Concept explainers
(a)
Show that an electron in a classical hydrogen atom spirals into the nucleus at a rate of
(a)
Answer to Problem 6P
An electron in a classical hydrogen atom spirals into the nucleus at a rate of
Explanation of Solution
The uniform circular motion of the electron as a particle about the proton in the hydrogen atom experiences a force which can be expressed as,
Here,
The force can be expressed in terms of Newton’s second law,
Here,
Use equation (I) in equation (II) and solve for
Write the expression for the centripetal acceleration.
Here,
Use equation (II) in equation (IV),
The value of the
Write the expression for the total energy,
Here,
Write the expression for the kinetic energy.
Write the expression for the potential energy.
Use equation (VIII) and (IX) in equation (VII),
Use equation (VI) in equation (X),
The given expression connecting
Use equation (XI) and equation (III) in equation (XII),
Conclusion:
Therefore, from equation (XIII) it is shown that an electron in a classical hydrogen atom spirals into the nucleus at a rate as
(b)
The time interval over which the electron reaches
(b)
Answer to Problem 6P
The time interval over which the electron reaches
Explanation of Solution
Write the expression for the time interval in terms of
Solve equation (XIII) for
Conclusion:
Substitute
Therefore, the time interval over which the electron reaches
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Chapter 42 Solutions
Physics for Scientists and Engineers With Modern Physics
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