Problem 2 - University Physics with Modern Physics

Question

Fields within the Atom. In the Bohr model of the hydrogen atom, the electron moves in a circular orbit of radius \(5.3 \times 10^{-11} \mathrm{m}\) with a speed of \(2.2 \times 10^{6} \mathrm{m} / \mathrm{s}\) . If we are viewing the atom in such a way that the electron's orbit is in the plane of the paper with the electron moving clockwise, find the magnitude and direction of the electric and magnetic fields that the electron produces at the location of the nucleus (treated as a point).

Step-by-Step Solution

Answer

Electric field: \(5.12 \times 10^{11} \mathrm{N/C}\), to the electron; Magnetic field: \(12.5 \times 10^{-6} \mathrm{T}\), into the plane.

Step 1: Understanding the Problem

In the Bohr model, the electron orbits the nucleus in a circular path, creating both an electric and magnetic field. We need to calculate the magnitude and direction of these fields at the nucleus's location.

Step 2: Calculate the Electric Field

The electric field created by a charge is calculated using Coulomb's law: \[ E = \frac{k_e \cdot q}{r^2} \] where \(k_e = 8.99 \times 10^9 \mathrm{N} \cdot \mathrm{m}^2/\mathrm{C}^2\), \(q = -1.6 \times 10^{-19} \mathrm{C}\) is the charge of the electron, and \(r = 5.3 \times 10^{-11} \mathrm{m}\) is the radius of the orbit. Substituting the values, we find: \[ E = \frac{8.99 \times 10^9 \times (-1.6 \times 10^{-19})}{(5.3 \times 10^{-11})^2} \approx -5.12 \times 10^{11} \mathrm{N}/\mathrm{C} \] The negative sign indicates the field direction is towards the electron.

Step 3: Calculate the Magnetic Field

The magnetic field created by a moving charge is calculated using: \[ B = \frac{\mu_0}{4\pi} \cdot \frac{qv}{r^2} \] where \(\mu_0 = 4\pi \times 10^{-7} \mathrm{T} \cdot \mathrm{m}/\mathrm{A}\), \(v = 2.2 \times 10^6 \mathrm{m/s}\), and \(q = -1.6 \times 10^{-19} \mathrm{C}\). Calculate: \[ B = \frac{10^{-7} \times (-1.6 \times 10^{-19}) \times 2.2 \times 10^6}{(5.3 \times 10^{-11})^2} \approx -12.5 \times 10^{-6} \mathrm{T} \] The direction using the right-hand rule for a clockwise-moving charge shows the field is into the plane of the paper.

Step 4: Summarize the Fields

The electric field at the nucleus is approximately \(5.12 \times 10^{11} \mathrm{N/C}\), directed towards the electron. The magnetic field is \(12.5 \times 10^{-6} \mathrm{T}\), directed into the plane.

Key Concepts

Electric Field

The electric field is a fundamental concept in physics, describing how charged particles influence each other in space. It originates from a charged particle, like an electron, and extends outward. The strength of this field is determined by Coulomb's law, a vital principle that allows us to calculate the electric field magnitude at a given point.
- In the context of the Bohr model, the electric field exerted by the electron can be thought of as a force per unit charge experienced by any charge that's near it, including the nucleus of the hydrogen atom.
- By using the equation \[ E = \frac{k_e \cdot q}{r^2} \]where \(k_e\) stands for Coulomb’s constant, we calculate the field strength at the nucleus.
This relationship shows that the field strength decreases with the square of the distance from the charge, emphasizing that as you move away from a charged particle, its influence wanes.
In our exercise, we found the field's magnitude to be approximately \(5.12 \times 10^{11} \mathrm{N/C}\), indicating a strong attraction back towards the electron, highlighting its role in the atom's stability.

Magnetic Field

A magnetic field arises from the motion of charged particles. When an electron, carrying a charge, moves in an orbit, it creates a magnetic field that affects the surrounding area.
- In the Bohr model of the hydrogen atom, this magnetic effect is due to the electron moving at a speed of \(2.2 \times 10^6 \mathrm{m/s}\).
- Using the formula for a moving charge's magnetic field: \[ B = \frac{\mu_0}{4\pi} \cdot \frac{qv}{r^2} \] you can determine its influence on the hydrogen atom's nucleus.
The right-hand rule is a handy tool to predict the direction of this field. In this scenario, because the electron is moving clockwise, this results in a magnetic field directed into the plane of the paper.
For this specific calculation, we derived a field strength of approximately \(12.5 \times 10^{-6} \mathrm{T}\). The small magnitude signifies a subtle but crucial influence, contributing to the interplay of forces holding the atom together.

Hydrogen Atom

The hydrogen atom, the simplest of all atoms, becomes an excellent model to explore fundamental physics concepts like electric and magnetic fields. In the Bohr model, the hydrogen atom consists of one electron rotating around a heavy and positively charged nucleus, which is a proton.
- By examining the forces and fields in a hydrogen atom, scientists can better understand atomic stability and electron behavior.
- The electron's orbit, in this model, allows physicists to calculate theoretical electric and magnetic fields that originate due to its charge and motion.
These fields provide insight into how the atom's structural integrity is maintained, supporting the concept that electric attraction keeps the electron bound, while its velocity and route determine the magnetic effects.
This atom exemplifies the intersection of classical physics and quantum mechanics, combining to give a depiction that's simple yet deeply informative about atomic interactions.

Coulomb's Law

Coulomb's Law forms the bedrock of electrostatics, describing how two charged particles interact in space. It posits that the force between two charges is proportional to the product of their magnitudes and inversely proportional to the square of the distance between them:
\[ F = \frac{k_e \cdot |q_1 \, q_2|}{r^2} \]
- This principle extends to calculating electric fields, crucial when considering the electron and proton within the hydrogen atom.
- In our problem, Coulomb's law facilitates determining the intensity of the electric field generated by the electron.
By understanding this inverse-square law, we envisage how changes in proximity between charges influence their interactions dramatically.
This concept helps clarify why electrons remain in specific orbits; the force of attraction described by Coulomb's law balances with the kinetic energy of the electron's orbit, supporting the overarching structure of atomic theory.