Problem 57 - Physics

Question

Concept Questions (a) In a nucleus, each proton experirences a repulsive electrostatic force from each of the other protons. Write an expression for the magnitude of the force that one proton (charge \(=+e\) ) applies to another proton that is located a distance \(r\) away. (b) The force that acts on either of two particular protons in the nucleus has the smallest possible magnitude. Relative to one another, where in the nucleus must these two protons be located? Explain.

Step-by-Step Solution

Answer

(a) The force is \( F = k \frac{e^2}{r^2} \). (b) For minimal force, protons should be as far apart as possible.

Step 1: Identify the Formula for Electrostatic Force

The electrostatic force between two charges is given by Coulomb's Law. The formula for the magnitude of the force between two point charges is: \[ F = k \frac{|q_1 q_2|}{r^2} \]where \( F \) is the magnitude of the force, \( k \) is Coulomb's constant \( (8.988 \times 10^9 \text{ N m}^2/\text{C}^2) \), \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance between the charges.

Step 2: Apply Coulomb's Law to Two Protons

In the nucleus, each proton has a charge of \( +e \), where \( e \) is the elementary charge (\( 1.602 \times 10^{-19} \text{ C} \)). For two protons separated by a distance \( r \), the force one proton exerts on another is: \[ F = k \frac{e^2}{r^2} \]This expression represents the magnitude of the electrostatic force between two protons.

Step 3: Determine the Location for Minimal Force

The magnitude of the electrostatic force between two protons decreases as the distance between them increases. Therefore, the force that acts on either of the two protons will be smallest when the distance \( r \) between them is largest. Within the constraints of a nucleus, this implies that the protons must be located as far apart as possible, which typically means at opposite ends of the nucleus if it were an ideal sphere.

Key Concepts

Electrostatic Force

When we think about the interactions between charges, electrostatic force comes into play. Coulomb's Law guides us in understanding this force between two charged objects.

For instance, in a nucleus with protons, each having a positive elementary charge, they repel each other due to their like charges. This repulsive force is calculated using the equation:

• \[ F = k \frac{|q_1 q_2|}{r^2} \]

where \( F \) is the force magnitude, \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges (in this case, both are \(+e\) for protons), and \( r \) is the distance between the charges.

This formula essentially tells us that the force magnitude increases as the charges get closer. That's why protons in a nucleus, being quite close, generate substantial electrostatic force against each other.

Elementary Charge

A really interesting aspect of protons is their charge, measured as an elementary charge, denoted by \( e \). It's the basic unit of electric charge in the atomic scale.

Protons possess a positive elementary charge valued at \( +1.602 \times 10^{-19} \text{ C} \). Electrons carry the same magnitude but with negative sign, showing their opposite nature to protons.

• Why is this tiny charge important? Because it's a fundamental building block of matter.
• It's part of what dictates how atoms and molecules form, affecting chemical interactions at the most fundamental level.
• In nuclear chemistry and physics, understanding the elementary charge is crucial in explaining behaviors like proton interactions.

Nuclear Forces

Despite the strong electrostatic repulsion between protons, they are held together in the nucleus, which poses a curious question. How?

• This is where nuclear forces, also called strong forces, come into effect.
• These forces are much stronger than the electrostatic forces but act only over short distances, typically within the nucleus itself.
• They are crucial because without these forces, the nucleus wouldn't be stable due to the electrostatic repulsion between protons.

The profound strength and the limited range of nuclear forces ensure that atoms can remain intact, allowing for the existence of all matter as we know it.

These forces play a significant role in various nuclear reactions and processes, including fission and fusion.

Proton Interactions

When analyzing proton interactions in the nucleus, we see a blend of forces at work. Protons are positive, leading to the known repulsive electrostatic force. Yet, they sit together in the nucleus due to the stronger nuclear forces holding them in place.

The balance between these two forces determines the structure of the nucleus:

• Electrostatic forces try to push protons apart.
• Nuclear forces hold them close together.
• Proper balance is key; if forces were out of this balance, nuclei could be unstable, affecting atom stability.

Understanding these interactions is critical not just for physics but also for advancements in nuclear energy research and applications, as well as in medicine, through techniques like proton therapy.