Problem 75 - Fundamentals of Physics

Question

The charges of an electron and a positron are \(-e\) and \(+e .\) The mass of each is \(9.11 \times 10^{-31} \mathrm{~kg} .\) What is the ratio of the electrical force to the gravitational force between an electron and a positron?

Step-by-Step Solution

Answer

The ratio \( \frac{F_e}{F_g} \) is approximately \( 2.27 \times 10^{39} \).

Step 1: Understand the Forces

The electrical force between two charges is given by Coulomb's Law: \( F_{e} = \frac{k imes |q_1 \times q_2|}{r^2} \). For an electron and a positron, \( q_1 = -e \) and \( q_2 = +e \), so \( F_e = \frac{k \times e^2}{r^2} \), where \( k = 8.99 \times 10^9 \frac{N \cdot m^2}{C^2} \) and \( e = 1.6 \times 10^{-19} C \).

Step 2: Calculate Electrical Force

Substitute the values of \( k \) and \( e \) into the formula: \( F_e = \frac{8.99 \times 10^9 \times (1.6 \times 10^{-19})^2}{r^2} \).

Step 3: Understand Gravitational Force

The gravitational force between two masses is given by Newton's Law of Universal Gravitation: \( F_g = \frac{G \times m_1 \times m_2}{r^2} \). For an electron and positron, \( m_1 = m_2 = 9.11 \times 10^{-31} \text{ kg} \). The gravitational constant \( G = 6.67 \times 10^{-11} \frac{Nm^2}{kg^2} \).

Step 4: Calculate Gravitational Force

Substitute the values of \( G \), \( m_1 \), and \( m_2 \) into the formula: \( F_g = \frac{6.67 \times 10^{-11} \times (9.11 \times 10^{-31})^2}{r^2} \).

Step 5: Find the Ratio of Forces

To find the ratio of the electrical force to the gravitational force, calculate \( \frac{F_e}{F_g} \): \( \frac{F_e}{F_g} = \frac{\frac{8.99 \times 10^9 \times (1.6 \times 10^{-19})^2}{r^2}}{\frac{6.67 \times 10^{-11} \times (9.11 \times 10^{-31})^2}{r^2}} \). The \( r^2 \) terms cancel out.

Step 6: Simplify the Expression

Simplify the expression to find: \( \frac{F_e}{F_g} = \frac{8.99 \times 10^9 \times (1.6 \times 10^{-19})^2}{6.67 \times 10^{-11} \times (9.11 \times 10^{-31})^2} \).Calculate this expression to find the value of \( \frac{F_e}{F_g} \).

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in physics that describes the electric force between two charged objects. It states that the magnitude of the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The formula for this is:

• \( F_{e} = \frac{k \times |q_1 \times q_2|}{r^2} \)

where:

• \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \frac{N \cdot m^2}{C^2} \)
• \( q_1 \) and \( q_2 \) are the charges
• \( r \) is the distance between the charges

In the case of an electron and a positron, the charges are \( -e \) and \( +e \), respectively. When plugged into the formula, Coulomb's Law helps us calculate the electrical force acting between these two particles. This force is significant due to both the small magnitude of charges and the large constant \( k \), making the electric force generally much stronger than gravitational forces.

Newton's Law of Universal Gravitation

Newton's Law of Universal Gravitation explains the gravitational force between two masses. It posits that every mass attracts every other mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance separating their centers. The expression of this law is:

• \( F_{g} = \frac{G \times m_1 \times m_2}{r^2} \)

where:

• \( G \) is the gravitational constant, about \( 6.67 \times 10^{-11} \frac{Nm^2}{kg^2} \)
• \( m_1 \) and \( m_2 \) are the masses
• \( r \) is the distance between the masses

For tiny particles like an electron and a positron, each with a mass of \( 9.11 \times 10^{-31} \text{ kg} \), the gravitational force is calculated using this formula. While significant on a cosmic scale, gravitational forces between particles are exceedingly small and often negligible compared to electromagnetic forces because of the lesser value of \( G \).

Electron and Positron Interaction

An electron and a positron interact through both electrical and gravitational forces. However, the electrical force, governed by Coulomb's Law, predominantly dictates their interaction. This is because the magnitudes of electric charges are much larger compared to their masses when considering fundamental particles.
An electron carries a charge of \(-e\), and a positron carries a charge of \(+e\). Thus, both attract each other due to the opposite charges, resulting in a force that is strong and noticeable at smaller scales.

• The very nature of their charges—opposite yet equal in magnitude—ensures powerful attraction.
• They can undergo annihilation when they meet, converting their mass to energy.

On the other hand, the gravitational force between them is minuscule. Though present, it is practically imperceptible compared to the electrical force. Therefore, in particle physics, while both forces are accounted for, electrical interactions take the forefront when studying electron-positron interactions.