Ch 37: The Foundations of Modern Physics

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 37: The Foundations of Modern Physics

Problem 46b

Chapter 37, Problem 46b

A classical atom that has an electron orbiting at frequency ⨍ would emit electromagnetic waves of frequency ⨍ because the electron's orbit, seen edge-on, looks like an oscillating electric dipole. What is the total mechanical energy of this atom?

Verified step by step guidance

Understand the problem: The total mechanical energy of an atom is the sum of its kinetic energy and potential energy. For a classical atom, the electron is assumed to move in a circular orbit around the nucleus under the influence of the Coulomb force, which acts as the centripetal force.

Express the centripetal force: The centripetal force required to keep the electron in circular motion is provided by the Coulomb force. Using Coulomb's law, the force is given by:

\frac{k e_{2}}{r^{2}}

, where

k

is Coulomb's constant,

e_{2}

is the charge of the electron squared, and

r

is the radius of the orbit.

Relate the centripetal force to the electron's motion: The centripetal force is also equal to the mass of the electron times its centripetal acceleration. This can be written as:

F = \frac{m^{v}}{r}

, where

m

is the mass of the electron,

v

is its velocity, and

r

is the radius of the orbit.

Equate the two expressions for force: Set the Coulomb force equal to the centripetal force to find the relationship between the electron's velocity and the radius of its orbit:

\frac{k e_{2}}{r^{2}} = \frac{m^{v}}{r}

. Simplify this equation to solve for

v

in terms of

r

Calculate the total mechanical energy: The total mechanical energy is the sum of the kinetic energy and potential energy. The kinetic energy is given by

\frac{1}{2} m v^{2}

, and the potential energy due to the Coulomb force is

- \frac{k}{e_{2}} r

. Substitute the expression for

v

from the previous step into these formulas, and combine them to find the total mechanical energy.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electromagnetic Waves

Electromagnetic waves are oscillations of electric and magnetic fields that propagate through space. In the context of atomic physics, when an electron transitions between energy levels, it can emit or absorb electromagnetic radiation at a frequency corresponding to the energy difference between those levels. This is fundamental to understanding how atoms interact with light and other forms of electromagnetic radiation.

Mechanical Energy in Atoms

The total mechanical energy of an atom is the sum of its kinetic and potential energy. For an electron in an atom, kinetic energy arises from its motion, while potential energy is due to the electrostatic attraction between the negatively charged electron and the positively charged nucleus. Understanding this energy balance is crucial for analyzing the stability and behavior of atoms.

Electric Dipole Moment

An electric dipole moment is a measure of the separation of positive and negative charges within a system. In the case of an electron orbiting a nucleus, the oscillation of the electron creates a time-varying dipole moment, which is responsible for the emission of electromagnetic radiation. This concept is key to understanding how atoms emit light and the nature of their interactions with electromagnetic fields.

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