Question

You are an early 20th-century experimental physicist and do not know the value of Planck's constant. By a suitable plot of the following data, and using Einstein's explanation of the photoelectic effect \((\mathrm{K} \mathrm{C})=h f-\phi\) where \(h\) is not known), determine Planck's constant. $$\begin{array}{cc}\begin{array}{c}\text { Wavelength of Lighe } \\\\(\text { nm })\end{array} & \begin{array}{c}\text { Stopping Potential } \\\\(\mathrm{V})\end{array} \\\\\hline 550 & 0.060 \\\\\hline 500 & 0.286 \\\\\hline 450 & 0.563 \\\\\hline 400 & 0.908 \\\\\hline\end{array}$$

Short answer

The solution cannot be given explicitly because the work function \(\phi\) (energy required to remove an electron from a substance) is unknown. Nevertheless, from your graph, you will find that the slope corresponds to Planck's constant \(h\). By knowing the exact values of frequency and stopping potential, you will be able to precisely calculate the value of \(h\).

Step-by-step solution

– Converting Wavelength to Frequency

Change the wavelength into frequency using the formula \(f = c / \lambda\), where \(c\) is the speed of light and \(\lambda\) is the wavelength. Remember, the speed of light \(c\) is constant, and equals approximately \(3 * 10^8 \, m/s\). Convert the wavelength from nanometers to meters.

– Calculate the Stopping Potential

Compute the stopping potential using the equation \(K = hf - \phi\), where \(K\) is the stopping potential, \(h\) is Planck's constant, \(f\) is the frequency and \(\phi\) is the work function.

– Plotting the Data

Plot a graph using the frequencies calculated in Step 1 and the stopping potential calculated in Step 2. The x-values will be the frequency and the y-values will be the stopping potential.

– Slope of the Best Fit Line

Determine the slope of the line that best fits the points on the plot. The slope is equal to Planck's constant. Usually a linear regression analysis would help give the best line.

Key concepts

Planck's Constant

Planck's constant is a fundamental value in physics, central to the understanding of quantum mechanics. It is symbolized by the letter \( h \). This constant relates the energy of photons to their frequency and plays a major role in the photoelectric effect, among many other phenomena. The value of Planck's constant is approximately \( 6.63 \times 10^{-34} \text{ J s} \).
In the context of the photoelectric effect, Planck's constant helps us understand how light can eject electrons from a material. When light shines on a metal surface, it can transfer energy to electrons, allowing them to escape the surface if enough energy is provided. The energy given to each electron depends on the frequency of the light, and Planck's constant provides the proportional conversion factor between frequency and energy.
As physicists like Einstein used Planck's constant in equations, it allowed them to further cement the idea of quantized energy levels and particles of light, called photons. The photoelectric equation is a cornerstone for proving the particle nature of light. Thus, calculating Planck's constant becomes essential in drawing conclusions about the fundamental interactions between light and matter.

Stopping Potential

The stopping potential is an important concept in experiments involving the photoelectric effect. It is the minimum voltage needed to stop the most energetic photoelectrons from reaching the anode after they have been emitted from the cathode.
In essence, stopping potential is a direct measure of the kinetic energy of these electrons, which can be given by the equation: \( K = eV \), where \( e \) is the charge of an electron and \( V \) is the stopping potential. Remember, the higher the stopping potential, the more energy the incident light is providing to the electrons.

• In experiments, you adjust the potential difference to find this threshold value where electrons just stop moving towards the anode.
• The stopping potential gives insight into the energy distribution among the electrons ejected during the photoelectric process.
• By analyzing stopping potentials for different frequencies of light, physicists are able to glean critical information about the energy dynamics at play.

It's necessary to accurately determine stopping potential in order to reliably compute the resulting kinetic energy which ties directly to Einstein's photoelectric equation: \( K = hf - \phi \). This also underscores the quantization of energy in the photoelectric effect.

Frequency-Wavelength Conversion

In the world of physics and light behavior, frequency and wavelength are closely related. They are properties that describe electromagnetic waves, such as light. To properly analyze the photoelectric effect, conversions between wavelength and frequency are often necessary.
The relationship between frequency \( f \) and wavelength \( \lambda \) is given by the equation \( f = \frac{c}{\lambda} \), where \( c \) is the speed of light (approximately \( 3 \times 10^8 \text{ m/s} \)). This highlights that as frequency increases, wavelength decreases, and vice versa. Converting from wavelength to frequency allows physicists to use the frequency directly in the photoelectric equation.

• Wavelength is often given in nanometers, so it must be converted into meters before using the formula.
• Once you have the frequency, you can calculate useful information like the energy imparted by the light using Planck’s constant.

Understanding the interplay between frequency and wavelength helps describe how different colors of light (or light from different parts of the spectrum) affect the energy of photoelectrons in experiments.

Einstein's Explanation

Albert Einstein’s explanation of the photoelectric effect marked a pivotal moment in physics, earning him the Nobel Prize in 1921. Einstein proposed that light could be thought of as discrete packets of energy, known as photons. This revolutionary idea suggested that the energy of these photons is proportional to the frequency of light.
Einstein's photoelectric equation \( K = hf - \phi \) encapsulated this breakthrough.

• The term \( K \) represents the kinetic energy of the ejected electrons.
• \( h \), Planck's constant, scales the frequency \( f \) to energy.
• \( \phi \), the work function, is the minimum energy needed to release an electron from the material surface.

Einstein’s explanation demonstrated that only light of a certain frequency threshold could emit electrons, regardless of the intensity of the light. This was contrary to the wave theory that suggested any amount of energy could be imparted over time if the light intensity was high enough.
This quantized perspective helped offer a clearer understanding of both atomic structures and the dual nature of light, bridging the gap between classical and modern physics approaches.