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In developing night-vision equipment, you need to measure the work function for a metal surface, so you perform a photoelectric-effect experiment. You measure the stopping potential \(V_{0}\) as a function of the wavelength \(\lambda\) of the light that is incident on the surface. You get the results in the table. $$ \begin{array}{l|llllll} \boldsymbol{\lambda}(\mathbf{n m}) & 100 & 120 & 140 & 160 & 180 & 200 \\ \hline \boldsymbol{V}_{0}(\mathbf{V}) & 7.53 & 5.59 & 3.98 & 2.92 & 2.06 & 1.43 \end{array} $$ In your analysis, you use \(c=2.998 \times 10^{8} \mathrm{~m} / \mathrm{s}\) and \(e=1.602 \times 10^{-19} \mathrm{C}\) which are values obtained in other experiments. (a) Select a way to plot your results so that the data points fall close to a straight line. Using that plot, find the slope and \(y\) -intercept of the best-fit straight line to the data. (b) Use the results of part (a) to calculate Planck's constant \(h\) (as a test of your data) and the work function (in eV) of the surface. (c) What is the longest wavelength of light that will produce photoelectrons from this surface? (d) What wavelength of light is required to produce photoelectrons with kinetic energy \(10.0 \mathrm{eV} ?\)

Step by step solution

01

Data Arrangement and Graph Plotting

Arrange the given wavelengths \( \lambda \) and the corresponding stopping potentials \( V_0 \) in two columns, and calculate values for \( 1/\lambda \) in another column. Plot a graph of stopping potentials \( V_0 \) against \( 1/\lambda \), which ought to result in a straight line.

02

Determining the Slope and Intercept

The slope and y-intercept of the best-fit line for the data can be found using any line of best fit method, for example, the least squares method. The slope corresponds to the value of \( h \) and the y-intercept corresponds to \( -\Phi / e \).

03

Calculation of Planck's constant and the Work function

Planck's Constant \( h \) is determined by the slope from the plot, keeping in mind that you'll have to unconventionally use units of electronvolts for the elementary charge \( e \) so that \( h \) will be in Joule-seconds. Multiply the negative y-intercept by the charge of an electron \( e \) to obtain the work function \( \Phi \) in electronvolts.

04

Calculation of Longest Wavelength of Light

The longest wavelength of light that will produce photoelectrons from the surface can be found using the formula \( \lambda_{max} = h/(\Phi c) \). Substituting the values of Planck's constant \( h \), speed of light \( c \) and work function \( \Phi \) in this formula will yield the longest wavelength \( \lambda_{max} \).

05

Wavelength Required for Kinetic Energy of 10eV

The wavelength of light that is necessary to produce photoelectrons with a kinetic energy of 10.0eV can be determined via the formula \( \lambda = h/(c(E_{k} + \Phi)) \). Substitute the values of Planck's constant \( h \), speed of light \( c \), given kinetic energy \( E_{k} \), and work function \( \Phi \) into this formula to find the required wavelength.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Work Function

The concept of the work function is pivotal in understanding the photoelectric effect. It refers to the minimum amount of energy needed to remove an electron from the surface of a metal. This energy is specific to each material due to its unique atomic structure and bonding. In the context of a photoelectric effect experiment, the work function is symbolized as \( \Phi \), and its value is usually measured in electronvolts (eV). The relationship between the work function, the energy of the incident photons, and the kinetic energy of the emitted electrons is given by the equation:

• \( h u = \Phi + KE \), where \( h u \) is the energy of the incident light, and \( KE \) is the kinetic energy of the ejected electrons.

By plotting stopping potential \( V_0 \) against the inverse of wavelength \( 1/\lambda \), one can determine the work function. The y-intercept of the line derived from such a plot provides \( -\Phi/e \), allowing calculation of \( \Phi \) after multiplying by the elementary charge \( e \). This step is critical for determining the material's propensities in photoelectric interactions.

Planck's Constant

Planck's constant, denoted by \( h \), is a fundamental constant in physics that relates the energy of a photon to its frequency. It plays a crucial role in quantum mechanics and the photoelectric effect, providing a bridge between wave and particle theories of light. The formula \( E = h u \) is used to calculate the energy \( E \) carried by a photon when its frequency \( u \) is known. In practice, the slope of the line when plotting stopping potential \( V_0 \) against \( 1/\lambda \) provides a value for \( hc/e \), making it possible to calculate \( h \) by rearranging terms.
Once the slope is determined from the best-fit line in a photoelectric experiment, we can solve for Planck's constant given that:

• \( \text{Slope} = \frac{hc}{e} \)

Multiplying the slope by the elementary charge \( e \) gives the product \( hc \), from which \( h \) is isolated. Understanding Planck's constant is fundamental for anyone studying photoelectric phenomena and quantum physics overall.

Stopping Potential

Stopping potential is a key concept in the study of the photoelectric effect. It represents the minimum voltage needed to stop the most energetic photoelectrons emitted from a metal surface when light shines on it. This potential, denoted as \( V_0 \), is directly related to the maximum kinetic energy \( KE_{max} \) of these photoelectrons by the equation:

• \( eV_0 = KE_{max} \)

By measuring \( V_0 \), we can infer important aspects about the energy levels of the incident photons and the material's work function. The stopping potential is practically determined through experiments by adjusting an external voltage supply until photoelectrons cease to flow.
Analyzing the plot of \( V_0 \) against \( 1/\lambda \) provides critical insight into the interaction of light and matter, confirming theories established in quantum mechanics. In particular, it's through this analysis that concepts like Planck's constant and the work function are experimentally verified.

Wavelength

Wavelength is an essential term when discussing light and other forms of electromagnetic radiation. It describes the distance between successive crests of a wave. In the context of the photoelectric effect, the wavelength \( \lambda \) of incident light is inversely related to its frequency and, consequently, its energy. As per the equation \( c = \lambda u \), where \( c \) is the speed of light, any change in wavelength affects the photon's energy, impacting electron ejection from a surface.

• The shortest wavelength translates to the highest photon energy.
• The longest wavelength sufficient to release photoelectrons corresponds to meeting the work function \( \Phi \).

To determine the longest wavelength of light that can still produce photoelectrons, we use the formula \( \lambda_{\text{max}} = \frac{hc}{\Phi} \). For any application involving the photoelectric effect, understanding and manipulating the wavelength of light is vital to achieving desired experimental outcomes.