Ch 22: Electric Charges and Forces

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 22: Electric Charges and Forces

Problem 55

Chapter 22, Problem 55

In a simple model of the hydrogen atom, the electron moves in a circular orbit of radius 0.053 nm around a stationary proton. How many revolutions per second does the electron make?

Verified step by step guidance

Start by identifying the centripetal force acting on the electron. In this case, the centripetal force 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.

Relate the centripetal force to the centripetal acceleration of the electron. The centripetal force is also given by

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

, where

m

is the mass of the electron,

v

is its velocity, and

r

is the radius of the orbit. Set the two expressions for force equal to each other:

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

Solve for the velocity

v

of the electron. Rearrange the equation to isolate

v

v = \sqrt{\frac{k e_{2}}{m}}

. Substitute the known values for

k

e_{2}

m

, and

r

to calculate

v

Determine the time it takes for the electron to complete one revolution. The circumference of the circular orbit is

C = 2 π r

. The time for one revolution is

T = \frac{C}{v}

, where

v

is the velocity of the electron.

Finally, calculate the number of revolutions per second. The frequency of revolution is the reciprocal of the period:

f = \frac{1}{T}

. Substitute the expression for

T

to find

f

f = \frac{v}{2}

. Use the previously calculated value of

v

to find the frequency.

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Key Concepts

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

Centripetal Force

Centripetal force is the net force acting on an object moving in a circular path, directed towards the center of the circle. In the case of the hydrogen atom, the electron experiences centripetal force due to the electrostatic attraction between the negatively charged electron and the positively charged proton. This force is essential for maintaining the electron's circular orbit.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around a central point, expressed in radians per second or revolutions per second. For the electron in the hydrogen atom, angular velocity can be calculated using the relationship between linear velocity and the radius of the circular path. Understanding angular velocity is crucial for determining how many revolutions the electron makes in a given time frame.

Quantization of Energy Levels

In quantum mechanics, the energy levels of electrons in an atom are quantized, meaning they can only occupy specific energy states. For the hydrogen atom, the electron's orbit corresponds to discrete energy levels, which influence its speed and the radius of its orbit. This concept is fundamental in understanding the behavior of electrons in atoms and how they relate to the frequency of revolutions.

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Textbook Question

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