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Compute the maximum energy of a photon with a wavelength of 400 nm for a material with a work function of 2.3 eV.

Given:
- Wavelength, [tex] \lambda = 400 \, \text{nm} [/tex]
- Work function, [tex] \phi = 2.3 \, \text{eV} [/tex]

(Note: Remember to use Planck's constant and the speed of light in your calculations.)

Hint: The maximum energy of the photon can be calculated using the formula:
[tex]\[ E = \frac{hc}{\lambda} \][/tex]
Where:
- [tex] h [/tex] is Planck's constant ([tex]6.626 \times 10^{-34} \, \text{Js}[/tex])
- [tex] c [/tex] is the speed of light ([tex]3 \times 10^8 \, \text{m/s}[/tex])



Answer :

GinnyAnswer
Let's solve the given problem step-by-step.

### Problem:

We are given:
- The wavelength of the incident light: [tex]\( \lambda = 400 \)[/tex] nm
- The work function of the metal: [tex]\( \phi = 2.3 \)[/tex] eV

We need to find:
1. The wavelength in meters.
2. The energy of a photon with this wavelength.
3. The maximum kinetic energy of the photoelectrons ejected.

### Step 1: Convert the Wavelength to Meters
The wavelength is given in nanometers. We need to convert this to meters.

[tex]\[ \lambda = 400 \text{ nm} = 400 \times 10^{-9} \text{ m} = 4.0 \times 10^{-7} \text{ m} \][/tex]

### Step 2: Calculate the Energy of a Photon

The energy of a photon can be calculated using the formula:

[tex]\[ E = \frac{h \cdot c}{\lambda} \][/tex]

Where:
- [tex]\( h \)[/tex] is Planck's constant, approximately [tex]\( 4.135667696 \times 10^{-15} \)[/tex] eV·s
- [tex]\( c \)[/tex] is the speed of light, approximately [tex]\( 3.0 \times 10^8 \)[/tex] m/s

Substituting the values, we get:

[tex]\[ E = \frac{4.135667696 \times 10^{-15} \text{ eV·s} \times 3.0 \times 10^8 \text{ m/s}}{4.0 \times 10^{-7} \text{ m}} \][/tex]

Calculating this gives us the photon energy:

[tex]\[ E \approx 3.101750772 \text{ eV} \][/tex]

### Step 3: Calculate the Maximum Kinetic Energy of the Ejected Photoelectron

The maximum kinetic energy ([tex]\( K_{\text{max}} \)[/tex]) of the photoelectrons can be found using the photoelectric equation:

[tex]\[ K_{\text{max}} = E - \phi \][/tex]

Where:
- [tex]\( \phi \)[/tex] is the work function (2.3 eV)

Substituting the values:

[tex]\[ K_{\text{max}} = 3.101750772 \text{ eV} - 2.3 \text{ eV} \][/tex]

Calculating this gives us:

[tex]\[ K_{\text{max}} \approx 0.801750772 \text{ eV} \][/tex]

### Summary

In summary, given a wavelength of 400 nm, we have:

1. Converted the wavelength to meters: [tex]\( 4.0 \times 10^{-7} \)[/tex] meters.
2. Calculated the energy of a photon with this wavelength: approximately [tex]\( 3.101750772 \)[/tex] eV.
3. Determined the maximum kinetic energy of the photoelectrons considering the work function of the metal: approximately [tex]\( 0.801750772 \)[/tex] eV.

These calculations are crucial in understanding the photoelectric effect and the behavior of light interacting with materials.

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