Question

What is meant by threshold wavelength in photoelectric effect?

Hint:

A common point of confusion is the difference between threshold frequency and threshold wavelength. Remember their inverse relationship: \begin{itemize} \item Threshold Frequency (\(f_0\)): The minimum frequency required for photoemission. \item Threshold Wavelength (\(\lambda_0\)): The maximum wavelength that allows photoemission. \end{itemize}

Answer 1

Step 1: Understanding the Concept:
The photoelectric effect is the phenomenon where electrons are ejected from a metal surface when light shines on it. According to quantum theory, light consists of packets of energy called photons. For an electron to be ejected, it must absorb a photon with enough energy to overcome the binding force holding it to the metal. This minimum energy required for ejection is called the work function (\(\Phi\)) of the material.

Step 2: Key Formula or Approach:
The energy of a single photon (\(E\)) is given by the Planck-Einstein relation: \[ E = hf = \frac{hc}{\lambda} \] where \(h\) is Planck's constant, \(c\) is the speed of light, \(f\) is the frequency, and \(\lambda\) is the wavelength of the light. For the photoelectric effect to occur, the energy of the incident photon must be greater than or equal to the work function: \[ E \ge \Phi \text{or} \frac{hc}{\lambda} \ge \Phi \]

Step 3: Detailed Explanation:
The condition for photoemission, \(\frac{hc}{\lambda} \ge \Phi\), reveals an important relationship between wavelength and electron emission. Since wavelength \(\lambda\) is in the denominator, a larger wavelength corresponds to lower photon energy. This implies that there is a maximum wavelength beyond which the photon energy will be insufficient to overcome the work function. This maximum possible wavelength for which photoemission can occur is called the threshold wavelength, denoted by \(\lambda_0\).
The threshold wavelength corresponds to the exact point where the photon energy is just equal to the work function: \[ E = \Phi \implies \frac{hc}{\lambda_0} = \Phi \] If the incident light has a wavelength \(\lambda > \lambda_0\), its photons will have less energy than the work function (\(E < \Phi\)), and no electrons will be emitted, no matter how intense the light is.

Step 4: Final Answer:
The threshold wavelength (\(\lambda_0\)) is defined as the maximum wavelength of incident radiation that can cause the photoelectric effect for a specific material. It is a characteristic property of the material and is related to its work function by \(\lambda_0 = hc/\Phi\).