Problem 11 - College Physics

Question

An electron and a proton are separated by \(2.0 \mathrm{nm}\). (a) What is the magnitude of the force on the electron? (b) What is the net force on the system?

Step-by-Step Solution

Answer

(a) 5.75 x 10^-10 N, (b) Net force is zero.

Step 1: Identify Known Values

We are given the distance between the electron and the proton as \( r = 2.0 \text{ nm} = 2.0 \times 10^{-9} \text{ m} \). The charge of an electron \( e = -1.6 \times 10^{-19} \text{ C} \), and a proton has the same magnitude of charge but positive \( +1.6 \times 10^{-19} \text{ C} \).

Step 2: Understand Coulomb's Law

Coulomb's Law allows us to calculate the magnitude of the force between two point charges. The formula is \( F = k \frac{|q_1 q_2|}{r^2} \), where \( k \) is Coulomb's constant, \( 8.99 \times 10^9 \text{ Nm}^2/\text{C}^2 \), \( |q_1| \) and \( |q_2| \) are the absolute charges, and \( r \) is the separation distance.

Step 3: Calculate the Force Using Coulomb's Law

Substitute the known values into Coulomb's Law: \[F = 8.99 \times 10^9 \frac{(1.6 \times 10^{-19})^2}{(2.0 \times 10^{-9})^2} \]Calculate this to find the magnitude of the force. The calculation gives:\[F \approx 5.75 \times 10^{-10} \text{ N}\]

Step 4: Compute the Net Force on the System

In this two-charge system, the forces on each particle are equal in magnitude and opposite in direction due to Newton's third law. Thus, there is no net external force acting on the system as it is isolated.

Key Concepts

Electrostatic Force

The electrostatic force is a type of force that occurs between two charged particles. It's a key concept when analyzing interactions at the atomic level, specifically between electrons and protons. This force can either be attractive or repulsive depending on the charges of the particles involved.

• When charges are of opposite types (positive and negative), such as an electron and a proton, the force is attractive, pulling the particles towards each other.
• If the charges are of the same type (both positive or both negative), the force is repulsive, pushing the particles apart.

Understanding the nature of the electrostatic force is crucial because it determines how particles, like atoms and molecules, bond together. The strength of this force is quantified using Coulomb's Law, which states that the force between two point charges varies directly with the product of their charges and inversely with the square of the distance between them.
This force is responsible for much of what we see in chemical bonding and reactions in physics and chemistry.

Point Charges

The concept of point charges is fundamental in electromagnetism. Point charges refer to charged particles that are considered to have no size for the purposes of analysis, which simplifies calculations significantly. They are treated as single points with a certain charge.
This idea allows physicists to model real-world objects, such as electrons or protons, more effectively. When calculating forces between point charges, like those in our exercise between an electron and a proton, we assume the following criteria:

• The charges are negligible in size relative to the distances between them.
• Their positions can be treated as coordinates in space.

By understanding point charges, students can more effectively apply the principles of electrostatics, using equations like Coulomb's Law to determine the forces and fields acting on these particles in various scenarios.

Newton's Third Law

Newton's Third Law of Motion states, "For every action, there is an equal and opposite reaction." In the context of electrostatics and the exercise provided, this law plays a crucial role in understanding the interactions between the electron and proton.
When the electron exerts an attractive force on the proton due to their opposite charges, the proton simultaneously exerts an equal force back on the electron, but in the opposite direction. This is a perfect embodiment of Newton's Third Law.

• The magnitudes of these forces are equal, ensuring that the system remains in equilibrium unless acted upon by an external force.
• This reciprocal nature of force application is why there is no net force in an isolated two-particle system, like our electron-proton pair.

Understanding Newton's Third Law in the context of electrostatic forces helps students grasp how forces typically occur in pairs, maintaining balance in isolated systems unless disrupted by an outside force.