The work function of a metal is ϕ eV. If λ nm UV light is incident on the photocathode, what is
the stopping voltage of the photoelectrons in volts?

The stopping voltage of photoelectrons can be determined using the **photoelectric effect equation**. According to Einstein's photoelectric equation:

\[
E_{\text{photon}} = \text{KE}_{\text{max}} + \phi
\]

Where:
- \( E_{\text{photon}} \) is the energy of the incident photon.
- \( \text{KE}_{\text{max}} \) is the maximum kinetic energy of the emitted electrons.
- \( \phi \) is the work function of the metal (in electron volts, eV).

### 1. Photon Energy
The energy of the incident photon \( E_{\text{photon}} \) is related to the wavelength \( \lambda \) by the equation:

\[
E_{\text{photon}} = \frac{hc}{\lambda}
\]

Where:
- \( h = 6.626 \times 10^{-34} \, \text{J·s} \) is Planck’s constant.
- \( c = 3 \times 10^8 \, \text{m/s} \) is the speed of light.
- \( \lambda \) is the wavelength in meters.

To convert the energy to electron volts (eV), we use the conversion factor: \( 1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J} \).

So:

\[
E_{\text{photon}} (\text{in eV}) = \frac{1240}{\lambda \, (\text{in nm})}
\]

### 2. Maximum Kinetic Energy of the Electrons
From the photoelectric equation:

\[
\text{KE}_{\text{max}} = E_{\text{photon}} - \phi
\]

### 3. Stopping Voltage
The stopping voltage \( V_s \) is the voltage needed to stop the photoelectrons and is related to the maximum kinetic energy by:

\[
V_s = \text{KE}_{\text{max}} \, \text{(in eV)}
\]

Thus, the stopping voltage can be calculated as:

\[
V_s = \frac{1240}{\lambda} - \phi
\]

### Final Formula:
\[
V_s = \left( \frac{1240}{\lambda} - \phi \right) \, \text{volts}
\]

Where:
- \( \lambda \) is in nanometers (nm),
- \( \phi \) is in electron volts (eV),
- \( V_s \) is in volts.

Would you like to calculate the result for specific values of \( \lambda \) and \( \phi \)? 

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Here are five related questions to expand on this concept:
1. How does the stopping voltage change if the wavelength of the incident light is halved?
2. What happens to the photoelectric emission if the frequency of light is below the threshold frequency?
3. How would increasing the work function \( \phi \) of the material affect the photoelectric current?
4. Can electrons be emitted if the energy of the incident photons is less than the work function?
5. How would you experimentally determine the stopping voltage for a particular metal?

**Tip:** The energy of light is inversely proportional to its wavelength; shorter wavelengths correspond to higher energies.

The work function of a metal is ϕ eV. If λ nm UV light is incident on the photocathode, what is the stopping voltage of the photoelectrons in volts?

Learn how to calculate the stopping voltage of photoelectrons when UV light is incident on a metal photocathode. This problem covers the photoelectric effect, energy of photons, and kinetic energy of electrons. Suitable for advanced high school and undergraduate physics students.

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