12.2 The photoelectric effect

History and expectations as to the photoelectric effect (ESCQK)

In 1887, Heinrich Hertz (a German physicist) noticed that ultraviolet light incident on a metal plate could cause sparks. Metals were known to be good conductors of electricity, because the electrons are more able to move. They should be able to be dislodged if energy were added through the incident light. The problem was that different metals requred different minimum frequencies of light.

The expectation at the time was that electrons would be emitted for any frequency of light, after a delay (for low intensities) during which the electrons absorbed sufficient energy to escape from the metal surface. The higher the intensity the shorter the delay was be as they would absorb energy faster. This was based on the idea that light was a wave continuously delivering energy to the electrons.

It is important to remember that higher frequency light corresponds to higher energy.

The next piece of the puzzle came from Philipp Lenard (a Hungarian physicist) in 1902 when he discovered that the maximum velocity with which electrons are ejected by ultraviolet light is entirely independent of the intensity of light.

His expectation was that at high intensities the electrons would absorb more energy and so would have a greater velocity.

A paradox existed as the expectations and the observations did not match.

Albert Einstein (a German physicist) solved this paradox by proposing that light is made up of packets of energy called quanta (now called photons) which interacted with the electrons in the metal like particles instead of waves. Each incident photon would transfer all its energy to one electron in the metal.

The photoelectric effect

The photoelectric effect is the process whereby an electron is emitted by a substance when light shines on it.

Einstein received the 1921 Nobel Prize for his contribution to understanding the photoelectric effect. His explanation wasn't very popular and took a while to be accepted, in fact, some scientists at the time felt that is was a big mistake.

In the motivation letter for Einstein to be accepted into the Prussian Academy of Science it was specifically mentioned as a mistake:

In sum, one can say that there is hardly one among the great problems in which modern physics is so rich to which Einstein has not made a remarkable contribution. That he may sometimes have missed the targeting his speculations, as, for example, in his hypothesis of light-quanta, cannot really be held too much against him, for it is not possible to introduce really new ideas even in the most exact sciences without sometimes taking a risk - A. Pais, “Subtle is the Lord: The Science and the Life of Albert Einstein,” New York: Oxford University Press, 1982, p. 382

Implications of Einstein's model (ESCQM)

Einstein's model is consistent with the observation that the electrons were emitted immediately when light was shone on the metal and that the intensity of the light made no difference to the maximum kinetic energy of the emitted electrons.

The energy needed to knock an electron out of the substance is called the work function (symbol \(W_{0}\)) of the substance. This is a characteristic of the substance. If the energy of the photon is less than the work function then no electron can be emitted, no matter how many photons strike the substance. We know that the frequency of light is related to the energy, that is why there is a minimum frequency of light that can eject electrons. This minimum frequency we call the cut-off frequency, \(f_0\). For a specific colour of light (i.e. a certain frequency or wavelength), the energy of the photons is given by \(E=hf=\frac{hc}{\lambda}\), where \(h\) is Planck's constant. This tells us that the \(W_{0} = hf_0\) .

Work function

The minimum energy needed to knock an electron out of a metal is called the work function (symbol \(W_{0}\)) of the metal. As it is energy, it measured in joules (J).

Energy is conserved so if the photon has a higher energy than \(W_{0}\) then the excess energy goes into the kinetic energy \({E}_{k}\) of the electron that was emitted from the substance.

The excess over and above the binding energy is actually the maximum kinetic energy the emitted electron can have. This is because not all electrons are on the surface of the substance. For electrons below the surface there is additional energy required to eject the electron from the material which then cannot contribute to the kinetic energy of the electron.

This equation is known as the photoelectric equation.

The last piece of the puzzle is now clear, the question was `why does increasing the intensity of the light not affect the maximum kinetic energy of the emitted photons?'. The answer is that each emitted electron has absorbed one photon, increasing the intensity just increases the number of photons (we expect more electrons but we don't expect their maximum kinetic energy to change).

The discovery and understanding of the photoelectric effect was one of the major breakthroughs in science in the twentieth century as it provided concrete evidence of the particle nature of light. It overturned previously held views that light was composed purely of a continuous transverse wave. On the one hand, the wave nature is a good description of phenomena such as diffraction and interference for light, and on the other hand, the photoelectric effect demonstrates the particle nature of light. This is now known as the 'dual-nature' of light. (dual means two)

Einstein won the 1921 Nobel Prize for Physics for this quantum theory and his explanation of the photoelectric effect.

Units of energy (ESCQN)

When dealing with calculations at a small scale (like at the level of electrons) it is more convenient to use different units for energy rather than the joule (J). We define a unit called the electron-volt (eV) as the kinetic energy gained by an electron passing through a potential difference of one volt.

where q is the charge of the electron and V is the potential difference applied. The charge of 1 electron is \(\text{1,6} \times \text{10}^{-\text{19}}\) \(\text{C}\), so \(\text{1}\) \(\text{eV}\) is calculated to be:

You can see that \(\text{1,6} \times \text{10}^{-\text{19}}\) \(\text{J}\) is a very small amount of energy and so using electron-volts (eV) at this level is easier.

Hence, \(\text{1}\) \(\text{eV}\) = \(\text{1,6} \times \text{10}^{-\text{19}}\) \(\text{J}\) which means that \(\text{1}\) \(\text{J}\) = \(\text{6,241} \times \text{10}^{\text{18}}\) \(\text{eV}\)

Using the photoelectric effect equation (ESCQP)

It is useful to observe the photoelectric effect equation represented graphically.

It can be seen from the graph that \(E_{k}\) is plotted on the \(y-\)axis and \(f\) is plotted on the \(x-\)axis. Using the straight line equation, \(y=mx+c\), we can identify

\begin{align*} {E}_{k\max}& = hf-W_{0} \\ \underbrace{{E}_{k\max}}_{y}& = h\underbrace{f}_x-hf_{0} \end{align*} This allows us to conclude that the slope of the graph \(m\) is Planck's constant \(h\). Also, the \(x\) intercept is the cut-off frequency \(f_{0}\).

Worked example 3: [NSC 2011 Paper 1]

A metal surface is illuminated with ultraviolet light of wavelength \(\text{330}\) \(\text{nm}\). Electrons are emitted from the metal surface.

The minimum amount of energy required to emit an electron from the surface of this metal is \(\text{3,5} \times \text{10}^{-\text{19}}\) \(\text{J}\).

1. Name the phenomenon illustrated above.

   (1 mark)

2. Give ONE word or term for the underlined sentence in the above paragraph.

   (1 mark)

3. Calculate the frequency of the ultraviolet light.

   (4 marks)

4. Calculate the kinetic energy of a photoelectron emitted from the surface of the metal when the ultraviolet light shines on it.

   (4 marks)

5. The intensity of the ultraviolet light illuminating the metal is now increased. What effect will this change have on the following:

   1. Kinetic energy of the emitted photoelectrons (Write down only INCREASES, DECREASES or REMAINS THE SAME.)

      (1 mark)

   2. Number of photoelectrons emitted per second (Write down only INCREASES, DECREASES or REMAINS THE SAME.)

      (1 mark)

6. Overexposure to sunlight causes damage to skin cells.

   1. Which type of radiation in sunlight is said to be primarily responsible for this damage?

      (1 mark)

   2. Name the property of this radiation responsible for the damage.

      (1 mark)

[TOTAL: 14 marks]

Question 1

Photo-electric effect

(1 mark)

Question 2

Work function

(1 mark)

Question 3

\begin{align*} c & = f \lambda \\ \text{3} \times \text{10}^{\text{8}} & = f(\text{330} \times \text{10}^{-\text{9}})\\ \therefore f & = \text{9,09} \times \text{10}^{\text{14}}\text{ Hz} \end{align*}

OR

\begin{align*} E & = \frac{hc}{\lambda} \\ & = \frac{(\text{6,63} \times \text{10}^{-\text{34}})(\text{3} \times \text{10}^{\text{8}})}{(\text{330} \times \text{10}^{-\text{9}})} \\ & = \text{6,03} \times \text{10}^{-\text{19}}\text{ J} \end{align*}\begin{align*} E & = hf \\ \text{6,03} \times \text{10}^{-\text{19}}\text{ J} & = (\text{6,63} \times \text{10}^{-\text{34}})f\\ \therefore f & = \text{9,09} \times \text{10}^{\text{14}}\text{ Hz} \end{align*}

(4 marks)

Question 4

Option 1:

\begin{align*} E & = W_{o} + K \\ \frac{hc}{\lambda} & = W_{o} + K \\ \therefore \frac{(\text{6,63}\times10^{-34})(3\times10^8)}{330\times10^{-9}} & = \text{3,5}\times10^{-19} + K\\ \therefore K & = \text{2,53}\times10^{-19}\text{ J} \end{align*}

Option 2:

\begin{align*} E & = W_{o} + K \\ hf & = W_{o} + K \\ \therefore (\text{6,63}\times10^{-34})(\text{9,09}\times10^{14} & = \text{3,5}\times10^{-19} + K\\ \therefore K & = \text{2,53}\times10^{-19}\text{ J} \end{align*}

(4 marks)

Question 5.1

Remains the same.

(1 mark)

Question 5.2

Increases

(1 mark)

Question 6.1

Ultraviolet radiation

(1 mark)

Question 6.2

High energy or high frequency

(1 mark)

[TOTAL: 12 marks]

Solar cells (ESCQQ)

The generation of electricity using solar cells isn't due to the photoelectric effect but is similar in nature. Photons in sunlight hit the solar panel and are absorbed by semiconducting materials, such as silicon.

Electrons (negatively charged) are knocked loose from their atoms, allowing them to flow through the material to produce electricity. This is called the photovoltaic effect. The photovoltaic effect was first observed by French physicist Antoine E. Becquerel in 1839.

Due to the special composition of solar cells, the electrons are only allowed to move in a single direction. An array of solar cells converts solar energy into a usable amount of direct current (DC) electricity.