An electron orbits a proton at constant speed in a circle of...

An electron orbits a proton at constant speed in a circle of radius $r$. (a) Using Coulomb's law, write an expression for the magnitude of the electric force on the electron in terms of $r$, the elementary charge $e$, and the Coulomb constant $k$. (b) Apply Newton's second law to the electron and use it to show that the electron's speed is $$ v=\sqrt{\frac{k e^{2}}{m_{\mathrm{e}} r}} $$ [Hint: The electron is in uniform circular motion.] (c) Use the Bohr assumption about the electron's angular momentum, Eq. ( $27-17$ ), to show that the radius of the $n$ th Bohr orbit is $$ r_{n}=\frac{n^{2} \hbar^{2}}{m_{\mathrm{e}} k e^{2}} $$

An electron orbits a proton at constant speed in a circle of radius $r$. (a) Using Coulomb's law, write an expression for the magnitude of the electric force on the electron in terms of $r$, the elementary charge $e$, and the Coulomb constant $k$. (b) Apply Newton's second law to the electron and use it to show that the electron's speed is
$$
v=\sqrt{\frac{k e^{2}}{m_{\mathrm{e}} r}}
$$
[Hint: The electron is in uniform circular motion.] (c) Use the Bohr assumption about the electron's angular momentum, Eq. ( $27-17$ ), to show that the radius of the $n$ th Bohr orbit is
$$
r_{n}=\frac{n^{2} \hbar^{2}}{m_{\mathrm{e}} k e^{2}}
$$

Key Concepts

Coulomb's Law

Coulomb's law describes the electric force between two charged particles. It states that the magnitude of the force is directly proportional to the product of the magnitudes of the two charges and inversely proportional to the square of the distance between them. This law is fundamental in understanding interactions between charged particles in an electrostatic context.

Uniform Circular Motion and Centripetal Force

Uniform circular motion refers to the motion of an object traveling at constant speed along a circular path. To maintain this motion, a constant inward (centripetal) force is required. Newton's second law, when applied to circular motion, equates the net inward force to the mass of the object multiplied by the centripetal acceleration, which is given by the square of the speed divided by the radius of the circle.

Newton's Second Law

Newton's second law of motion establishes the relationship between force, mass, and acceleration. In the context of circular motion, this law is used by equating the net force acting on an object to the mass times its centripetal acceleration. This provides the framework to derive expressions for variables such as the speed of an object moving in a circular path.

Angular Momentum Quantization and the Bohr Model

The Bohr model of the atom incorporates the quantization of angular momentum to explain the discrete energy levels of electrons in an atom. The theory postulates that the angular momentum of an electron in orbit is restricted to integer multiples of a fundamental constant (typically expressed in terms of Planck's constant). This quantization condition leads to discrete allowed orbit radii and energy levels, which are consistent with observed atomic spectra.

Transcript

00:01 So this question is about the form model of an atom, hydrogen atom.

00:08 And we first want to get an expression for the electron speed.

00:14 And so we want to set the electric force equal to the mass times the acceleration.

00:22 And then so force for circular motion is mb squared over r.

00:31 And then the electric force has a magnitude of qe, and therefore v is equal to qe times r divided by m square rooted.

00:44 Just doing the algebra for that.

00:49 And then we want to get the kinetic energy.

00:53 Next, kinetic energy is equal to one -half mv squared.

01:01 So we want to do one half m times this squared, which is just q -e -r over m.

01:10 The m's cancel, so it's just one -half q -e times r.

01:24 Okay.

01:26 I'm just kind of curious that these are the terms that the question wants it in.

01:30 In terms of the electric field, should we use? i guess we can always plug in for the electric field.

01:40 Of a proton.

01:44 So i guess we can, let me modify these answers.

01:46 So we know e is equal to k, q, and then these are the same q.

01:53 So i'll go ahead and use that.

01:55 They have the charge of an electron and the charge of a proton have the same magnitude is what i meant over r squared...

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