29.7

Concepts & Calculations

In the photoelectric effect, electrons can be emitted from a metal surface when light shines on it, as we have seen. Einstein explained this phenomenon in terms of photons and the conservation of energy. The same basic explanation applies when the light has a single wavelength or when it consists of a mixture of wavelengths, as sunlight does. Our earlier discussion in Section 29.3 deals with a single wavelength. Example 8, in contrast, considers a mixture of wavelengths and reviews the basic concepts that come into play in this important phenomenon.

| Concepts & Calculations Example 8 | | | Sunlight and the Photoelectric Effect

Sunlight, whose visible wavelengths range from 380 to 750 nm, is incident on a sodium surface. The work function is . Find the maximum kinetic energy KEmax (in joules) of the photoelectrons emitted from the surface and the range of wavelengths that will cause photoelectrons to be emitted.
Concept Questions and Answers Will electrons with the greatest value of KEmax be emitted when the incident photons have a relatively greater or a relatively smaller amount of energy?

> Answer When a photon ejects an electron from the sodium, the photon energy is used for two purposes. Some of it is used to do the work of ejecting the electron, and what remains is carried away as kinetic energy of the emitted electron. The minimum work to eject an electron is equal to the work function W0, and in this case the electron carries away a maximum amount KEmax of kinetic energy. According to the principle of conservation of energy, the energy of an incident photon is equal to the sum of W0 and KEmax. Thus, for a given value of W0, greater values of KEmax will occur when the incident photon has a greater energy to begin with.

In the range of visible wavelengths, which wavelength corresponds to incident photons that carry the greatest energy?

> Answer According to Equation 29.2, the energy of a photon is related to the frequency f and Planck’s constant h according to . But the frequency is given by Equation 16.1 as , where c is the speed of light and is the wavelength. Therefore, we have , so the smallest value of the wavelength corresponds to the greatest energy.

What is the smallest value of KEmax with which an electron can be ejected from the sodium?

> Answer The smallest value of KEmax with which an electron can be ejected from the sodium is 0 J. This occurs when an incident photon carries an energy that exactly equals the work function W0, with no energy left over to be carried away by the electron as kinetic energy.

Problem solving insight Values for the work function are usually given in electron volts (eV). Be sure to convert the value into joules before combining it with other quantities that are specified in joules.

Solution The value given for the work function is in electron volts, so we first convert it into joules by using the fact that :

The incident photons that have the greatest energy are those with the smallest wavelength or . Then, according to Equation 29.3 and Equation 16.1 (), the greatest value for KEmax is

As the wavelength increases, the energy of the incident photons decreases until it equals the work function, at which point electrons are ejected with zero kinetic energy. Thus, we have

The range of wavelengths over which sunlight ejects electrons from the sodium surface is, then, .

Kinetic energy and momentum, respectively, are discussed in Chapters 6 and 7. In the present chapter we have seen that moving particles of matter not only possess kinetic energy and momentum but are also characterized by a wavelength that is known as the de Broglie wavelength. We conclude with an example that reviews kinetic energy and momentum and focuses on the relation between these fundamental physical quantities and the de Broglie wavelength.

| Concepts & Calculations Example 9 | | | The De Broglie Wavelength

An electron and a proton have the same kinetic energy and are moving at speeds much less than the speed of light. Determine the ratio of the de Broglie wavelength of the electron to that of the proton. Concept Questions and Answers How is the de Broglie wavelength related to the magnitude p of the momentum?

> Answer According to Equation 29.8, the de Broglie wavelength is given by , where h is Planck’s constant. The greater the momentum, the smaller the de Broglie wavelength, and vice versa.

How is the magnitude of the momentum related to the kinetic energy of a particle of mass m that is moving at a speed that is much less than the speed of light?

> Answer According to Equation 7.2 the magnitude of the momentum is , where v is the speed. Equation 6.2 gives the kinetic energy as , which can be solved to show that . Substituting this result into Equation 7.2 shows that .

Which has the greater de Broglie wavelength, the electron or the proton?

> Answer According to , the particle with the greater wavelength is the one with the smaller momentum. But indicates that, for a given kinetic energy, the particle with the smaller momentum is the one with the smaller mass. The masses of the electron and proton are and . Thus, the electron, with its smaller mass, has the greater de Broglie wavelength.

Solution Using Equation 29.8 for the de Broglie wavelength and the fact that the magnitude of the momentum is related to the kinetic energy by , we have

Applying this result to the electron and the proton gives

As expected, the wavelength for the electron is greater than that for the proton.

At the end of the problem set for this chapter, you will find homework problems that contain both conceptual and quantitative parts. These problems are grouped under the heading Concepts & Calculations.

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