How To Calculate The Distance Between Two Protons Such That The Electrical Repulsive Force Equals The Weight Of A Proton On Earth's Surface?

Introduction

In the realm of physics, understanding the fundamental forces that govern the interactions between particles is crucial. One such force is the electrical repulsive force, which arises between particles with like charges, such as protons. Protons, being positively charged subatomic particles, naturally repel each other due to this force. This article delves into a fascinating scenario where we explore the distance required between two protons for the electrical repulsive force acting on either one to equal its weight on the Earth's surface. This exploration combines concepts from electrostatics and mechanics, providing a comprehensive understanding of the interplay between these fundamental forces. To tackle this problem, we will employ Coulomb's Law, which quantifies the electrical force between charged particles, and the concept of weight, which is the force exerted on an object due to gravity. We will also consider the mass of a proton and the acceleration due to gravity on Earth's surface. By equating the electrical repulsive force to the weight of the proton, we can derive an equation that allows us to calculate the distance between the protons. This calculation will shed light on the magnitude of the electrical force at such small scales and its significance compared to gravitational forces. Furthermore, this exploration will highlight the immense strength of the electrical force compared to gravity at the subatomic level. The results will provide insights into the behavior of charged particles and the delicate balance of forces that govern the stability of matter.

Understanding the Forces at Play

To begin, let's first understand the forces involved in this scenario. We have the electrical repulsive force acting between the two protons and the weight of each proton due to Earth's gravity. The electrical force, also known as the electrostatic force, is described by Coulomb's Law, which states that the force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. Mathematically, this is expressed as:

$$F_e = k \frac{q_1 q_2}{r^2}$$

where:

• ${ F_e }$ is the electrical force,
• ${ k }$ is Coulomb's constant (approximately $8.99 imes 10^9 , \text{N} \cdot \text{m}^2/\text{C}2$),
• ${ q_1 }$ and ${ q_2 }$ are the magnitudes of the charges, and
• ${ r }$ is the distance between the charges.

In our case, both charges are protons, each carrying a charge (${ e }$) of approximately $1.602 imes 10^{-19} , \text{C}$. Therefore, ${ q_1 = q_2 = e $. The weight (( W }$) of an object is the force exerted on it due to gravity and is given by:

$$W = mg$$

where:

• ${ m }$ is the mass of the object, and
• ${ g }$ is the acceleration due to gravity (approximately $9.8 , \text{m/s}^2$ on Earth's surface).

For a proton, the mass (${ m }$) is given as $1.7 imes 10^{-27} , \text{kg}$. Thus, the weight of a proton on Earth's surface is:

$$W = (1.7 imes 10^{-27} \, \text{kg})(9.8 \, \text{m/s}^2) \approx 1.666 imes 10^{-26} \, \text{N}$$

Now, we need to find the distance (${ r }$) at which the electrical repulsive force (${ F_e }$) equals the weight (${ W }$) of the proton. This condition will allow us to balance the electrostatic repulsion with the gravitational force acting on the proton.

Setting Up the Equation

To determine how far apart the two protons must be, we need to set the electrical repulsive force equal to the weight of a proton. This condition represents a balance between the electrostatic repulsion and the gravitational force acting on the proton. By equating these two forces, we can derive an equation that allows us to calculate the distance (${ r }$) between the protons.

We have the electrical repulsive force given by Coulomb's Law:

$$F_e = k \frac{e^2}{r^2}$$

where:

• ${ k = 8.99 imes 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2 }$ is Coulomb's constant,
• ${ e = 1.602 imes 10^{-19} \, \text{C} }$ is the charge of a proton, and
• ${ r }$ is the distance between the protons.

The weight of the proton is given by:

$$W = mg$$

where:

• ${ m = 1.7 imes 10^{-27} \, \text{kg} }$ is the mass of the proton, and
• ${ g = 9.8 \, \text{m/s}^2 }$ is the acceleration due to gravity.

Setting the electrical force equal to the weight, we get:

$$k \frac{e^2}{r^2} = mg$$

This equation allows us to solve for the distance (${ r }$) at which the electrical repulsion equals the gravitational force on the proton. By rearranging this equation, we can isolate (${ r^2 }$) and subsequently find (${ r }$). This step is crucial in quantifying the separation required for the forces to balance, providing insight into the interplay between electrostatic and gravitational forces at the subatomic level.

Solving for the Distance

Now, let's solve the equation for the distance (${ r }$):

$$k \frac{e^2}{r^2} = mg$$

First, we rearrange the equation to isolate (${ r^2 }$):

$$r^2 = k \frac{e^2}{mg}$$

Next, we substitute the known values:

• ${ k = 8.99 imes 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2 }$
• ${ e = 1.602 imes 10^{-19} \, \text{C} }$
• ${ m = 1.7 imes 10^{-27} \, \text{kg} }$
• ${ g = 9.8 \, \text{m/s}^2 }$

$$r^2 = (8.99 imes 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2) \frac{(1.602 imes 10^{-19} \, \text{C})^2}{(1.7 imes 10^{-27} \, \text{kg})(9.8 \, \text{m/s}^2)}$$

$$r^2 = (8.99 imes 10^9) \frac{(2.566404 imes 10^{-38})}{(1.7 imes 10^{-27})(9.8)}$$

$$r^2 = (8.99 imes 10^9) \frac{2.566404 imes 10^{-38}}{1.666 imes 10^{-26}}$$

$$r^2 = (8.99 imes 10^9)(1.539 imes 10^{-12})$$

$$r^2 = 1.3835 imes 10^{-2} \, \text{m}^2$$

Now, we take the square root to find (${ r }$):

$$r = \sqrt{1.3835 imes 10^{-2} \, \text{m}^2}$$

$$r \approx 0.1176 \, \text{m}$$

Therefore, the distance between the two protons must be approximately 0.1176 meters for the electrical repulsive force acting on either one to be equal to its weight on the Earth's surface. This calculation demonstrates the significant distance required to balance the electrostatic force with gravity, highlighting the relative strengths of these forces at this scale.

Significance of the Result

The result we obtained, approximately 0.1176 meters, reveals a crucial insight into the balance of forces at the subatomic level. This distance signifies the separation required for the electrical repulsive force between two protons to equal the gravitational force acting on them on Earth's surface. At this distance, the electrostatic repulsion is counteracted by the relatively weak gravitational attraction, creating a state of equilibrium.

This outcome highlights the immense strength of the electrical force compared to gravity at the subatomic scale. While gravity is the dominant force governing large-scale phenomena such as planetary motion, the electrical force is far more potent at the level of individual particles. The fact that two protons must be separated by more than 10 centimeters for their electrical repulsion to match their weight underscores this disparity.

Furthermore, this calculation provides a tangible example of Coulomb's Law in action. By equating the electrical force derived from Coulomb's Law with the weight of the proton, we have demonstrated the practical application of this fundamental law of physics. The result reinforces the inverse square relationship between electrical force and distance, as even a small change in separation can significantly alter the force between the protons.

In the context of atomic and nuclear physics, this finding has implications for understanding the stability of matter. Protons, being positively charged, naturally repel each other. However, within the nucleus of an atom, protons are held together by the strong nuclear force, which overcomes this electrical repulsion. The balance between these forces is critical for the stability of atomic nuclei. Understanding the magnitude of the electrical repulsive force is essential for comprehending the conditions under which nuclei can exist and the energies involved in nuclear reactions.

In summary, the distance of approximately 0.1176 meters represents a critical threshold where the electrical repulsive force between two protons equals their weight on Earth. This result underscores the strength of the electrical force at the subatomic level, provides a practical application of Coulomb's Law, and offers insights into the forces governing the stability of matter.

Conclusion

In conclusion, we have successfully calculated the distance at which the electrical repulsive force between two protons equals the weight of a proton on Earth's surface. By applying Coulomb's Law and equating the electrical force with the gravitational force, we determined that the protons must be approximately 0.1176 meters apart for this condition to be met. This calculation underscores the significant strength of the electrical force compared to gravity at the subatomic level.

The process involved understanding the fundamental forces at play, setting up the appropriate equation, and solving for the distance. We began by defining the electrical force using Coulomb's Law, which describes the force between charged particles. We then calculated the weight of a proton on Earth's surface using the mass of a proton and the acceleration due to gravity. By equating these two forces, we derived an equation that allowed us to solve for the distance at which they balance.

The result highlights the importance of considering both electrical and gravitational forces when analyzing the interactions of charged particles. While gravity is the dominant force at macroscopic scales, the electrical force plays a crucial role at the atomic and subatomic levels. The relatively large distance required for the electrical force to equal the weight of a proton demonstrates the immense strength of the electrical force in comparison.

This exploration has provided valuable insights into the behavior of charged particles and the balance of forces that govern the stability of matter. The principles and calculations discussed here are fundamental to understanding a wide range of phenomena in physics, from atomic structure to nuclear reactions. The interplay between electrical and gravitational forces is a cornerstone of our understanding of the universe, and this example serves as a clear illustration of their relative magnitudes and effects.

By delving into this problem, we have not only determined a specific distance but also reinforced our understanding of the fundamental forces that shape the world around us. The concepts and methods employed here can be applied to a variety of similar problems, making this a valuable exercise in understanding the interplay of forces in physics.