Problem 7 - College Physics

Question

In the photoelectric effect, what is the relationship between the threshold frequency \(f_{0}\) and the work function \(\phi ?\)

Step-by-Step Solution

Answer

\(\phi = h f_{0} \); work function depends linearly on threshold frequency.

Step 1: Understanding the Photoelectric Effect

The photoelectric effect occurs when light shining on a metal surface ejects electrons. For this to happen, the light must have a minimum frequency, known as the threshold frequency \(f_0\).

Step 2: Threshold Frequency and Work Function

The work function \(\phi\) is the minimum energy required to eject an electron from the metal surface. This energy can be expressed in terms of frequency, using the equation \(\phi = h f_{0}\), where \(h\) is Planck's constant.

Step 3: Equation for Relationship

The relationship between the threshold frequency \(f_{0}\) and the work function \(\phi\) can be defined by the equation: \(\phi = h f_{0}\). This indicates that the work function is directly proportional to the threshold frequency.

Key Concepts

Threshold Frequency

In the phenomenon of the photoelectric effect, threshold frequency (\(f_{0}\)) is key. It's the minimum frequency of light that can cause electrons to be ejected from the surface of a metal. If the frequency of the incident light is below this point, no electrons will be emitted. Even if the intensity of light is increased, ejection won't happen without reaching the threshold frequency. Here's how it works:

• Each metal has its own specific threshold frequency.
• The threshold frequency (\(f_{0}\)) depends on the type of metal and its atomic structure.

Understanding threshold frequency helps demystify why some materials seem "invisible" to specific light if its frequency is too low. It's not about the light's brightness, but its frequency that matters for electron ejection.

Work Function

The work function, symbolized as \(\phi\), represents the minimum energy needed to eject an electron from the metal surface in the photoelectric effect. This energy is what binds the electrons to the metal. A key observation is:

• Work function is unique for different materials.
• It signifies the strength with which the electrons are bound to the metal.

To calculate the work function, we use the equation \(\phi = h f_{0}\). Planck's constant (\(h\)) is involved, linking the concept directly to frequency. Thus, for a fixed threshold frequency, the work function reflects how strongly electrons are held inside the metal.

Planck's Constant

Planck’s constant (\(h\)) is a fundamental constant which plays a pivotal role in the world of quantum mechanics. It connects the energy of a photon to its frequency in the equation \(E = hf\). When talking about the photoelectric effect, it comes into play in the relationship \(\phi = hf_{0}\). Let’s break it down:

• Planck's constant is approximately \(6.626 \times 10^{-34}\) Js.
• It unifies energy and frequency into a single equation format.
• It helps quantify the amount of energy carried by photons based on their frequency.

Understanding Planck’s constant helps to bridge classical physics concepts with quantum ideas. It solidifies the framework that tells us how energy is quantized in the realm of photons and electrons.