Problem 14 - College Physics

Question

Light with a wavelength range of \(145-295 \mathrm{nm}\) shines on a silicon surface in a photoelectric effect apparatus, and a reversing potential of \(3.50 \mathrm{~V}\) is applied to the resulting photoelectrons. (a) What is the longest wavelength of the light that will eject electrons from the silicon surface? (b) With what maximum kinetic energy will electrons reach the anode?

Step-by-Step Solution

Answer

(a) 263 nm (b) 3.5 eV

Step 1: Understand the Photoelectric Effect

The photoelectric effect occurs when light hits a material and ejects electrons from it. The energy of the photons must overcome the material's work function to release electrons. For silicon, this work function is approximately 4.7 eV.

Step 2: Calculate Maximum Wavelength for Electron Ejection

The energy of a photon is given by the equation \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ Js}) \), \( c \) is the speed of light \( (3 \times 10^8 \text{ m/s}) \), and \( \lambda \) is the wavelength. The work function \( \phi \) of silicon is 4.7 eV, which is \( 4.7 \times 1.6 \times 10^{-19} \text{ J} \). Setting \( E = \phi \), solve for \( \lambda \):\[ \lambda = \frac{hc}{\phi} = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{4.7 \times 1.6 \times 10^{-19}} \approx 263 \text{ nm} \]

Step 3: Determine Maximum Kinetic Energy

With a reversing potential of 3.5 V, the stopping energy \( eV_0 \) is \( 3.5 \times 1.6 \times 10^{-19} \text{ J} = 5.6 \times 10^{-19} \text{ J} \). The kinetic energy of photoelectrons \( KE_{max} = E - \phi \). The longest wavelength relevant for maximum kinetic energy still results in zero KE because it just matches the work function, thus kinetic energy increases for shorter wavelengths. Here, for maximum KE, combine photon energy and reversing potential effects:\[ KE_{max} = eV_0 = 3.5 \text{ eV} \]

Step 4: Conclude With the Values

From the calculations, the longest wavelength that can eject an electron from silicon is approximately 263 nm. The maximum kinetic energy that an electron will have upon reaching the anode, with consideration of the reverse potential, is 3.5 eV.

Key Concepts

Photon Energy

Photon energy is crucial in understanding the photoelectric effect. Photons are tiny packets of light energy, and the energy they carry is determined by their frequency or wavelength. The equation for photon energy is:\[ E = \frac{hc}{\lambda} \]Where:

• \( E \) is the photon energy
• \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \text{ Js} \))
• \( c \) is the speed of light (\( 3 \times 10^8 \text{ m/s} \))
• \( \lambda \) is the wavelength of the light

In essence, photons with shorter wavelengths (higher frequency) have more energy. During the photoelectric effect, light shines on a material like silicon and transfers its energy to electrons. This energy must be sufficient to overcome the material's work function to free electrons. The longer the wavelength, the less energy it carries.

Work Function

The work function is the minimum energy needed to eject an electron from the surface of a material. Each material has a unique work function based on its atomic and electronic structure. For silicon, the work function is about 4.7 eV. Our formula for the work function related to photon energy is intuitive: \[ \phi = \frac{hc}{\lambda} \]Before electrons can be ejected, the incoming photon must have energy equal to or greater than the work function. If the photon energy exactly matches the work function, electrons are released with zero kinetic energy. If the photon energy exceeds the work function, the excess is converted into the kinetic energy of the ejected electrons. Therefore, the wavelength at which photon energy equals the work function is crucial for understanding electron ejection.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. In the context of the photoelectric effect, it's the energy that electrons have once they are ejected from a material surface. The maximum kinetic energy of an ejected electron is calculated using the equation:\[ KE_{max} = E - \phi \]Where:

• \( KE_{max} \) is the maximum kinetic energy
• \( E \) is the energy of the incoming photon
• \( \phi \) is the work function of the material

In practical experiments, a reverse potential is often applied, which affects the maximum kinetic energy of the electrons. For example, if a reverse potential of 3.5 V is applied, this potential energy needs to be counteracted by the kinetic energy of the electron before they reach the anode. Therefore, the kinetic energy must be higher than the work function to overcome any such additional potential barrier and successfully eject an electron.

Silicon Surface

Silicon is a commonly used material in photoelectric effect experiments due to its semiconducting properties. Its work function, which represents the energy needed to eject electrons, is approximately 4.7 eV. This makes it a practical material for studies involving light in the ultraviolet range (like wavelengths from 145 to 295 nm) because it requires just enough photon energy for electron ejection without excessive energy loss. Here's why understanding the material's role is important:

• It helps in setting the appropriate wavelength range for experiments.
• It determines the photon energy required to overcome the work function.
• It affects the efficiency of electron ejection based on photon energy and wavelength.

By knowing these specifics, scientists can tailor experiments to ensure electrons are efficiently ejected, enabling the study of the fundamental principles of quantum mechanics and material properties.

Electron Ejection

Electron ejection is a critical step in the photoelectric effect, where electrons absorb energy from photons and are released from the material. For electrons to be ejected from a silicon surface, the photons need to carry energy that surpasses the work function. If photon energy matches the work function, electrons are simply ejected without any additional motion energy. When photon energy exceeds the work function, electrons are not only ejected but are provided with kinetic energy to move, which is what we measure. The entire process involves:

• Photon absorption by electrons.
• Energy transfer exceeding the work function.
• Overcoming any applied potential that may try to stop the electron.

The phenomenon highlights the dual nature of light, demonstrating both wave-like and particle-like properties. Understanding how different wavelengths and energies affect electron ejection helps in numerous applications, from photovoltaic cells to quantum computing. By studying electron ejection, we uncover deeper insights into atomic and electronic interactions, paving the way for advanced technologies.