Problem 10 - College Physics

Question

\(\bullet\) What would the minimum work function for a metal have to be for visible light (having wavelengths between 400 \(\mathrm{nm}\) and 700 \(\mathrm{nm} )\) to eject photoelectrons?

Step-by-Step Solution

Answer

The minimum work function must be about 3.1 eV.

Step 1: Understand the Problem Context

The photoelectric effect occurs when photons strike a metal surface and eject electrons. The energy of the photons must be greater than or equal to the minimum work function of the metal for electrons to be ejected.

Step 2: Identify the Range of Wavelengths

The given wavelengths for visible light range from 400 nm to 700 nm. We need to consider the energy associated with these wavelengths to determine the minimum work function required.

Step 3: Determine the Maximum Energy from Visible Light

The energy of a photon is related to its wavelength by the formula \(E = \frac{hc}{\lambda}\), 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 in meters. The shortest wavelength (400 nm) will have the maximum energy.

Step 4: Convert Wavelength to Meters

Convert 400 nm to meters: \(400 \, \text{nm} = 400 \times 10^{-9} \, \text{m}\).

Step 5: Calculate Photon Energy for 400 nm Wavelength

Substitute the values into the energy formula: \[ E = \frac{(6.626 \times 10^{-34}) \cdot (3 \times 10^8)}{400 \times 10^{-9}} \]. This will give the energy in joules.

Step 6: Solve for Photon Energy

Perform the calculation: \[ E = \frac{1.9878 \times 10^{-25}}{400 \times 10^{-9}} = 4.97 \times 10^{-19} \, \text{J} \].

Step 7: Convert Energy from Joules to Electronvolts

Since work function is typically given in electronvolts (eV), convert the energy: \(1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J}\). Thus, \[ 4.97 \times 10^{-19} \, \text{J} = \frac{4.97 \times 10^{-19}}{1.602 \times 10^{-19}} \, \text{eV} \approx 3.1 \, \text{eV} \].

Step 8: Determine Minimum Work Function

The minimum work function should be at most equal to the photon energy calculated for the shortest wavelength: approximately 3.1 eV.

Key Concepts

Work Function

The work function is a crucial concept in understanding the photoelectric effect. It is the minimum amount of energy needed to eject an electron from the surface of a metal. When light hits the metal, only photons with energy equal to or greater than the work function can dislodge electrons.

• Measured in electronvolts (eV).
• Specific to each metal and dependent on its surface properties.
• In the context of visible light, only the higher energy photons, typically corresponding to the blue or violet end of the spectrum, have enough energy to overcome the work function of many metals.

Understanding the work function helps in designing sensors and light-detecting devices by selecting appropriate materials based on the desired wavelength sensitivity.

Wavelength

Wavelength is an essential characteristic of light, representing the distance between successive peaks of a wave. It's key in determining the energy of the photons involved in the photoelectric effect.

• Typically measured in nanometers (nm) for visible light.
• Inverse relationship with energy: shorter wavelengths correspond to higher energy photons.
• For visible light, the spectrum ranges from 400 nm (violet) to 700 nm (red).

This range is critical when calculating the minimum work function because it helps identify which wavelengths (or colors of light) can induce the photoelectric effect for a given material.

Photon Energy

Photon energy is what drives the photoelectric effect. It is the energy carried by a single photon and is directly related to its wavelength.

• Calculated using the formula: \(E = \frac{hc}{\lambda}\), where \(h\) is Planck's constant and \(c\) is the speed of light.
• Measured in joules or electronvolts (eV).
• Higher energy photons (shorter wavelengths) can eject electrons from metals with a higher work function.

Converting the photon energy from joules to electronvolts allows for comparisons with the work function, which is crucial in applications like photovoltaic cells and photo-detectors.

Visible Light

Visible light is the portion of the electromagnetic spectrum that is visible to the human eye. It ranges from violet (around 400 nm) to red (around 700 nm).

• Photons within this range are used in the photoelectric effect to study and harness electron emissions.
• Different colors correspond to different photon energies; violet light has the most energy, while red has the least within this range.
• The ability of visible light to drive the photoelectric effect depends on whether the light contains enough energy to overcome the work function of the specific metal.

Understanding these properties of visible light can inform the selection of materials and design of devices that rely on light detection and energy conversion.

Planck's Constant

Planck's constant is a fundamental constant in physics, pivotal in the calculation of photon energy. It links the energy of a photon to its frequency.

• Denoted by \(h\), its value is approximately \(6.626 \times 10^{-34} \text{ Js}\).
• Appears in the photon energy formula \(E = \frac{hc}{\lambda}\).
• Helps in converting wavelength measurements into energy values.

This constant is a cornerstone of quantum mechanics and is indispensable for understanding phenomena like the photoelectric effect. By incorporating Planck's constant into calculations, you can accurately determine whether a particular wavelength of light is capable of knocking electrons off a metal surface.