91Ó°ÊÓ

A helium nucleus, also known as an \(\alpha\) (alpha) particle, consists of two protons and two neutrons and has a diameter of \(10^{-15} \mathrm{~m}\) \(=1 \mathrm{fm} .\) The protons, with a charge of te, are subject to a repulsive Coulomb force. since the neutrons have zero charge, there must be an attractive force that counteracts the electric repulsion and keeps the protons from flying apart. This so-called strong force plays a central role in particle physics. (a) As a crude model, assume that an \(\alpha\) particle consists of two pointlike protons attracted by a Hooke's-law spring with spring constant \(k,\) and ignore the neutrons. Assume further that in the absence of other forces, the spring has an equilibrium separation of zero. Write an expression for the potential energy when the protons are separated by distance \(d\). (b) Minimize this potential to find the equilibrium separation \(d_{0}\) in terms of \(e\) and \(k .\) (c) If \(d_{0}=1.00 \mathrm{fm}\), what is the value of \(k ?\) (d) How much energy is stored in this system, in terms of electron volts? (e) A proton has a mass of \(1.67 \times 10^{-27} \mathrm{~kg}\). If the spring were to break, the \(\alpha\) particle would disintegrate and the protons would fly off in opposite directions. What would be their ultimate speed?

Step by step solution

01

Start by modeling the potential energy

Due to the repulsion between the protons, the potential energy of the system between two protons with a spring can be modeled by Hooke's Law and Coulomb's Law. The spring potential energy is \(-\frac{1}{2}k \cdot si^{2}\), the coulomb potential energy is given by \(\frac{k_e \cdot e^{2}}{d}\). Adding the two potential energies, we have total energy \(U = \frac{1}{2}k \cdot d^{2} - \frac{k_e \cdot e^{2}}{d}\)

02

Minimize the potential energy

The equilibrium distance of separation \(d_0\) can be calculated by setting the derivative of potential energy to zero. The derivative is given by \( 0 = k \cdot d - \frac{k_e \cdot e^{2}}{d^{2}} \). Solving this equation gives \(d_0 = \sqrt{\frac{k_e \cdot e^{2}}{k}}\)

03

Calculate the spring constant

Given \(d_{0}=1.00 \mathrm{fm}\), substitute it into the equation from step 2, we obtain the spring constant \(k = \frac{k_e \cdot e^{2}}{d_{0}^{2}}\)

04

Determine the energy

The potential energy stored in the system can be calculated by plugging the equilibrium distance \(d_0\) into the equation of the potential energy. This gives \(U = - \frac{k_e \cdot e^{2}}{d_0}\)

05

Calculate the proton speed

Due to conservation of energy, all the potential energy stored in the system is converted to kinetic energy when the spring breaks. This gives the equation \(K = -U\). Since \(K = \frac{1}{2} m v^{2}\), where \(m\) is the mass and \(v\) the speed of the protons, we can get their ultimate speed by solving for \(v\) in the equation \(\frac{1}{2} m v^{2} = -U\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coulomb's Law

Coulomb's Law is a fundamental principle in electromagnetism describing the force between two point charges. According to this law, the electric force (\( F \)) between two charges (\( q_1 \) and and say, it explains the underlying reasons for the interaction between charged particles like protons in the nucleus of an atom.

For example, in a helium nucleus, the repulsive force keeping two positively charged protons apart is explained by Coulomb's Law, which states that the magnitude of this force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. Mathematically, it is represented as \( F = k_e \frac{|q_1 \cdot q_2|}{r^2} \) where \( k_e \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the separation distance.

When two protons in a helium nucleus are at a distance \( d \) apart, the potential energy due to the Coulomb force can be derived from this law and is given by \( U = \frac{k_e \cdot e^2}{d} \), where \( e \) is the charge of a proton. This formula plays a crucial role in determining the balance of forces and the potential energy within such nuclear systems.

Hooke's Law

Hooke's Law is a principle of physics that relates to the behavior of springs and other elastic objects. It states that the force needed to extend or compress a spring by some distance is proportional to that distance. This is represented by the equation \( F = -kx \) where \( F \) is the force applied to the spring, \( x \) is the displacement of the spring from its equilibrium position, and \( k \) is the spring constant which measures the stiffness of the spring.

When applying Hooke's Law to the helium nucleus model, it suggests that an invisible 'spring' is what holds together the two protons despite their mutual electric repulsion. The potential energy stored in this 'spring' when the protons are a distance \( d \) apart is \( U = \frac{1}{2}k \cdot d^2 \). Such a model helps to conceptualize the strong nuclear force that acts over short distances within the atomic nucleus to keep it stable and balanced against the electrical repulsion between protons.

Potential Energy Minimization

Potential energy minimization is a concept in physics that states that a system will naturally move towards a state of minimal potential energy. In the context of forces within an atom, this principle allows us to understand the stability of atomic structures.

In the modeling of the helium nucleus, potential energy minimization is crucial for finding the equilibrium separation of protons. By doing so, we are essentially finding the point at which the forces of attraction (modeled by Hooke's Law) and repulsion (described by Coulomb's Law) are balanced. Mathematically, this involves taking the derivative of the total potential energy function and setting it to zero to find the minimum value.

The equilibrium distance \( d_0 \) is the separation at which the potential energy of the system reaches its minimal value. According to the step-by-step solution, \( d_0 \) is given by \( d_0 = \sqrt{\frac{k_e \cdot e^2}{k}} \), where \( k_e \) is Coulomb's constant, \( e \) is the charge of the proton, and \( k \) is the spring constant. At this separation, the forces are in perfect balance, thus explaining the relative stability of the helium nucleus.