Modern Physics for Scientists and Engineers
Modern Physics for Scientists and Engineers
4th Edition
ISBN: 9781133103721
Author: Stephen T. Thornton, Andrew Rex
Publisher: Cengage Learning
bartleby

Concept explainers

Question
Book Icon
Chapter 4, Problem 18P
To determine

The time taken by the electron of the hydrogen atom to radiate its energy and crash into the nucleus according to classical laws.

Expert Solution & Answer
Check Mark

Answer to Problem 18P

The time taken by the electron of the hydrogen atom to radiate its energy and crash into the nucleus is 1.56×1011s.

Explanation of Solution

Write the expression for E0.

    E0=e28πε0a0

Here, E0 is the Energy, a0 is the Bohr’s radius and e is the charge in electron.

Differentiate above equation with respect to time.

    dE0dt=e28πε0r2drdt        (I)

Write the expression for radiated Power in electromagnetic theory.

    P=14πε0(2Q23c3)(d2rdt2)2        (II)

Here, P is the power radiated, Q  is the charge, c is the speed of light, t is the time and r is the position vector of the electron from the center of the atom.

Power radiated is equal to Energy radiated per unit time.

Compare equation (I) with equation (II).

  e28πε0r2drdt=14πε0(2Q23c3)(d2rdt2)2

Simplify above equation.

    e22r2drdt=(2Q23c3)(d2rdt2)2        (III)

In a circular orbit, centripetal acceleration is given by: d2rdt2 .

Write the expression for centripetal acceleration.

    a=Fm        (IV)

Here, a is the radial acceleration, F is the centripetal force, and m is the mass.

Write the expression for force.

    F=e24πε0r2        (V)

Here, F is the force.

Substitute  e24πε0r2 for F in equation (IV).

    a=e24πε0mr2

Substitute e24πε0mr2 for d2rdt2 in equation (III).

    e22r2drdt=(2Q23c3)(e24πε0mr2)2drdt=4e4(4πε0)23c3m2r2

Rearrange above equation.

    dt=(4πε0)23c3m2r24e4dr

Integrate above equation.

    dt=(4πε0)23c3m24e4a00r2drt=(4πε0)23c3m24e4[r33]a00=(4πε0)2c3m2a034e4        (VI)

Conclusion:

Substitute 9×109 for 14πε0 , 1.6×1019 for e , 3×108m/s for c9.1×1031 for m  and 5.29×1011m for a0 in

    t=(3×108m/s)3(9.1×1031)2(5.29×1011m)3(9×109)24(1.6×1019)4=1.56×1011s

Thus, the time taken by the electron of the hydrogen atom to radiate its energy and crash into the nucleus is 1.56×1011s.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Students have asked these similar questions
suggest a reason ultrasound cleaning is better than cleaning by hand?
Checkpoint 4 The figure shows four orientations of an electric di- pole in an external electric field. Rank the orienta- tions according to (a) the magnitude of the torque on the dipole and (b) the potential energy of the di- pole, greatest first. (1) (2) E (4)
What is integrated science. What is fractional distillation What is simple distillation
Knowledge Booster
Background pattern image
Physics
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.
Similar questions
SEE MORE QUESTIONS
Recommended textbooks for you
  • Text book image
    Modern Physics
    Physics
    ISBN:9781111794378
    Author:Raymond A. Serway, Clement J. Moses, Curt A. Moyer
    Publisher:Cengage Learning
Text book image
Modern Physics
Physics
ISBN:9781111794378
Author:Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher:Cengage Learning