Ch 23: The Electric Field

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 23: The Electric Field

Problem 59

Chapter 23, Problem 59

INT In a classical model of the hydrogen atom, the electron orbits the proton in a circular orbit of radius 0.053 nm. What is the orbital frequency in rev/s? The proton is so much more massive than the electron that you can assume the proton is at rest.

Verified step by step guidance

Identify the forces acting on the electron. The centripetal force required to keep the electron in circular motion is provided by the electrostatic force between the electron and the proton. Use Coulomb's law to express the electrostatic force:

F = \frac{k e_{2}}{r^{2}}

, where

k

is Coulomb's constant,

e_{2}

is the charge of the electron squared, and

r

is the radius of the orbit.

Set the centripetal force equal to the electrostatic force to find the velocity of the electron. The centripetal force is given by

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

, where

m

is the mass of the electron and

v

is its velocity. Equating the two forces gives:

\frac{k e_{2}}{r^{2}} = \frac{m v^{2}}{r}

. Solve for

v

Express the orbital frequency in terms of the velocity and radius. The orbital frequency

f

is the number of revolutions per second, which can be calculated using the relationship

f = \frac{v}{2 π r}

, where

v

is the orbital velocity and

r

is the radius of the orbit.

Substitute the expression for

v

obtained from the force balance equation into the formula for

f

. This will give the orbital frequency in terms of known constants:

k

e_{2}

m

, and

r

Simplify the expression to calculate the orbital frequency. Ensure all units are consistent (e.g., convert the radius from nanometers to meters) before substituting numerical values for the constants. The final expression will yield the orbital frequency in revolutions per second.

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Key Concepts

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

Orbital Frequency

Orbital frequency refers to the number of complete revolutions an object makes around a central point per unit time, typically measured in revolutions per second (rev/s). In the context of an electron orbiting a proton, it can be calculated using the relationship between the orbital radius and the centripetal force acting on the electron due to the electrostatic attraction between the charged particles.

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, directed towards the center of the circle. For an electron in a hydrogen atom, this force is provided by the electrostatic attraction between the negatively charged electron and the positively charged proton. The balance of this force with the electron's inertia allows it to maintain a stable orbit.

Bohr Model of the Atom

The Bohr model describes the hydrogen atom as having electrons in fixed orbits around the nucleus, with quantized energy levels. This model simplifies the complex interactions in atomic physics by allowing calculations of properties like orbital radius and frequency. It provides a foundational understanding of atomic structure and behavior, particularly for hydrogen, which is the simplest atom.

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Related Practice

Textbook Question

The two parallel plates in FIGURE P23.53 are 2.0 cm apart and the electric field strength between them is 1.0×10⁴ N/C. An electron is launched at a 45° angle from the positive plate. What is the maximum initial speed v₀ the electron can have without hitting the negative plate?

243

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Textbook Question

CALC A uniform electric field’s strength is increasing with time as $E = (1.5 \times 10^{4} + (5.0 \times 10^{10} s^{- 1}) t) N/C$ . A proton is released in the field from rest at t = 0. What is the proton’s speed 1.0 μs later?

1113

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Textbook Question

A problem of practical interest is to make a beam of electrons turn a 90° corner. This can be done with the parallel-plate capacitor shown in FIGURE P23.55. An electron with kinetic energy 3.0×10⁻¹⁷ J enters through a small hole in the bottom plate of the capacitor. Should the bottom plate be charged positive or negative relative to the top plate if you want the electron to turn to the right? Explain.

229

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Textbook Question

A rod of length $L$ lies along the $y$ -axis with its center at the origin. The rod has a nonuniform linear charge density $λ = a ∣ y ∣$ , where a is a constant with the units C/m². Draw a graph of $λ$ versus $y$ over the length of the rod.

363

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Textbook Question

A rod of length $L$ lies along the $y$ -axis with its center at the origin. The rod has a nonuniform linear charge density $λ=a|y|$ , where a is a constant with the units C/m². Determine the constant a in terms of $L$ and the rod's total charge $Q$ .

1860

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Textbook Question

An electric field can induce an electric dipole in a neutral atom or molecule by pushing the positive and negative charges in opposite directions. The dipole moment of an induced dipole is directly proportional to the electric field. That is, $\vec{p} = α \vec{E}$ , where α is called the polarizability of the molecule. A bigger field stretches the molecule farther and causes a larger dipole moment. An ion with charge q is distance r from a molecule with polarizability α. Find an expression for the force _{$\vec{E}$ ion on dipole}.

2209

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