How to Calculate the Threshold Frequency When Given Only the Wavelength

Understanding the principles behind the photoelectric effect is essential for numerous applications in physics and engineering. One key aspect is the threshold frequency, which determines the minimum energy required to eject an electron from a metal surface. This article will guide you through the process of calculating the threshold frequency when you only have the wavelength at hand. Let's dive into the details.

Understanding the Photoelectric Effect

The photoelectric effect is a phenomenon where electrons are ejected from a metal surface when light of a sufficient frequency shines on it. The core concept here is the conservation of energy. When a photon hits an electron, the photon's energy is transferred to the electron, allowing it to break free from the metal's surface, and potentially gain kinetic energy as well.

Key Components of the Photoelectric Effect

Photon Energy: Energy (E hf), where h is Planck's constant and f is the frequency of the light. Work Function: The energy required (Phi) to remove an electron from the metal lattice. Kinetic Energy: The energy left over after removing the electron, (KE frac{1}{2}mv^2).

Threshold Frequency Calculation

The threshold frequency ((f_0)) is the minimum frequency of light that can eject an electron from the metal. It is the point where all the photon's energy is used to overcome the work function, leaving no excess energy for the electron's kinetic energy.

Given the relationship between frequency and wavelength, (f frac{c}{lambda}), where (c) is the speed of light and (lambda) is the wavelength, we can find the threshold frequency if we know the wavelength. However, the direct calculation of the threshold frequency requires knowing the work function and Planck's constant.

The mathematical relationship for the energy of a photon can be written as:

[E hf frac{hc}{lambda} frac{1.24}{lambda(text{in eV})}]

Here, 1.24 eV middot; nm hc in eV middot; nm units (inverted)

Step-by-Step Calculation

Identify the work function of the material. The work function, (Phi), is the energy required to free an electron from the material, usually given in electron volts (eV).

Use the relationship (hf Phi) to find the threshold frequency:

[f_0 frac{Phi}{h}]

Convert the work function into joules (J) if it is given in eV. Use the conversion factor: 1 eV 1.6 x 10-19 J.

[Phi_{text{J}} Phi_{text{eV}} times 1.6 times 10^{-19} text{ J/eV}]

Substitute the values into the frequency equation:

[f_0 frac{Phi_{text{J}}}{6.63 times 10^{-34} text{ Js}}]

Example Calculation

Let's solve for the threshold frequency when the work function is 3 eV and the wavelength is 500 nm.

Convert the work function to joules:

[Phi_{text{J}} 3 text{ eV} times 1.6 times 10^{-19} text{ J/eV} 4.8 times 10^{-19} text{ J}]

Use the threshold frequency formula:

[f_0 frac{4.8 times 10^{-19} text{ J}}{6.63 times 10^{-34} text{ Js}}]

Calculate the frequency:

[f_0 7.24 times 10^{14} text{ Hz}]

Conclusion

Understanding and calculating the threshold frequency is crucial for grasping the photoelectric effect. With the right formulas and a bit of algebra, you can determine the minimum frequency of light needed to eject electrons from a material. Whether you're a student, researcher, or professional in the field, this knowledge is invaluable.

References:

Photoelectric Effect - Wikipedia Photoelectric Effect - The Physics Classroom