Question

An electron moves away from a proton. Describe how the potential it encounters changes. Describe how its potential energy is changing.

Short answer

Answer: As an electron moves away from a proton, both the electric potential and potential energy increase (i.e., they become less negative).

Step-by-step solution

Understand Electric Potential

Electric potential (V) is defined as the electric potential energy (U) per unit charge (q) at a particular location in an electric field. Mathematically, it can be expressed as: V = \frac{U}{q} In the context of the electron-proton system, the electric potential will help us determine the change in potential energy as the electron moves away from the proton.

Understand Electric Potential Energy

Electric potential energy (U) of two charged particles is given by Coulomb's Law, which can be expressed as: U = \frac{k*q1*q2}{r} Here, - k is the electrostatic constant (8.99 * 10^9 Nm^2/C^2) - q1 and q2 are the magnitudes of the charges of the two interacting particles (in this case, they are the electron and the proton) - r is the distance between the two charges

Describe the change in potential

Now, let's focus on the change in electric potential (V) as the electron moves away from the proton. Since the electron and proton have opposite charges, the potential between them is negative. As the electron moves away from the proton (i.e., the distance r increases), the magnitude of the electric potential (V) decreases, as it is inversely proportional to the distance (r). However, since the potential is negative, a decrease in its magnitude means an increase in the potential value (i.e., it becomes less negative).

Describe the change in potential energy

To describe the change in potential energy (U) as the electron moves away from the proton, we refer back to the formula for electric potential energy: U = \frac{k*q1*q2}{r} As mentioned earlier, the charges of the electron and proton are opposite (q1 = -e, q2 = +e), where e is the elementary charge (1.6 * 10^{-19} C). As the electron moves away from the proton (i.e., the distance r increases), the magnitude of the potential energy (U) decreases. However, since the potential energy is negative (due to opposite charges), a decrease in its magnitude means an increase in the potential energy value (i.e., it becomes less negative). In summary, as the electron moves away from the proton, the electric potential and potential energy both increase (i.e., they become less negative).

Key concepts

Electric Potential

Imagine standing on a hill with a ball in your hand. As you hold the ball, it has gravitational potential energy due to its position in the Earth's gravitational field. Electric potential works in a similar way but in the context of electric charges and electric fields instead of gravity. It is a measure of potential energy per unit charge.

As an electron moves away from a proton, the scenario is akin to walking down a hill, decreasing the gravitational potential energy of the ball, but in terms of electric potential. Since an electron has a negative charge, the electric potential it encounters initially is also negative due to the attractive force toward the proton. As the distance between the electron and the proton increases, the negative electric potential decreases in magnitude. This means the electron is experiencing a rise in potential, albeit in a negative realm; it's like the potential is becoming 'less negative' as the electron moves further away. Mathematically, this is shown by the formula V = \( \frac{U}{q} \), revealing how potential energy and electric potential relate.

Coulomb's Law

When it comes to the universe of charged particles, think of Coulomb's Law as a fundamental rule detailing how it's all about attraction and repulsion. This law allows us to calculate the electric force between two charged particles. It states that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. It's given by the equation \( F = k \times \frac{q1 \times q2}{r^2} \), where k is Coulomb's constant.

In our textbook example, the law helps us understand the electric force and potential energy between an electron and a proton. So, as they drift apart, this law indicates the strength of their interaction wanes—similar to how the sound of a friend's voice fades as they walk away from you.

Charge Interaction

Diving deeper into our universe of charged particles, charge interaction is akin to a dance between particles. It's about how particles with electrical charge attract or repel each other. This interaction is driven by electric forces that follow Coulomb's Law.

Electrons, with their negative charge, and protons, with their positive charge, are attracted to each other, like two dancers drawn together on the dance floor. As the electron in our example ventures away from its proton partner, the force that initially pulled them together weakens, and so does their electric potential energy, as encapsulated by the formula \( U = \frac{k \times q1 \times q2}{r} \). The further apart they get, the weaker the attraction—akin to dancers separating and the energy of their interaction diminishing.

Electric Field

Picture a field of invisible lines that you can't see or touch, but you can definitely feel if you're a charged particle—that's an electric field. It is a vector field around a charged object where a force would be exerted on other charged particles. A positive charge creates a field radiating outward, while a negative charge creates an inward pull.

In our scenario with an electron moving away from a proton, the electron is moving within the proton's electric field. As the electron begins its journey outward, the intensity of the electric field it feels diminishes. This relationship can be represented as field lines growing more distant from one another as we move away from the charge source. This weakening field correlates with a decrease in electric potential energy and potential, much like the way a boat experiences less water resistance the further it sails from the shore.