Textbook Question
FIGURE EX39.14 is a graph of |ψ(x)|2 for an electron. What is the probability that the electron is located between x = 1.0 nm and x = 2.0 nm?
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35. Special Relativity
Inertial Reference Frames
Problem 41d
Textbook Question
Potassium and gold cathodes are used in a photoelectric-effect experiment. For each cathode, find: The stopping potential if the wavelength is 220 nm.
Verified step by step guidance1
Step 1: Recall the photoelectric equation: \( eV_s = h \nu - \phi \), where \( e \) is the charge of an electron, \( V_s \) is the stopping potential, \( h \) is Planck's constant, \( \nu \) is the frequency of the incident light, and \( \phi \) is the work function of the material.
Step 2: Convert the given wavelength (220 nm) into frequency using the relationship \( \nu = \frac{c}{\lambda} \), where \( c \) is the speed of light and \( \lambda \) is the wavelength. Ensure the wavelength is converted to meters before substitution.
Step 3: Look up the work functions (\( \phi \)) for potassium and gold. These values are material-specific and typically given in electron volts (eV). Convert them to joules if necessary using \( 1 \text{ eV} = 1.602 \times 10^{-19} \text{ J} \).
Step 4: Substitute the calculated frequency (\( \nu \)) and the work function (\( \phi \)) for each material into the photoelectric equation. Solve for the stopping potential \( V_s \) using \( V_s = \frac{h \nu - \phi}{e} \).
Step 5: Ensure the units are consistent throughout the calculation. The stopping potential \( V_s \) will be in volts (V). Perform the calculation separately for potassium and gold to find their respective stopping potentials.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Photoelectric Effect
The photoelectric effect is the phenomenon where electrons are emitted from a material when it absorbs light or electromagnetic radiation. This effect demonstrates the particle nature of light, as photons must have sufficient energy to overcome the work function of the material to release electrons. The energy of the incoming photons is inversely related to their wavelength, which is crucial for calculating the stopping potential.
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Stopping Potential
Stopping potential is the minimum voltage needed to stop the most energetic photoelectrons emitted from a cathode in a photoelectric experiment. It is directly related to the kinetic energy of the emitted electrons, which can be calculated using the equation E_k = eV_s, where E_k is the kinetic energy, e is the charge of the electron, and V_s is the stopping potential. This concept is essential for determining how much energy is required to halt the emitted electrons.
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Wavelength and Energy Relationship
The relationship between wavelength and energy is described by the equation E = hc/λ, where E is the energy of a photon, h is Planck's constant, c is the speed of light, and λ is the wavelength. Shorter wavelengths correspond to higher energy photons, which is critical in the context of the photoelectric effect, as the energy of the incident light must exceed the work function of the material to liberate electrons. This relationship is fundamental for calculating the stopping potential based on the given wavelength.
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