Question

The threshold frequency is the minimum frequency of light needed to release electrons
from a material. Determine the threshold frequency for this apparatus. What is the
associated threshold wavelength (the longest wavelength which releases electrons from
the apparatus)? Quote your result with uncertainty. (You may need to review the
rules for propagating uncertainty)

Answer 1

The photoelectric effect is the emission of electrons (called photoelectrons) from the surface of a conductor when light shines on the surface.The energy of a photon is given by

_Ephoton = hf = hc/ lamda

where we have used the wave relation

c = fλ.

The constant h is called Planck's constant. In the photoelectric effect light of wavelength λ falls on a metal surface. A photon is absorbed and all of the photon energy is transferred to an electron in the metal. The minimum energy required to free an electron from the surface of the metal is called the work function ϕ. Conservation of energy gives

hf = K_max + ϕ

where K_max is the maximum kinetic energy of the electrons emitted by the surface. Electrons from the interior of the metal can also be ejected but they require more energy than ϕ to be released from the metal and therefore have less kinetic energy than K_max.The rate at which photoelectrons are produced can be determined by collecting them at a metal anode and measuring the resulting current. The value of K_max can be measured by imposing a potential barrier, V, that converts part of the kinetic energy of the electrons to electric potential energy, eV. The stopping potential, V₀, is the value of V for which all the kinetic energy is removed, so

eV₀ = K_max

and the conservation of energy equation for the photoelectric effect becomes

hf = eV₀ + ϕ.

This model for the photoelectric effect predicts a threshold frequency f_th that is the minimum frequency the light must have in order to produce any photoelectrons. The threshold frequency is calculated by setting

K_max = 0

If the wavelength of the light is greater than λ_max no photoelectrons are produced.

Laws of Photoelectric Effect

1. For a light of any given frequency; (γ > γ Th) photoelectric current is directly proportional to the intensity of light
2. For any given material, there is a certain minimum (energy) frequency, called threshold frequency, below which the emission of photoelectrons stops completely, no matter how high is the intensity of incident light.
3. The maximum kinetic energy of the photoelectrons is found to increase with the increase in the frequency of incident light, provided the frequency (γ > γ Th) exceeds the threshold limit. The maximum kinetic energy is independent of the intensity of light.
4. The photo-emission is an instantaneous process.

The photoelectric effect occurs when light above a certain frequency (the threshold frequency) is shone on metals like zinc, this causes electrons to escape from the zinc. The escaping electrons are called photoelectrons.

It was shown in experiments that;

• the frequency of the light needed to reach a particular minimum value (depending on the metal) for photoelectrons to start escaping the metal
• the maximum kinetic energy of the photoelectrons depended on the frequency of the light not the intensity of the light

The above two observation can only be explained if the electromagnetic waves are emitted in packets of energy (quanta) called photons, the photoelectric effect can only be explained by the particle behaviour of light.

The photoelectric equation involves;

• h = the Plank constant 6.63 x 10^-34 J s
• f = the frequency of the incident light in hertz (Hz)
• &phi = the work function in joules (J)
• E_k = the maximum kinetic energy of the emitted electrons in joules (J)

The energy of a photon of light = hf   and the work function (f)is the minimum energy required to remove an electron from the surface of the material. So we can see from the equation above that if the light does not have a big enough frequency (f) so that the photon has enough energy to overcome the work function (f) then no photoelectrons will be emitted.

Threshold Frequency (γth)

It is the minimum frequency of the incident light or radiation that will produce a photoelectric effect i.e. ejection of photoelectrons from a metal surface is known as threshold frequency for the metal. It is constant for a specific metal but may be different for different metals.

If γ = frequency of incident photon and γth= threshold frequency, then,

• If γ < γTh, there will be no ejection of photoelectron and, therefore, no photoelectric effect.
• If γ = γTh, photoelectrons are just ejected from the metal surface, in this case, the kinetic energy of the electron is zero
• If γ > γTh, then photoelectrons will come out of the surface along with kinetic energy

Associated Thershold wavelength:

Threshold Wavelength (λth)

During the emission of electrons, a metal surface corresponding to the greatest wavelength to incident light is known threshold wavelength.

λth = c/γth

For wavelengths above this threshold, there will be no photoelectron emission. For λ = wavelength of the incident photon, then

• If λ < λTh, then the photoelectric effect will take place and ejected electron will possess kinetic energy.
• If λ = λTh, then just photoelectric effect will take place and kinetic energy of ejected photoelectron will be zero.
• If λ > λTh, there will be no photoelectric effect.

Uncertainty Analysis
There were a number of causes for uncertainty, but most were relatively minor. Essentially the same uncertainties existed for each experiment. The material used to obtain
the electrons was not an elemental substance, and there is some chance that the work
function may not be compltetly homogeneous in the material. Additionally, there was
some spreading of light from individual spectra, causeing very slight imperfections in
measurement. Most of this, however, was accounted for by using light filters. Some
degree of uncertainty exists in the exact wavelengths of the atomic spectra themselves.
There was a small degree of systematic error, because the resistance in the zero gain amplifier, while high, is not infinite, so some charge was able to leak off. This was especially
apparent in the intensity experiment, where the stopping potential dropped off slightly
at low transmission percentages.
Most of these uncertainties were very small compared to the uncertainty in the stopping potential. The voltmeter has a precision of but one millivolt, and there were some
slight fluctuations. There were slight movements of the apparatus as it was handled
during the experiment, which seemed to cause a marginal change in voltage readings.
Moreover, though enclosed in a dark setting, some ambient light from the room was
allowed into the experimental environment, so there may be some minor effects from
ambient light. However, the readings from the voltmeter seemed fairly consistent with a
high degree of precision, so the uncertainty for each measurement was taken to be only
±2 mV.
The real numerical uncertainty, however, came from the scattering of data about the
regression line (and the fact that there were just twelve data points). The uncertainty
σVmax
calculated from the greatly superseded the uncertainty measured from the data,
and by far contributed the most to measured uncertainties in h and φ/e.