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 48a

Chapter 37, Problem 48a

Physicists first attempted to understand the hydrogen atom by applying the laws of classical physics. Consider an electron of mass m and charge −e in a circular orbit of radius r around a proton of charge +e. Use Newtonian physics to show that the total energy of the atom is E =−e²/8πϵ₀𝓇

Verified step by step guidance

Start by analyzing the forces acting on the electron. The electron is in a circular orbit around the proton, so the centripetal force required to keep the electron in orbit is provided by the electrostatic (Coulomb) force between the electron and proton. Write the equation for the Coulomb force:

F = \frac{e^{2}}{4 π ε 0 r^{2}}

Equate the Coulomb force to the centripetal force, which is given by

F = \frac{m v^{2}}{r}

, where

m

is the mass of the electron and

v

is its velocity. This gives the equation:

\frac{e^{2}}{4 π ε 0 r^{2}} = \frac{m v^{2}}{r}

Solve for the velocity

v

of the electron in terms of the given quantities. Rearrange the equation to get:

v = \sqrt{\frac{e^{2}}{4 π ε 0 m r}}

Next, calculate the total energy of the system. The total energy

E

is the sum of the kinetic energy

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

and the potential energy

U = - \frac{e^{2}}{4 π ε 0 r}

. Substitute the expression for

v

into the kinetic energy formula.

Combine the kinetic and potential energy expressions. After substituting and simplifying, you will find that the total energy is:

E = - \frac{e^{2}}{8 π ε 0 r}

. This shows that the total energy of the atom is negative, indicating a bound system.

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Key Concepts

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

Coulomb's Law

Coulomb's Law describes the electrostatic force between two charged particles. It states that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This law is fundamental in understanding the interaction between the electron and proton in a hydrogen atom.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. For an electron in circular motion, its kinetic energy can be expressed as KE = (1/2)mv², where m is the mass of the electron and v is its velocity. This concept is crucial for calculating the total energy of the hydrogen atom, as it contributes to the overall energy balance.

Potential Energy in Electrostatics

In electrostatics, the potential energy (PE) between two point charges is given by PE = k(q₁q₂/r), where k is Coulomb's constant, q₁ and q₂ are the charges, and r is the distance between them. For the hydrogen atom, the potential energy of the electron-proton system is negative, indicating a bound state, and is essential for deriving the total energy of the atom.

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Related Practice

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Textbook Question

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