E18M.6 The Rutherford model of the hydrogen atom (pro- posed by Ernst Rutherford in roughly 1910) imagines the electron as orbiting the proton in a circle (a) Use Newton's second law, Coulomb's law, and what you know about acceleration in circular motion to show that the electron's acceleration in a circular orbit of radius r is ã/(4me,mr) and its orbital kinetic energy is K-72/ (8πεο r), where e is the proton's charge and m is the electron's mass. (b) Show that the electron's total orbital energy (kinetic + potential) is E/(8mEoP) (c) Because the electron is accelerating, it will radiate energy in the form of electromagnetic waves. Assum- ing that it does so slowly enough so that its orbit remains essentially circular, show that the Larmor for- mula predicts that the rate at which it radiates energy is (d) This has to come at the expense of the electron's orbital energy E, so (since E decreases with time) P-dE/dt Show using the result of part (b) that we also have (E18.21) dt STEr di (e) Set P in equation E18.20 to -dE/dt given above and rearrange things to show that (f) Define t 0 to be when0.053 nm (the approx- imate measured radius of the hydrogen atom). As it radiates energy, the electron will spiral inward until it reaches r 0 at a time we will define to be t T Integrate the left side of equation E18.22 fromr r to r 0 and the right side from 0 tot Tand solve for T to show that (E18.23) (g) Show that eシ(4 πε。mc2-2.8 × 10 15 m, and then cal- culate T. Is the Rutherford model plausible? E18M.6 The Rutherford model of the hydrogen atom (pro- posed by Ernst Rutherford in roughly 1910) imagines the electron as orbiting the proton in a circle (a) Use Newton's second law, Coulomb's law, and what you know about acceleration in circular motion to show that the electron's acceleration in a circular orbit of radius r is ã/(4me,mr) and its orbital kinetic energy is K-72/ (8πεο r), where e is the proton's charge and m is the electron's mass. (b) Show that the electron's total orbital energy (kinetic + potential) is E/(8mEoP) (c) Because the electron is accelerating, it will radiate energy in the form of electromagnetic waves. Assum- ing that it does so slowly enough so that its orbit remains essentially circular, show that the Larmor for- mula predicts that the rate at which it radiates energy is (d) This has to come at the expense of the electron's orbital energy E, so (since E decreases with time) P-dE/dt Show using the result of part (b) that we also have (E18.21) dt STEr di (e) Set P in equation E18.20 to -dE/dt given above and rearrange things to show that (f) Define t 0 to be when0.053 nm (the approx- imate measured radius of the hydrogen atom). As it radiates energy, the electron will spiral inward until it reaches r 0 at a time we will define to be t T Integrate the left side of equation E18.22 fromr r to r 0 and the right side from 0 tot Tand solve for T to show that (E18.23) (g) Show that eシ(4 πε。mc2-2.8 × 10 15 m, and then cal- culate T. Is the Rutherford model plausible?