Question

According to Coulomb's law, rank the interactions between charged particles from lowest potential energy to highest potential energy. a. a \(1+\) charge and a \(1-\) charge separated by \(100 \mathrm{pm}\) b. a \(2+\) charge and a \(1-\) charge separated by \(100 \mathrm{pm}\) c. a \(1+\) charge and a \(1+\) charge separated by \(100 \mathrm{pm}\) d. a \(1+\) charge and a \(1-\) charge separated by \(200 \mathrm{pm}\)

Short answer

(d) < (a) < (b) < (c)

Step-by-step solution

Understand Coulomb's Law

Coulomb's law states that the potential energy (PE) between two point charges is directly proportional to the product of the charges and inversely proportional to the distance between them. The formula to calculate the potential energy is PE = k * (q1 * q2) / r, where k is Coulomb's constant, q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.

Analyze Interactions in Terms of Coulomb's Law

To rank the interactions, we need to compare the product of the charges divided by the distance for each pair. Because we're not given a specific value for the constant k and we only need a relative ranking, we can ignore k in our comparisons.

Calculate PE for Each Pair

For (a), PE = (1+ * 1-) / 100pm = -1 / 100; for (b), PE = (2+ * 1-) / 100pm = -2 / 100; for (c), PE = (1+ * 1+) / 100pm = 1 / 100; for (d), PE = (1+ * 1-) / 200pm = -1 / 200. Remember that opposite charges have negative potential energy due to their attractive interaction, while like charges have positive potential energy due to their repulsive interaction.

Rank the Interactions

The interactions can now be ranked by their potential energy values from lowest to highest: (d) has the lowest potential energy, (a) has the second lowest, (b) is next, and (c) has the highest potential energy.

Key concepts

Electrostatic Potential Energy

The concept of electrostatic potential energy is essential when we deal with the forces between charged particles. It is the energy that results from the position of one charge relative to others in space. Imagine every charge creating an invisible energy field around itself, influencing potentials in its vicinity. This energy is more than just a number; it's indicative of the ability to do work, either pulling or pushing other charges.

The crucial idea here is that potential energy doesn't just depend on the amount of charge but also the distance between the charges. Closer charges mean higher potential energy if they repel, and lower, negative values if they attract. A good analogy could be considering gravitational potential energy, where objects have potential energy due to their positions in a gravitational field, which depends on their height and mass. In the case of electrostatics, the 'height' is the distance between charges, and the 'mass' equates to the magnitude of the charges.

Charge Interactions

Understanding charge interactions helps reveal the mysteries behind how matter behaves on an atomic level. Charges can either attract or repel each other, and the rule is simple: opposites attract, and likes repel. This is similar to how magnets work; north and south poles attract, while two north or two south poles repel each other.

When two opposite charges come into proximity, there's an attractive force pulling them together. This interaction stores negative potential energy due to the nature of the attractive force. On the flip side, like charges attempt to drive each other away, storing positive potential energy in their repulsion. These interactions don't just impact the charges, but they can cause movements, influencing the world at the microscopic level.

Physics of Charged Particles

Diving into the physics of charged particles opens a window into understanding how charges maneuver and influence their environment. A single charged particle can seem insignificant, but when combined with others, it can create powerful forces and fields. These charged particles, when free to move, give rise to electric currents, which are the basis for much of our technology.

To understand how these charged particles behave, we must consider the forces they experience and the resultant potential energy in different configurations. For example, in a conductor, free electrons move until the electrostatic forces are balanced, which is when we reach an equilibrium. This simple concept is foundational in the design of everything from batteries to computer chips.

Inverse Square Law

The inverse square law is a physical principle that tells us how the strength of a force diminishes as distance increases. Imagine the force as light from a lamp; the further you move from the lamp, the dimmer the light because it spreads over a larger area. Similarly, for charged particles, as the distance between them increases, the force they exert on each other weakens drastically.

This law applies to the interactions between charges as described in Coulomb's law. Consequentially, if you double the distance between two charges, the force (and thus the potential energy) is not merely halved; it is reduced to a quarter of its original value. Remember, when dealing with forces and their related potentials, this square of the distance has a profound effect and is a cornerstone concept in classical physics.