Photons and the Photoelectric Effect

CONCEPTS AT A GLANCE The total energy E and the linear momentum

are fundamental concepts in physics. We have seen in Chapters 6 and 7 how they apply to moving particles, such as electrons and protons. The total energy of a (nonrelativistic) particle is the sum of its kinetic energy (KE) and potential energy (PE), or

. The magnitude p of the particle’s momentum is the product of its mass m and speed v, or

. These particle concepts are listed in the upper-right portion of the Concepts-at-a-Glance chart in Figure 29-3. We will now discuss the fact that electromagnetic waves are composed of particle-like entities called photons, and the lower-right portion of the chart shows that the ideas of energy and momentum also apply to them. However, as we will see, the equations defining photon energy (

) and momentum (

) are different from those for a particle, as the chart indicates.

Figure 29-3 CONCEPTS AT A GLANCE A moving particle has energy E and momentum p. An electromagnetic wave is composed of particle-like entities called photons, each of which also has energy and momentum. Although the spotlight beams in the photograph look like continuous beams of light, each is composed of discrete photons. (Cameron Spencer/Getty Images News and Sport Services)

Experimental evidence that light consists of photons comes from a phenomenon called the photoelectric effect, in which electrons are emitted from a metal surface when light shines on it. Figure 29-4 illustrates the effect. The electrons are emitted if the light being used has a sufficiently high frequency. The ejected electrons move toward a positive electrode called the collector and cause a current to register on the ammeter. Because the electrons are ejected with the aid of light, they are called photoelectrons. As will be discussed shortly, a number of features of the photoelectric effect could not be explained solely with the ideas of classical physics.

Figure 29-4 In the photoelectric effect, light with a sufficiently high frequency ejects electrons from a metal surface. These photoelectrons, as they are called, are drawn to the positive collector, thus producing a current.

In 1905 Einstein presented an explanation of the photoelectric effect that took advantage of Planck’s work concerning blackbody radiation. It was primarily for his theory of the photoelectric effect that he was awarded the Nobel Prize in physics in 1921. In his photoelectric theory, Einstein proposed that light of frequency f could be regarded as a collection of discrete packets of energy (photons), each packet containing an amount of energy E given by


	(29.2)

where h is Planck’s constant. The light energy given off by a light bulb, for instance, is carried by photons. The brighter the bulb, the greater is the number of photons emitted per second. Example 1 estimates the number of photons emitted per second by a typical light bulb.

Example 1

Photons from a Light Bulb

In converting electrical energy into light energy, a sixty-watt incandescent light bulb operates at about 2.1% efficiency. Assuming that all the light is green light (

), determine the number of photons per second given off by the bulb.

Reasoning The number of photons emitted per second can be found by dividing the amount of light energy emitted per second by the energy E of one photon. The energy of a single photon is

, according to Equation 29.2. The frequency f of the photon is related to its wavelength

by Equation 16.1 as

Solution At an efficiency of 2.1%, the light energy emitted per second by a sixty-watt bulb is

. The energy of a single photon is

Therefore,

According to Einstein, when light shines on a metal, a photon can give up its energy to an electron in the metal. If the photon has enough energy to do the work of removing the electron from the metal, the electron can be ejected. The work required depends on how strongly the electron is held. For the least strongly held electrons, the necessary work has a minimum value W₀ and is called the work function of the metal. If a photon has energy in excess of the work needed to remove an electron, the excess appears as kinetic energy of the ejected electron. Thus, the least strongly held electrons are ejected with the maximum kinetic energy KE_max. Einstein applied the conservation-of-energy principle and proposed the following relation to describe the photoelectric effect:


	(29.3)

According to this equation,

, which is plotted in Figure 29-5, with KE_max along the y axis and f along the x axis. The graph is a straight line that crosses the x axis at

. At this frequency, the electron departs from the metal with no kinetic energy (

). According to Equation 29.3, when

the energy hf₀ of the incident photon is equal to the work function W₀ of the metal:

Figure 29-5 Photons can eject electrons from a metal when the light frequency is above a minimum value f₀. For frequencies above this value, ejected electrons have a maximum kinetic energy KE_max that is linearly related to the frequency, as the graph shows.

Example 2

The Photoelectric Effect for a Silver Surface

The work function for a silver surface is

. Find the minimum frequency that light must have to eject electrons from this surface.

Reasoning The minimum frequency f₀ is that frequency at which the photon energy equals the work function W₀ of the metal, so the electron is ejected with zero kinetic energy. Since

, the work function expressed in joules is

Using Equation 29.3, we find

Solution The minimum frequency f₀ is

Photons with frequencies less than f₀ do not have enough energy to eject electrons from a silver surface. Since

, the wavelength of this light is

, which is in the ultraviolet region of the electromagnetic spectrum.

A N A L Y Z I N G M U L T I P L E - C O N C E P T P R O B L E M S

Example 3

The Maximum Speed of Ejected Photoelectrons

Light with a wavelength of 95 nm shines on a selenium surface, which has a work function of 5.9 eV. The ejected electrons have some kinetic energy. Determine the maximum speed with which electrons are ejected.

Reasoning The maximum speed of the ejected electrons is related to their maximum kinetic energy. Conservation of energy dictates that this maximum kinetic energy is related to the work function of the surface and the energy of the incident photons. The work function is given. The energy of the photons can be obtained from the frequency of the light, which is related to the wavelength.

Knowns and Unknowns We have the following data:

Description	Symbol	Value	Comment
Wavelength of light		95 nm
Work function of selenium surface	W₀	5.9 eV	Will be converted to joules
Unknown Variable
Maximum speed of photoelectrons	v_max	?

Modeling the Problem

Step 1 Kinetic Energy and Speed

The maximum kinetic energy KE_max of the ejected electrons is

where m is the mass of an electron. Solving for the maximum speed v_max gives Equation 1 at the right. The mass of the electron is

(see inside of front cover). The maximum kinetic energy is unknown, but we will evaluate it in Step 2.


	(1)

Step 2 Conservation of Energy

According to the principle of conservation of energy, as expressed by Equation 29.2, we have

where f is the frequency of the light. Solving for KE_max gives


	(2)

which can be substituted into Equation 1 as shown at the right. In this expression the work function W₀ is known, and we will deal with the unknown frequency f in Step 3.

Step 3 Relationship between Frequency and Wavelength

The frequency and wavelength of the light are related to the speed of light c according to

(Equation 16.6). Solving for the frequency gives

which we substitute into Equation (2), as shown at the right.

Solution Combining the results of each step algebraically, we find that

Thus, the maximum speed of the photoelectrons is

Note in this calculation that we have converted the value of the work function from electron volts to joules.

Related Homework: Problem 8, 44

Whereas the photon model of light explains the photoelectric effect satisfactorily, the electromagnetic wave model of light does not. Certainly, it is possible to imagine that the electric field of an electromagnetic wave would cause electrons in the metal to oscillate and tear free from the surface when the amplitude of oscillation becomes large enough. However, were this the case, higher-intensity light would eject electrons with a greater maximum kinetic energy, a fact that experiment does not confirm. Moreover, in the electromagnetic wave model, a relatively long time would be required with low-intensity light before the electrons would build up a sufficiently large oscillation amplitude to tear free. Instead, experiment shows that even the weakest light intensity causes electrons to be ejected almost instantaneously, provided the frequency of the light is above the minimum value f₀. The failure of the electromagnetic wave model to explain the photoelectric effect does not mean that the wave model should be abandoned. However, we must recognize that the wave model does not account for all the characteristics of light. The photon model also makes an important contribution to our understanding of the way light behaves when it interacts with matter.

Check Your Understanding 1

In the photoelectric effect, electrons are ejected from the surface of a metal when light shines on it. Which one or more of the following would lead to an increase in the maximum kinetic energy of the ejected electrons? (a) Increasing the frequency of the incident light, (b) Increasing the number of photons per second striking the surface. (c) Using photons whose frequency f₀ is less than W₀/h, where W₀ is the work function of the metal and h is Planck’s constant. (d) Selecting a metal that has a greater work function. (The answer is given at the end of the book.)

Background: The conservation of energy relates the maximum kinetic energy of the ejected electrons to the energy of the incident photons and the work function of the metal. This relation, Equation 29.3, holds the key to understanding the photoelectric effect.

For similar questions (including calculational counterparts), consult Self-Assessment Test 29.1, which is described at the end of this section.

Because a photon has energy, the photon can eject an electron from a metal surface when it interacts with the electron. However, a photon is different from a normal particle. A normal particle has a mass and can travel at speeds up to, but not equal to, the speed of light. A photon, on the other hand, travels at the speed of light in a vacuum and does not exist as an object at rest. The energy of a photon is entirely kinetic in nature, because it has no rest energy and no mass. To show that a photon has no mass, we rewrite Equation 28.4 for the total energy E as

The term

is zero because a photon travels at the speed of light,

. Since the energy E of the photon is finite, the left side of the equation above is zero. Thus, the right side must also be zero, so

and the photon has no mass.

One of the most exciting and useful applications of the photoelectric effect is the charge-coupled device (CCD). An array of these devices is used instead of film in digital cameras (see Figure 29-6) to capture images in the form of many small groups of electrons. CCD arrays are also used in digital camcorders and electronic scanners, and they provide the method of choice with which astronomers capture those spectacular images of the planets and the stars. For use with visible light, a CCD array consists of a sandwich of semiconducting silicon, insulating silicon dioxide, and a number of electrodes, as Figure 29-7 shows. The array is divided into many small sections, or pixels, sixteen of which are shown in the drawing. Each pixel captures a small part of a picture. Digital cameras can have up to eight million pixels, depending on price. The greater the number of pixels, the better is the resolution of the photograph. The blow-up in Figure 29-7 shows a single pixel. Incident photons of visible light strike the silicon and generate electrons via the photoelectric effect. The range of energies of the visible photons is such that approximately one electron is released when a photon interacts with a silicon atom. The electrons do not escape from the silicon, but are trapped within a pixel because of a positive voltage applied to the electrodes beneath the insulating layer. Thus, the number of electrons that are released and trapped is proportional to the number of photons striking the pixel. In this fashion, each pixel in the CCD array accumulates an accurate representation of the light intensity at that point on the image. Color information is provided using red, green, or blue filters or a system of prisms to separate the colors. Astronomers use CCD arrays not only in the visible region of the electromagnetic spectrum but in other regions as well.

Figure 29-6 (a) Digital cameras like this one use an array of charge-coupled devices instead of film to capture an image. (YOSHIKAZU TSUNO/AFP/Getty Images) (b) Images taken by a digital camera can be easily downloaded to a computer and sent to your friends via the Internet. (Ross Woodhall/Taxi/Getty Images)

Figure 29-7 A CCD array can be used to capture photographic images using the photoelectric effect.

In addition to trapping the photoelectrons, the electrodes beneath the pixels are used to read out the electron representation of the picture. By changing the positive voltages applied to the electrodes, it is possible to cause all of the electrons trapped in one row of pixels to be transferred to the adjacent row. In this fashion, for instance, row 1 in Figure 29-7 is transferred into row 2, row 2 into row 3, and row 3 into the bottom row, which serves a special purpose. The bottom row functions as a horizontal shift register, from which the contents of each pixel can be shifted to the right, one at a time, and read into an analog signal processor. This processor senses the varying number of electrons in each pixel in the shift register as a kind of wave that has a fluctuating amplitude. After another shift in rows, the information in the next row is read out, and so forth. The output of the analog signal processor is sent to an analog-to-digital converter, which produces a digital representation of the image in terms of the zeros and ones that computers recognize.

The physics of a safety feature of garage door openers. Another application of the photoelectric effect depends on the fact that the moving photoelectrons in Figure 29-4 constitute a current—a current that changes as the intensity of the light changes. All automatic garage door openers have a safety feature that prevents the door from closing when it encounters an obstruction (person, vehicle, etc.). As Figure 29-8 illustrates, a sending unit transmits an invisible (infrared) beam across the opening of the door. The beam is detected by a receiving unit that contains a photodiode. A photodiode is a type of p-n junction diode (see Section 23.5). When infrared photons strike the photodiode, electrons bound to the atoms absorb the photons and become liberated. These liberated, mobile electrons cause the current in the photodiode to increase. When a person walks through the beam, the light is momentarily blocked from reaching the receiving unit, and the current in the photodiode decreases. The change in current is sensed by electronic circuitry that immediately stops the downward motion of the door and then causes it to rise up.

Figure 29-8 When an obstruction prevents the infrared light beam from reaching the receiving unit, the current in the receiving unit drops. This drop in current is detected by an electronic circuit that stops the downward movement of the garage door and then causes it to rise.

Look back at the spectacular photograph at the beginning of this chapter. It shows the central portion of the Eagle Nebula, a giant star-forming region some 7000 light-years from earth. The photo was taken by the Hubble Space Telescope and reveals towering clouds of molecular gas and dust, in which there is dramatic evidence of the energy carried by photons. These clouds extend more than a light-year from base to tip and are the birthplace of stars. A star begins to form within a cloud when the gravitational force pulls together sufficient gas to create a high-density “ball.” When the gaseous ball becomes sufficiently dense, thermonuclear fusion (see Section 32.5) occurs at its core, and the star begins to shine. The newly born stars are buried within the cloud and cannot be seen from earth. However, the process of photoevaporation allows astronomers to see many of the high-density regions where stars are being formed. Photoevaporation is the process in which high-energy, ultraviolet (UV) photons from hot stars outside the cloud heat it up, much like microwave photons heat food in a microwave oven. Figure 29-9a is a reproduction of the upper left portion of the chapter-opening photo and shows streamers of gas photoevaporating from the cloud as it is illuminated by stars located beyond the photograph’s upper edge. As photoevaporation proceeds, globules of gas that are denser than their surroundings are exposed. The globules are known as evaporating gaseous globules (EGGs), and they are slightly larger than our solar system. The drawing in part b of Figure 29-9 shows that the EGGs shade the gas and dust behind them from the UV photons, creating the many finger-like protrusions seen on the surface of the cloud. Astronomers believe that some of these EGGs contain young stars within them. In some cases, so much gas has boiled off that a newborn star can be seen on the surface of an EGG (see the circled feature in the chapter-opening photograph).

Figure 29-9 (a) This is the upper left portion of the chapter-opening photograph. Photoevaporation produces finger-like projections on the surface of the gas clouds in the Eagle Nebula. At the fingertips are high-density evaporating gaseous globules (EGGs). (Courtesy NASA) (b) This drawing illustrates the photoevaporation that is occurring in the photograph in part (a).