. The formula is given by U = k * q1 * q2 / r, where k is Coulomb's constant, q1 and q2 are the charges of the particles, and r is the separation distance between the charges.

Calculate the potential energy for Option A

Using the Coulomb's law formula, calculate the potential energy for the pair of particles in option A: U(A) = k * (-1) * (2) / 150e-12 m.

Calculate the potential energy for Option B

Calculate the potential energy for the pair of particles in option B: U(B) = k * (1) * (1) / 150e-12 m.

Calculate the potential energy for Option C

Calculate the potential energy for the pair of particles in option C: U(C) = k * (-1) * (3) / 100e-12 m.

Determine the pair with the lowest potential energy

Compare the calculated potential energies, keeping in mind that the potential energy is negative for attractive forces (opposite charges) and positive for repulsive forces (like charges). The pair with the lowest (most negative) potential energy is the pair with the lowest potential energy according to Coulomb's law.

Key concepts

Electrostatic Potential Energy Calculation

Electrostatic potential energy is a key concept when understanding the forces between charged particles. It reflects the work done to bring a charge within an electric field to a specific point against the electric force. Interestingly, this energy is {\bf potential}, meaning it has the potential to do work in the future.

The calculation of this energy can be done through a specific formula derived from Coulomb's Law. This formula can be written in the form: \( U = \frac{k \cdot q_1 \cdot q_2}{r} \), where \( U \) represents the electrostatic potential energy, \( k \) is Coulomb's constant (approximately \( 8.988 \times 10^9 \) Nm²/C² in a vacuum), \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance between the centers of the two charges.

The numerator reflects the interactive effect of the charges: the larger the charges, the larger the energy. The denominator means that as the charges move further apart, the energy decreases. This is intuitive – the further apart charges are, the less they 'feel' each other's presence, leading to a lower potential energy.

Charge Interaction

The interaction between charges is governed by the basic principle that like charges repel, and unlike charges attract. This elemental rule underlies everything from the bonds between atoms to the way materials gain and lose electrical charge.

In terms of potential energy, an important concept is the sign of the energy value. When opposite charges interact (\( 1- \) and \( 2+ \) for instance), the potential energy is negative, indicating an attractive force, a stable configuration that the system favors. Conversely, when like charges interact (e.g. \( 1+ \) and \( 1+ \) charges), the potential energy is positive, reflecting a repulsive force, an unstable configuration that the system tends to move away from.

Recognizing the sign of the potential energy can help us rapidly determine the nature of charge interaction without detailed calculation, which is particularly handy in physics and chemistry. For instance, two electrons approaching each other would have an increasing positive potential energy, signifying their mutually repulsive interaction.

Coulomb's Constant

Understanding Coulomb's constant is essential in using Coulomb's law effectively. This constant, denoted by \( k \) and also called the electrostatic constant, plays a pivotal role in calculating the force and potential energy between two point charges. Its value in a vacuum is approximately \( 8.988 \times 10^9 \) newton square meters per coulomb squared (\( \text{Nm}^2/\text{C}^2 \)).

The magnitude of \( k \) signifies the strength of the electrostatic force and potential energy in a vacuum or air, which is practically the same for most intents and purposes. It's crucial to note that the value of \( k \) could vary in different mediums. For instance, in water, charges interact less strongly because water 'screens' the charges, and as a result, \( k \) would effectively have a smaller value.

Another interesting point is that \( k \) is related to the permittivity of free space (\( \varepsilon_0 \) – a measure of how an electric field affects and is affected by a medium), where \( k = \frac{1}{4 \pi \varepsilon_0} \). When you're solving problems involving charges in a vacuum, \( k \) remains constant, simplifying calculations.