Problem 15 - College Physics

Question

.. Light with a wavelength range of \(145-295\) 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) The longest wavelength is approximately 264 nm. (b) The maximum kinetic energy is 3.50 eV.

Step 1: Identify the work function of silicon

The work function for silicon is known to be approximately 4.7 eV. This will be used to find the longest wavelength of light capable of ejecting electrons from the surface.

Step 2: Convert work function to energy

Convert the work function of silicon from electronvolts to joules, using the conversion factor: \( 1 \, \mathrm{eV} = 1.602 \times 10^{-19} \, \mathrm{J} \). Thus, the energy in joules is: \[ 4.7 \, \mathrm{eV} \times 1.602 \times 10^{-19} \, \frac{\mathrm{J}}{\mathrm{eV}} = 7.5294 \times 10^{-19} \, \mathrm{J} \]

Step 3: Calculate the longest wavelength

Use the photoelectric equation \( E = \frac{hc}{\lambda} \) to solve for the longest wavelength \( \lambda \) that can eject electrons. Rearrange the equation: \[ \lambda = \frac{hc}{E} \] where \( h = 6.626 \times 10^{-34} \, \mathrm{Js} \) and \( c = 3.00 \times 10^8 \, \mathrm{m/s} \). Thus, substituting for \( E \): \[ \lambda = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{7.5294 \times 10^{-19}} \approx 2.64 \times 10^{-7} \, \mathrm{m} \] So, \( \lambda \approx 264 \, \mathrm{nm} \). This is the longest wavelength capable of ejecting electrons.

Step 4: Calculate the maximum kinetic energy of photoelectrons

The maximum kinetic energy \( K_\text{max} \) of the ejected electrons can be found using the equation: \[ K_\text{max} = eV \] where \( e = 1.602 \times 10^{-19} \, \mathrm{C} \) is the elementary charge and \( V = 3.50 \, \mathrm{V} \) is the reversing potential: \[ K_\text{max} = 1.602 \times 10^{-19} \times 3.50 \approx 5.607 \times 10^{-19} \, \mathrm{J} \] Alternatively, this is \[ \frac{K_\text{max}}{1.602 \times 10^{-19}} \approx 3.50 \, \mathrm{eV} \].

Key Concepts

Wavelength

Wavelength is a fundamental concept in understanding the photoelectric effect. It's the distance between two consecutive peaks of a wave, such as a light wave. In the context of the photoelectric effect, the wavelength of the incident light determines if electrons can be ejected from a surface when the light hits it.
The photoelectric equation, which links wavelength to energy, is given by:

• \( E = \frac{hc}{\lambda} \)

Here, \( E \) is the energy of the photons, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \, \text{Js}) \), and \( c \) is the speed of light \( (3.00 \times 10^{8} \, \text{m/s}) \).
For electrons to be emitted, the energy of the incident photons (light) must be greater than or equal to the work function of the material, which translates into a shorter wavelength being necessary. The longer the wavelength, the less energy the photon carries.

Work Function

The work function is a crucial parameter in the photoelectric effect and can be thought of as the threshold energy needed to eject an electron from the surface of a material. It's often symbolized as \( \phi \) and is usually measured in electronvolts (eV).

• For silicon, the work function is approximately 4.7 eV.

This value is the minimum energy a photon must have to release an electron. If the photon's energy is less than the work function, no electrons will be emitted.
Conversion of the work function to joules is often necessary for calculations, as in the given problem where 4.7 eV is converted to \( 7.5294 \times 10^{-19} \, \text{J} \), utilizing the conversion factor \( 1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J} \).
Understanding the work function helps in determining the threshold wavelength below which light will cause electron ejection.

Kinetic Energy

In the photoelectric effect, the kinetic energy of the ejected electrons is an important metric. It represents the excess energy that the electrons have after overcoming the work function. This kinetic energy can be calculated using the equation:

• \( K_{\text{max}} = hu - \phi \)

where \( u \) is the frequency of the light. Alternatively, when the stopping voltage \( V \) is known, the kinetic energy can be expressed as:

• \( K_{\text{max}} = eV \)

Where \( e = 1.602 \times 10^{-19} \, \text{C} \) is the elementary charge and \( V \) is the reversing potential applied to halt the photoelectrons.
In our exercise, the reversing potential is 3.50 V, yielding a kinetic energy \( K_{\text{max}} \approx 5.607 \times 10^{-19} \, \text{J} \), equivalent to 3.50 eV. This value quantifies how fast the ejected electrons are traveling when they reach the anode.

Silicon Surface

When discussing the photoelectric effect, knowing the material or surface from which electrons are being ejected is vital. Each material has a unique work function value that informs whether light of a particular wavelength can eject electrons. In this case, we are examining a silicon surface.
Silicon is widely used in electronics and photovoltaics, making its photoelectric properties significant for applications like solar cells. It requires photons with energies at least as large as its work function \( (4.7 \, \text{eV}) \) for electron emission.
In the photoelectric effect apparatus, silicon's work function affects the range of light wavelengths that can successfully free electrons from its surface. This is why understanding the specific properties of the silicon surface aids in configuring systems where the photoelectric effect is utilized for energy or information transfer.