Find the threshold wavelength for photoelectric effect from a copper surface, a sodium surface and a cesium surface. The work function of these metals are 4.5 eV, 2.3 eV and 1.9 eV respectively

To determine the threshold wavelength for the photoelectric effect from copper, sodium, and cesium, we need to use the concept of work function, which is the minimum energy required to eject an electron from the surface of a metal. The relationship between energy and wavelength can be described by the equation:

E = hc / λ

Where:

• E is the energy in electron volts (eV),
• h is Planck's constant (4.135667696 x 10^-15 eV·s),
• c is the speed of light (approximately 3 x 10^8 m/s),
• λ is the wavelength in meters (m).

To find the threshold wavelength (λ₀), we can rearrange the equation to:

λ₀ = hc / E

Now, let's plug in the values for each metal's work function and calculate the threshold wavelength.

Copper Surface

The work function for copper is 4.5 eV. Plugging this into our formula gives:

λ₀ = (4.135667696 x 10^-15 eV·s) * (3 x 10^8 m/s) / 4.5 eV

Calculating this results in:

λ₀ ≈ 2.75 x 10^-7 m or 275 nm

Sodium Surface

For sodium, with a work function of 2.3 eV, we calculate:

λ₀ = (4.135667696 x 10^-15 eV·s) * (3 x 10^8 m/s) / 2.3 eV

This gives us:

λ₀ ≈ 5.39 x 10^-7 m or 539 nm

Cesium Surface

Lastly, for cesium, which has a work function of 1.9 eV, the calculation is:

λ₀ = (4.135667696 x 10^-15 eV·s) * (3 x 10^8 m/s) / 1.9 eV

Resulting in:

λ₀ ≈ 6.56 x 10^-7 m or 656 nm

Summary of Threshold Wavelengths

To summarize, the threshold wavelengths for each metal are as follows:

• Copper: 275 nm
• Sodium: 539 nm
• Cesium: 656 nm

These values indicate the minimum wavelength of light needed to initiate the photoelectric effect in each respective metal. Shorter wavelengths (higher energy) are necessary for metals with higher work functions, while longer wavelengths (lower energy) can suffice for metals with lower work functions. This concept is crucial in understanding the applications of the photoelectric effect, such as in photovoltaic cells and light sensors.