Problem 71 - Fundamentals of Physics

Question

An old model of a hydrogen atom has the charge \(+e\) of the proton uniformly distributed over a sphere of radius \(a_{0},\) with the electron of charge \(-e\) and mass \(m\) at its center. (a) What would then be the force on the electron if it were displaced from the center by a distance \(r \leq a_{0} ?\) (b) What would be the angular frequency of oscillation of the electron about the center of the atom once the electron was released?

Step-by-Step Solution

Answer

The force is \( -\frac{e^2}{4\pi\varepsilon_0 a_0^3} r \) and the angular frequency is \( \sqrt{\frac{e^2}{4\pi\varepsilon_0 a_0^3 m}} \).

Step 1: Understand the Problem

We are given a hydrogen atom model with a proton's charge distributed over a sphere and an electron at the center. We need to find the force on the electron when it is displaced and the angular frequency of oscillation.

Step 2: Force Inside a Uniformly Charged Sphere

According to Gauss's law, the electric field inside a uniformly charged sphere is proportional to the distance from the center. The field inside is given by:\[ E = \frac{1}{4\pi\varepsilon_0} \cdot \frac{Q}{a_0^3} r \] where \(Q = e\) is the charge, and \(r\) is the displacement from the center.

Step 3: Calculate Force on Electron

The force on the electron can be calculated using the formula: \( F = -eE \). Substituting the electric field \(E\) from the previous step, we get:\[ F = -e \left( \frac{1}{4\pi\varepsilon_0} \cdot \frac{e}{a_0^3} \right) r = -\frac{e^2}{4\pi\varepsilon_0 a_0^3} r \]This force is directed towards the center, acting like a restoring force.

Step 4: Determine the Oscillation Frequency

For a simple harmonic motion, the restoring force \( F \) can be compared to the form: \( F = -k r \), where \(k\) is the effective spring constant. Thus, we identify \(k\) as:\[ k = \frac{e^2}{4\pi\varepsilon_0 a_0^3} \]The angular frequency \( \omega \) is given by \( \omega = \sqrt{\frac{k}{m}} \). Substituting \(k\), we get:\[ \omega = \sqrt{\frac{e^2}{4\pi\varepsilon_0 a_0^3 m}} \]

Key Concepts

Electric Field

The concept of electric fields is fundamental in understanding the behavior of charged particles. Imagine a charged object creating an invisible "force field" around it. This field is what affects other charges in its vicinity. In our exercise, the proton's charge is distributed over a sphere and creates an electric field within this region.
When you displace the electron from its central position, it experiences a force due to this electric field. The field inside a uniformly charged sphere behaves in a specific way. According to a principle called Gauss's Law, the strength of the electric field is directly proportional to the distance from the center of the sphere. This relationship is described by the formula:
\[ E = \frac{1}{4\pi\varepsilon_0} \cdot \frac{Q}{a_0^3} r \]

• \(E\) represents the electric field.
• \(Q\) is the total charge of the sphere – here equal to the charge of a proton, \(e\).
• \(a_0\) is the radius of the sphere.
• \(r\) is the distance of the electron from the center.

This electrical force is crucial, as it translates directly into the force felt by the electron when displaced.

Gauss's Law

Gauss's Law is a powerful tool in electrostatics. It simplifies complex problems involving electric fields. Named after the German mathematician Carl Friedrich Gauss, the law relates the electric field to the distribution of electric charge.
In our scenario, Gauss's Law helps us determine the electric field inside a uniformly charged sphere. The key takeaway is that inside such a sphere, the electric field is proportional to the distance from the center. This means that the closer you are to the center, the weaker the field becomes. Conversely, as you move towards the surface, the field strength increases.
For our hydrogen atom model, Gauss's Law allows us to calculate the specific electric field acting on the displaced electron. This foundation is crucial for understanding the nature of the force acting on the electron and, subsequently, its motion.

Simple Harmonic Motion

When the displaced electron experiences a force pulling it back towards the center, it enters into a type of motion known as simple harmonic motion. This is the oscillatory motion experienced by objects like pendulums or springs.
Simple harmonic motion is characterized by a linear restoring force that is proportional to the displacement but directed towards the equilibrium position. In simple terms, the farther an object is displaced, the stronger the force pulling it back. This form of motion can be seen as a type of "back and forth" motion, much like how a swing moves.
In the hydrogen atom model, the force acting on the electron can be represented as:

• \(F = -k r\)

Where \(k\) is the spring constant, related to how stiff the "spring" is. Here, this concept applies to the electric force pulling the electron back to the center.

Angular Frequency

Angular frequency is a concept used to describe how quickly an object in harmonic motion oscillates. It is often denoted by the symbol \(\omega\) and signifies the rate of oscillation in radians per second.
In the context of the hydrogen atom with the electron oscillating back and forth, the angular frequency is a measure of how fast these oscillations occur. The angular frequency can be determined from the relationship:

• \[ \omega = \sqrt{\frac{k}{m}} \]

Where \(k\) represents the effective spring constant derived from the electric field strength, and \(m\) is the mass of the oscillating electron.
This equation shows how both the electric forces and the electron's mass affect the frequency of the oscillation. It provides a way to link the physical properties of the atom to the observed oscillatory behavior of the electron, deepening our understanding of atomic models and oscillatory movements.