Problem 66 - Fundamentals of Physics

Question

An electron is in a vacuum near Earth's surface and located at \(y=0\) on a vertical \(y\) axis. At what value of \(y\) should a second electron be placed such that its electrostatic force on the first electron balances the gravitational force on the first electron?

Step-by-Step Solution

Answer

The second electron should be placed approximately at \( y = 5.08 \times 10^{-11} \, \text{m} \).

Step 1: Understanding the Forces

The gravitational force on the first electron is given by the formula \( F_g = m imes g \), where \( m \) is the mass of the electron and \( g \) is the acceleration due to gravity \( 9.8 \, \text{m/s}^2 \). The electrostatic force between two electrons is \( F_e = \frac{k imes e^2}{r^2} \), where \( e \) is the charge of an electron, \( k \) is Coulomb's constant \( 8.99 \times 10^9 \, \text{N m}^2/ ext{C}^2 \), and \( r \) is the distance between the electrons.

Step 2: Equating the Forces

Set the gravitational force equal to the electrostatic force: \( m imes g = \frac{k imes e^2}{r^2} \). We want \( r \) such that this equality holds, signifying the electrostatic force balances the gravitational force.

Step 3: Substituting Known Values

Replace \( m = 9.11 \times 10^{-31} \, \text{kg} \) and \( e = 1.602 \times 10^{-19} \, \text{C} \) into the equation: \( 9.11 \times 10^{-31} imes 9.8 = \frac{8.99 \times 10^9 imes (1.602 \times 10^{-19})^2}{r^2} \).

Step 4: Solving for Distance

Solve the equation for \( r^2 \): \( r^2 = \frac{8.99 \times 10^9 imes (1.602 \times 10^{-19})^2}{9.11 \times 10^{-31} imes 9.8} \). Calculate to find \( r \).

Step 5: Calculating the Value of \( r \)

Perform the calculation: \( r = \sqrt{\frac{8.99 \times 10^{9} \times (1.602 \times 10^{-19})^2}{9.11 \times 10^{-31} \times 9.8}} \). The result will give the distance \( y \) at which the second electron should be placed.

Key Concepts

Gravitational Force

Gravitational force is a fundamental force of nature that attracts two bodies with mass towards each other. Near Earth's surface, this force acts on all objects in a downward direction towards the center of the Earth.
For an electron, the gravitational force can be calculated using the formula:

• \( F_g = m \times g \)
• where \( m \) is the mass of the electron, approximately \( 9.11 \times 10^{-31} \text{ kg} \), and \( g \) is the acceleration due to gravity, \( 9.8 \text{ m/s}^2 \).

This force is usually very small for particles as light as electrons.
Despite its small magnitude, it's crucial when balancing forces with electrostatic force, especially in physics problems involving tiny particles.

Coulomb's Law

Coulomb's law describes the electrostatic interaction between two charged particles. It states that the force between these particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between their centers.
This is mathematically expressed as:

• \( F_e = \frac{k \times e^2}{r^2} \)
• where \( e \) is the charge of an electron, \( k \) is Coulomb's constant \( 8.99 \times 10^9 \text{ N m}^2/\text{C}^2 \), and \( r \) is the distance between the electrons.

Coulomb's law is key when determining how to balance forces in scenarios involving charged particles, such as this exercise with two electrons.

Electron Charge

An electron's charge is a fundamental property that affects how it interacts with other particles.
Specifically, the charge of an electron is negative and has a magnitude of \( 1.602 \times 10^{-19} \text{ C} \). This micro-unit charge plays a significant role when calculating forces utilizing Coulomb's law, influencing the magnitude of the force each electron exerts on another.
Understanding the role of electron charge is critical when dealing with electrostatic forces as it directly ties into the core calculations involving charged particles.

Force Balance Equation

The force balance equation in this context involves setting the gravitational force equal to the electrostatic force to determine the balance point between them.
The equation is:

• \( m \times g = \frac{k \times e^2}{r^2} \)
• where the values for \( m \), \( g \), \( e \), \( k \), and the unknown \( r \) or distance are used to solve this balance.

By equating these forces, one can identify the precise distance, or "y" value, needed to ensure the forces are balanced. The equation highlights the interplay between gravitational and electrostatic forces on the same object, like our electron pair in the problem.