Problem 1 - Fundamentals of Physics

Question

A \({ }^{7} \mathrm{Li}\) nucleus with a kinetic energy of \(3.00 \mathrm{MeV}\) is sent toward a \({ }^{232} \mathrm{Th}\) nucleus. What is the least center-to- center separation between the two nuclei, assuming that the (more massive) \({ }^{232} \mathrm{Th}\) nucleus does not move?

Step-by-Step Solution

Answer

The least center-to-center separation is approximately \(3.48 \times 10^{-14}\) meters.

Step 1: Understand the Concept of Coulomb's Repulsion

The closest approach between the two nuclei can be determined using Coulomb's Law, which explains the electrostatic force between charged particles. The Li nucleus will come to a halt due to this electrostatic repulsion, so its initial kinetic energy will be converted to potential energy at the point of closest approach.

Step 2: Set Up the Energy Conservation Equation

Write the energy conservation equation for the kinetic and potential energy: \[\text{Initial Kinetic Energy} = \text{Coulomb Potential Energy at closest approach}\]The formula for Coulomb Potential Energy is:\[U = \frac{k_e \cdot Q_1 \cdot Q_2}{r}\]Where \(k_e\) is Coulomb's constant, \(Q_1\) and \(Q_2\) are the charges, and \(r\) is the distance between charges at the least separation.

Step 3: Calculate the Charges

For \(^{7} \text{Li}\), it has 3 protons, so charge \(Q_1 = 3e\). For \(^{232} \text{Th}\), it has 90 protons, so charge \(Q_2 = 90e\). Here, \(e\) is the charge of a single proton \(e = 1.6 \times 10^{-19} \text{ C}\).

Step 4: Solve for the Least Center-to-Center Distance

Substitute the given kinetic energy \(3.00 \text{ MeV} = 3 \times 10^6 \times 1.6 \times 10^{-13} \text{ J}\), \(k_e = 8.988 \times 10^9 \text{ N m}^2/\text{C}^2\), \(Q_1 = 3e\), and \(Q_2 = 90e\) in:\[3.00 \times 10^6 \times 1.6 \times 10^{-13} = \frac{8.988 \times 10^9 \times (3 \times 1.6 \times 10^{-19}) \times (90 \times 1.6 \times 10^{-19})}{r}\]Solving this equation will give the least center-to-center separation \(r\).

Step 5: Perform the Calculations

Solving the equation gives:\[r = \frac{8.988 \times 10^9 \times 3 \times 90 \times (1.6 \times 10^{-19})^2}{3.00 \times 10^6 \times 1.6 \times 10^{-13}} \approx 3.48 \times 10^{-14} \text{ meters}\]

Key Concepts

Coulomb's Law

Coulomb's Law describes the electrostatic force between two charged objects. It states that the force, which is either attractive or repulsive depending on the charge signs, is proportional to the product of the two charges and inversely proportional to the square of the distance between them. The formula is typically expressed as:

• \[ F = \frac{k_e \cdot Q_1 \cdot Q_2}{r^2} \]
• Where \( F \) is the electrostatic force, \( k_e \) is Coulomb's constant \( (8.988 \times 10^9 \; \text{Nm}^2/\text{C}^2) \), \( Q_1 \) and \( Q_2 \) are the charges, and \( r \) is the distance between the centers of the two charges.

In nuclear physics, Coulomb's Law is essential for understanding interactions between charged particles, such as protons in atomic nuclei. When particles are close enough, this force becomes quite significant, influencing motion and energy within atomic interactions. Understanding this law helps explain why charged particles behave the way they do under certain conditions. In our specific example, the law explains the repulsion between a positively charged lithium nucleus and a thorium nucleus.

Kinetic Energy

Kinetic energy is the energy an object has because of its motion. It plays a critical role in nuclear physics, particularly in nuclear reactions where particles collide or are scattered. The kinetic energy \( (KE) \) of an object can be calculated using the formula:

• \[ KE = \frac{1}{2}mv^2 \]
• Where \( m \) is the mass and \( v \) is the velocity of the object.

For our lithium nucleus moving towards the thorium nucleus, this energy is initially high, given as \(3.00 \text{ MeV}\). The motion of the nucleus provides the kinetic energy that enables it to approach the thorium nucleus. This energy will be converted into potential energy as the nucleus is eventually stopped and repelled by the thorium nucleus's Coulomb force. In many nuclear processes, analyzing changes in kinetic energy helps us understand particle interactions and their impact on nuclear stability or reactions like fusion and fission.

Potential Energy

Potential energy in a system regulated by electrostatic forces, such as between atomic nuclei, is often referred to as Coulomb potential energy. It is the energy stored by the virtue of the position of charged particles. When dealing with charged particles, the potential energy \( (U) \) can be expressed with the equation:

• \[ U = \frac{k_e \cdot Q_1 \cdot Q_2}{r} \]
• Where \( k_e \) is Coulomb's constant, \( Q_1 \) and \( Q_2 \) are the charges of interacting particles, and \( r \) is the distance separating them.

In the context of the lithium and thorium nuclei, the potential energy becomes crucial at the closest approach where all the kinetic energy is transferred into potential energy, halting the lithium nucleus' travel. Understanding potential energy is fundamental when studying forces in nuclear physics, as it helps describe how energy can be stored and how charged particles interact with each other at varying distances.

Electrostatic Force

The electrostatic force is one of the fundamental interactions in physics, especially significant in understanding atomic and nuclear interactions. This force acts between charged particles, such as the protons within atomic nuclei. It is described by Coulomb's Law and can be either attractive or repulsive:

• Repulsive when two particles have like charges (both positive or both negative).
• Attractive when particles have opposite charges (one positive, one negative).

In nuclear physics, electrostatic forces influence the structural integrity of nuclei and play a key role in nuclear reactions. When the lithium nucleus approaches the thorium nucleus, they experience a repulsive electrostatic force because both are positively charged. This force is responsible for converting kinetic energy into potential energy, stopping the lithium nucleus at its closest point, and determining the minimal separation between the two nuclei. Understanding this force helps explain the dynamics of nuclear stability, radioactive decay, and the mechanisms of nuclear reactions.