Question

A particular ultraviolet laser produces radiation of wavelength \(355 \mathrm{nm}\). Suppose this is used in a photoelectric experiment with a calcium sample. What will the stopping potential be?

Short answer

#Answer# The stopping potential for the photoelectric experiment with a calcium sample and an ultraviolet laser of 355 nm wavelength is approximately 1.54 V.

Step-by-step solution

Calculate the energy of photons from the ultraviolet laser

Given the wavelength of the ultraviolet laser (355 nm), we want to calculate the energy of the photons. The formula to calculate the energy of a photon is: \[E = h \cdot f\] Where: - \(E\) is the energy of the photon - \(h\) is the Planck's constant (\(6.63 \times 10^{-34} \mathrm{Js}\)) - \(f\) is the frequency of light But the frequency can also be determined using the formula: \[f = \frac{c}{\lambda }\] Where: - \(c\) is the speed of light in vacuum (\(3.00 \times 10^{8} \mathrm{m/s}\)) - \(\lambda\) is the wavelength of the light First, converting the wavelength to meters: \[\lambda = 355 \mathrm{nm} = 355 \times 10^{-9} \mathrm{m}\] Now, finding the frequency (f) using wavelength and speed of light: \[f = \frac{3.00 \times 10^{8} \mathrm{m/s}}{355 \times 10^{-9} \mathrm{m}} \approx 8.45 \times 10^{14} \mathrm{Hz}\] Next, calculating the energy of the photon using Planck's constant (h) and frequency (f): \[E = (6.63 \times 10^{-34} \mathrm{Js}) \times (8.45 \times 10^{14} \mathrm{Hz}) \approx 5.61 \times 10^{-19} \mathrm{J}\]

Determine the work function of calcium

The work function of calcium is the energy required to remove an electron from the surface of a calcium atom. For calcium, the work function (\(\varphi\)) is approximately \(3.15 \times 10^{-19} \mathrm{J}\).

Apply the photoelectric effect equation to solve for stopping potential

The photoelectric effect equation is: \[E = e \cdot V + \varphi\] Where: - \(E\) is the energy of the photon (calculated in Step 1) - \(e\) is the elementary charge (\(1.60 \times 10^{-19} \mathrm{C}\)) - \(V\) is the stopping potential (which we want to find) - \(\varphi\) is the work function of calcium (given in Step 2) Rearrange the equation to solve for stopping potential (V): \[V = \frac{E - \varphi}{e}\] Now, substitute the calculated values for energy and work function, and the value for elementary charge: \[V = \frac{(5.61 \times 10^{-19} \mathrm{J} - 3.15 \times 10^{-19} \mathrm{J})}{1.60 \times 10^{-19} \mathrm{C}}\] Finally, calculate the stopping potential (V): \[V \approx 1.54 \mathrm{V}\] Thus, the stopping potential for the photoelectric experiment with the given ultraviolet laser and a calcium sample is approximately 1.54 V.

Key concepts

Stopping Potential

The concept of stopping potential is central to understanding the photoelectric effect and is a key area of study in introductory physics. It refers to the minimum voltage required to halt the flow of photoelectrons emitted from a material's surface when it is exposed to light. When light of a certain frequency strikes the surface of a metal, it can cause electrons to be ejected, a phenomenon known as the photoelectric effect.

However, if a reverse voltage, called the stopping potential, is applied, it can prevent the photoelectrons from reaching the anode in a photoelectric cell, effectively 'stopping' them. This potential gives us valuable insights into the energy of the ejected electrons and allows us to measure the work function of the material – that is, the minimum energy required to remove an electron from the surface of the metal.

The importance of stopping potential lies in its ability to provide a direct measure of the photon energy that exceeds the work function — anything left after accounting for the work function is converted into the kinetic energy of the photoelectrons.

Work Function

Work function plays a critical role in the photoelectric effect. It is defined as the minimum energy needed to remove an electron from the surface of a material, such as a metal. This quantity is unique to each material and reflects the strength of the bond between the electrons and the atomic structure of the material itself.

When light with sufficient energy — greater than the work function — is shone on the material, electrons can absorb photon energy and overcome the work function barrier to be ejected. This energy threshold must be met; if the energy of the incoming photons is less than the work function, no electrons will be emitted, regardless of the intensity of the light.

Understanding the work function is crucial for a multitude of applications, ranging from the design of photodetectors to the study of surface science and even the development of photovoltaic cells for solar energy harvesting.

Planck's Constant

Planck's constant, symbolized as \(h\), is a fundamental quantity in quantum mechanics that sets the scale of quantum effects. With the value of approximately \(6.63 \times 10^{-34} \mathrm{Js}\), Planck's constant relates the energy of a photon to its frequency through the equation \(E = h \cdot f\).

Max Planck, the namesake of the constant, introduced this idea to explain blackbody radiation, leading to the birth of quantum theory. Planck's constant not only quantifies the discrete nature of energy in quantum phenomena but is also integral to the Heisenberg uncertainty principle and the structure of atoms. It is fundamental to the precision measurement of physical quantities and underpins technologies such as atomic clocks.

Photon Energy Calculation

Calculating the energy of a photon is necessary to understand the dynamics of light-matter interactions in the photoelectric effect. The energy of a photon is given by the equation \(E = h \cdot f\), where \(E\) is the photon energy, \(h\) is Planck's constant, and \(f\) is the frequency of the photon.

To calculate the photon energy, one must first determine its frequency, which can be done using the speed of light \(c\) and the wavelength \(\lambda\) with the relation \(f = \frac{c}{\lambda}\). Once the frequency is known, multiplying it by Planck's constant gives the photon energy. This energy calculation is essential for predicting the outcome of the photoelectric effect, including the ejection of electrons and the determination of stopping potential in photoelectric experiments.

Furthermore, this knowledge allows students to grasp the principles behind various technologies, including solar panels, LED lights, and quantum computing systems.