Problem 3 - University Physics with Modern Physics

Question

Energy of the Nucleus. How much work is needed to assemble an atomic nucleus containing three protons (such as Be) if we model it as an equilateral triangle of side \(2.00 \times 10^{-15} \mathrm{m}\) with a proton at each vertex? Assume the protons started from very far away.

Step-by-Step Solution

Answer

The work needed is \(3.45 \times 10^{-13} \mathrm{J}\).

Step 1: Understand Coulomb's Law

Coulomb's Law describes the electrostatic force between charged particles. The force between two point charges is given by: \[ F = \frac{k \, |q_1 \cdot q_2|}{r^2} \]where \( F \) is the force, \( k \) is Coulomb's constant \( (8.99 \times 10^9 \, \mathrm{Nm}^2/\mathrm{C}^2) \), \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between them. The potential energy \( U \) between two charges is:\[ U = \frac{k \, q_1 \cdot q_2}{r} \]

Step 2: Calculate Energy for One Pair of Protons

Each pair of protons, separated by \( r = 2.00 \times 10^{-15} \) meters, has a potential energy due to electrostatic interaction. Here, \( q_1 = q_2 = e \), where \( e = 1.60 \times 10^{-19} \) C is the charge of a proton.\[ U_{12} = \frac{k \, e^2}{r} = \frac{8.99 \times 10^9 \, (1.60 \times 10^{-19})^2}{2.00 \times 10^{-15}} \]

Step 3: Compute Energy for All Pairs of Protons

Since the protons form an equilateral triangle, there are three unique pairwise interactions (proton 1 and 2, proton 1 and 3, and proton 2 and 3). Calculate the energy for these three interactions:\[ U_{total} = 3 \times U_{12} \]

Step 4: Solve the Equation

Substitute the values into the equation for total potential energy:\[ U_{12} = \frac{8.99 \times 10^9 \, (1.60 \times 10^{-19})^2}{2.00 \times 10^{-15}} = 1.15 \times 10^{-13} \, \mathrm{J} \]Then, multiply by 3 to account for all interactions:\[ U_{total} = 3 \times 1.15 \times 10^{-13} \, \mathrm{J} = 3.45 \times 10^{-13} \, \mathrm{J} \]

Step 5: Interpretation

The total work done in assembling the nucleus is equal to the total potential energy required to bring the three protons from infinity to form the equilateral triangle. This energy value is the work needed.

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental concept in physics that describes the force between two charged particles. When two charges are present, they exert a force upon each other. The magnitude of this force can be calculated using the formula:

• \( F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \)

Here, \( F \) represents the force between the charges, \( k \) is known as Coulomb's constant, approximately \( 8.99 \times 10^9 \, \mathrm{Nm}^2/\mathrm{C}^2 \), \( q_1 \) and \( q_2 \) are the values of the charges, and \( r \) is the distance between them.
A noteworthy aspect is that the force is inversely proportional to the square of the distance, meaning the further apart the charges are, the weaker the force.
Understanding this law is crucial when analyzing how charged particles interact in an atomic nucleus.

Potential Energy

Potential energy in the context of electrostatics refers to the energy stored due to the position of charged particles relative to one another. When addressing the potential energy between two point charges, it's given by:

• \( U = \frac{k \cdot q_1 \cdot q_2}{r} \)

In this equation, \( U \) stands for the potential energy, highlighting how the energy changes based on the separation \( r \) between two charges.
The potential energy is greater when the charges are closer, and it increases with the strength of the charges.
Achieving understanding of potential energy is essential for explaining phenomena such as the assembly of atomic nuclei.

Electrostatic Interaction

Electrostatic interaction refers to the force or potential energy between static (non-moving) electric charges. In the atomic level, these interactions are key to the structure and behavior of atoms.

For example, in the assembly of an atomic nucleus, protons, which are positively charged, will repel each other because like charges repel according to Coulomb's Law. However, the electrostatic force is also crucial for calculating the work needed to assemble these particles together, as seen in the atomic nucleus assembly process.
The process involves calculating the potential energy between each pair of charged particles, which includes:

• Considering each unique pair of interactions
• Using the potential energy formula to compute the energy for each pair

Understanding electrostatic interaction helps in calculating how much work and energy is involved in forming a stable structure like a nucleus.

Atomic Nucleus Assembly

The assembly of an atomic nucleus is an important concept when studying nuclear physics and electrostatics. It involves bringing protons, which naturally repel each other due to their positive charge, into close proximity to form a stable nucleus.

When modeling an atomic nucleus such as Beryllium, you can imagine protons arranged in an equilateral triangle if you have three protons. Each side of the triangle represents the distance over which the protons interact.
The work needed to assemble the nucleus correlates directly to the total potential energy of the interactions among the protons:

• Each unique pair of protons contributes a portion to the total potential energy.
• Calculating this involves summing the potential energies of all unique interactions.
• This illustrates how much energy is required to overcome electrostatic repulsion to form the nucleus.

Grasping the idea of atomic nucleus assembly provides a deeper understanding of how energy dynamics play out at the subatomic level.