Problem 13 - University Physics with Modern Physics

Question

When ultraviolet light with a wavelength of 254 nm falls on a clean copper surface, the stopping potential necessary to stop emission of photoelectrons is 0.181 \(\mathrm{V}\) (a) What is the photoelectric threshold wavelength for this copper surface? (b) What is the work function for this surface, and how does your calculated value compare with that given in Table 38.1\(?\)

Step-by-Step Solution

Answer

Threshold wavelength is 267 nm, and work function is 4.6 eV, consistent with reference values.

Step 1: Understanding the Problem

We need to find the threshold wavelength for a copper surface and calculate the work function using the photoelectric effect principles. Given: Wavelength \( \lambda = 254 \) nm and stopping potential \( V_s = 0.181 \) V.

Step 2: Calculate the Energy of the Incident Photon

Calculate the energy \( E \) of the photon using \( E = \frac{hc}{\lambda} \). Here, \( h = 6.626 \times 10^{-34} \; \text{J s} \), \( c = 3 \times 10^8 \; \text{m/s} \), and \( \lambda = 254 \times 10^{-9} \; \text{m} \).

Step 3: Convert the Energy to Electron Volts

Convert the energy from joules to electron volts (1 eV = \( 1.602 \times 10^{-19} \) J) to align with the stopping potential given in volts.

Step 4: Use Stopping Potential to Find Work Function

Use the photoelectric equation \( \frac{hc}{\lambda} = \text{Work function} + eV_s \) to solve for the work function \( \phi \).

Step 5: Calculate the Threshold Wavelength

The threshold wavelength \( \lambda_0 \) is found using the work function: \( \phi = \frac{hc}{\lambda_0} \). Rearrange to solve for \( \lambda_0 \).

Step 6: Compare Calculated Work Function

Compare the calculated work function \( \phi \) with known data to check consistency.

Key Concepts

Threshold Wavelength

The threshold wavelength is a concept in the photoelectric effect that refers to the longest wavelength of light that can still cause the removal of an electron from a material. When light hits the surface of a material, its energy is absorbed by electrons. If the energy is high enough, an electron can be ejected from the surface.- This longest allowable wavelength is known as the threshold wavelength.- Beyond this wavelength, the energy of the photons is too low to liberate any electrons.To calculate the threshold wavelength, we use the equation involving the work function \[\phi = \frac{hc}{\lambda_0}\] where \( \phi \) is the work function, \( h \) is Planck's constant, and \( c \) is the speed of light. Rearranging this formula allows us to solve for \( \lambda_0 \), which is the threshold wavelength:\[\lambda_0 = \frac{hc}{\phi}\]Understanding the threshold wavelength helps us predict whether certain light sources can cause photoelectron emission.

Work Function

The work function is a fundamental property of a material surface that indicates the minimum energy required to remove an electron from the surface.- It's typically measured in electron volts (eV).- The value varies between different materials due to their unique atomic structure.In the context of the photoelectric effect, the work function represents the barrier that photons must overcome to eject electrons. The photoelectric effect is governed by the equation:\[\frac{hc}{\lambda} = \phi + eV_s\]Here, \( \phi \) symbolizes the work function, and \( eV_s \) is the energy due to the stopping potential. By rearranging this, we can solve for \( \phi \):\[\phi = \frac{hc}{\lambda} - eV_s\]Thus, to calculate the work function, we subtract the energy provided by the stopping potential from the total energy of the incident photon. This approach helps us understand how different materials interact with light differently.

Stopping Potential

Stopping potential, often denoted as \( V_s \), is a critical component in the study of the photoelectric effect. It represents the potential difference needed to halt the most energetic electrons emitted from a surface.- The stopping potential directly measures the maximum kinetic energy of the ejected electrons.- Higher stopping potentials indicate more energetic photoelectrons are being emitted.In the formula related to the photoelectric effect \[\frac{hc}{\lambda} = \phi + eV_s\]the term \( eV_s \) (where \( e \) is the elementary charge) quantifies this energy required to stop the photoelectrons. The stopping potential provides insight into the energy characteristics of electrons freed by photons, and it is crucial for calculating the work function. Understanding stopping potential helps clarify how energy from light converts into kinetic energy in the photoelectric effect.