(a)(i) Explain the terms: photoelectric emission and threshold frequency; (ii) Einstein's photoelectric equation can be written as \(E = hf - hf_{o}\) What does each of the symbols used in the equation above represent?
(b) Calculate the frequency of the proton whose energy is required to eject a surface electron with a kinetic energy of \(1.97 x 10^{-16} eV\) if the work function of the metal is \(1.33 x 10^{-16}eV\). \((1 eV = 1.6 x 10^{-18}J; h = 6.60 x 10^{-34}JS)\).
(c) In a photoelectric cell, no electrons are emitted until the threshold frequency of light is reached. Explain what happens to the energy of the light before emission of electrons begins. State one factor that may affect the numbers of emitted electrons.
(d) Explain what is meant by the duality of matter, illustrating your answer with observation phenomena.
(a)(i) Photoelectric emission and threshold frequency
- Photoelectric emission is the ejection of electrons from the surface of a metal when electromagnetic radiation (light) of sufficiently high frequency falls on it.
- Threshold frequency is the minimum frequency of the incident radiation below which no electrons are emitted, no matter how intense the light.
(a)(ii) Symbols in \( E = hf - hf_o \)
- \( E \) is the maximum kinetic energy of the emitted (photo)electron.
- \( hf \) is the energy of the incident photon (h = Planck's constant, f = its frequency).
- \( hf_o \) is the work function of the metal (\( f_o \) is the threshold frequency).
(b) Frequency of the incident radiation
By Einstein's equation, the photon energy equals the work function plus the kinetic energy given to the electron:
\[ hf = W + KE = (1.33 \times 10^{-16}) + (1.97 \times 10^{-16}) = 3.30 \times 10^{-16}\,\text{J} \]
\[ f = \frac{hf}{h} = \frac{3.30 \times 10^{-16}}{6.60 \times 10^{-34}} = 5.0 \times 10^{17}\,\text{Hz} \]
The frequency of the incident radiation is \( 5.0 \times 10^{17}\,\text{Hz} \) (treating the given energies in joules).
(c) Below the threshold frequency, each photon does not carry enough energy to free an electron; the light energy absorbed by the surface electrons is simply re-radiated or converted into heat, so no electron gains sufficient energy to overcome the work function and escape. The number of electrons emitted (once above threshold) depends on the intensity (brightness) of the incident light.
(d) Duality of matter
The duality of matter means that matter (such as electrons) can behave both as particles and as waves. It behaves as a particle in phenomena such as the photoelectric effect and collisions, and as a wave in phenomena such as electron diffraction and interference, where a beam of electrons produces a diffraction pattern like that of light. The de Broglie relation \( \lambda = \dfrac{h}{mv} \) links the two aspects.
