Calculating the Threshold Wavelength Using the Photoelectric Effect

The threshold wavelength is a fundamental concept in the study of the photoelectric effect, relating the energy of incident light to the work function of a metal. To determine the threshold wavelength of a specific metal when given its work function, we can use the energy-electromagnetic equation derived from the photoelectric effect.

Understanding the Concept

The photoelectric effect is a phenomenon where electrons are ejected from a metal surface when light shines on it. The work function ((phi)) of a metal is the minimum amount of energy required to remove an electron from the metal's surface. Once the threshold wavelength is determined, it can be used to predict the behavior of electrons under different light conditions.

The Equation and Its Derivation

The relationship between the work function and the threshold wavelength can be mathematically expressed as:

(phi frac{hc}{lambda_{c}})

Where:

(phi) represents the work function in electron volts (eV) (h) is Planck's constant, approximately (4.135667696 times 10^{-15} eV cdot s) (c) is the speed of light, approximately (3 times 10^{8} , m/s) (lambda_{c}) is the threshold wavelength in meters

Example Calculation

Let's calculate the threshold wavelength for a metal with a work function of 2.8 eV.

Convert the work function from eV to joules:

[1 eV 1.602 times 10^{-19} , J]

[2.8 eV 2.8 times 1.602 times 10^{-19} J approx 4.49 times 10^{-19} J]

Substitute the values into the equation to find (lambda_{c}):

[lambda_{c} frac{hc}{phi}]

[lambda_{c} frac{4.135667696 times 10^{-15} eV cdot s times 3 times 10^{8} , m/s}{2.8 eV}]

Perform the multiplication and division:

[lambda_{c} approx frac{1.2407 times 10^{-6} , m}{2.8} approx 4.43 times 10^{-7} , m]

This can be expressed in angstroms, where (1 , text{angstrom} 1 times 10^{-10} , m):

[lambda_{c} approx 4430 , text{angstroms}]

Conclusion

The threshold wavelength for a metal with a work function of 2.8 eV is approximately (4430) angstroms. This information is crucial for understanding the behavior of metals in various photoelectric experiments and devices.