Problem 45 - Master Resource Book in JEE Main Physics

Question

The threshold wavelength for a metal having work function \(W_{0}\) is \(\lambda_{0}\). What is the threshold wavelength for a metal whose work function is \(\frac{W_{0}}{2} ?\) (a) \(4 \lambda_{0}\) (b) \(2 \lambda_{0}\) (c) \(\frac{\lambda_{0}}{2}\) (d) \(\frac{\lambda_{0}}{4}\)

Step-by-Step Solution

Answer

The threshold wavelength is \(2 \lambda_{0}\) (option b).

Step 1: Understand the Relationship

Work function \(W_0\) and threshold wavelength \(\lambda_0\) are related through the equation \(W_0 = \frac{hc}{\lambda_0}\), where \(h\) is Planck's constant and \(c\) is the speed of light. This formula relates the energy needed to emit an electron with the wavelength of incident photons.

Step 2: Calculate the New Work Function

For the metal with work function \(\frac{W_0}{2}\), its corresponding threshold wavelength \(\lambda_1\) can be found using \(\frac{W_0}{2} = \frac{hc}{\lambda_1}\).

Step 3: Set Up the Equation for New Wavelength

From the equation \(\frac{W_0}{2} = \frac{hc}{\lambda_1}\), solve for \(\lambda_1\). Rearranging gives \(\lambda_1 = \frac{2hc}{W_0}\).

Step 4: Express New Threshold Wavelength in Terms of \(\lambda_0\)

Substitute the expression for \(W_0\) from \(W_0 = \frac{hc}{\lambda_0}\) into \(\lambda_1 = \frac{2hc}{W_0}\). This becomes \(\lambda_1 = 2 \lambda_0\).

Step 5: Choose the Correct Answer

With \(\lambda_1 = 2 \lambda_0\), the correct answer is option (b) \(2 \lambda_0\).

Key Concepts

Threshold Wavelength

The threshold wavelength is a fundamental concept in understanding the photoelectric effect. It refers to the longest wavelength of light that can still release electrons from a metal surface.
This wavelength is closely related to the work function of the metal, which is the minimum energy needed to remove an electron.
The threshold wavelength is given by the formula:

• \[ W_0 = \frac{hc}{\lambda_0} \]

Where:

• \( W_0 \) is the work function of the metal,
• \( h \) is Planck's constant \( (6.626 \times 10^{-34} \, \text{Js}) \),
• \( c \) is the speed of light \( (3 \times 10^8 \, \text{m/s}) \),
• \( \lambda_0 \) is the threshold wavelength.

This equation shows the inverse relationship between the threshold wavelength and the work function. This means as the work function halves, the threshold wavelength doubles.
Understanding this relationship is crucial in solving physics problems involving light and metals.

Photoelectric Effect

The photoelectric effect is a key phenomenon in physics that demonstrates the particle nature of light. It occurs when light hits a metal surface and electrons are ejected as a result.
The basic principle is simple: when a photon with enough energy strikes the metal, it transfers its energy to an electron, allowing it to escape.
The energy of the incoming photon must be greater than the work function for this effect to occur. Important points about the photoelectric effect:

• It supports the quantum theory, showing light behaves as particles called photons.
• The kinetic energy of ejected electrons doesn't depend on the intensity of light, but on its frequency.
• This effect cannot be explained by classical wave theory of light, highlighting its revolutionary importance.

In essence, the photoelectric effect showcases the intriguing interplay between light and matter. It's a cornerstone for modern physics, helping to develop concepts like light quanta and quantum mechanics.

Planck's Constant

Planck's constant \((h)\) is a pivotal constant in quantum mechanics, defining the size of quantum effects.
It links the energy of a photon to its frequency: \( E = hf \), where \(E\) is energy, \(f\) is frequency, and \(h\) is Planck's constant. This equation is central to understanding quantum phenomena. Planck's constant's value is \(6.626 \times 10^{-34} \, \text{Js}\).Its significance lies in establishing the quantization of energy levels, meaning energy only comes in discrete units or 'quanta'.Role of Planck's Constant in the Photoelectric Effect:

• It helps determine the energy of photons that triggers the photoelectric effect.
  
• It provides insight into how different wavelengths of light impact electron ejection.
  
• The formula \( W_0 = \frac{hc}{\lambda_0} \) is founded on Planck's constant, showing its crucial role in connecting light's energy to wavelength.

Planck’s constant is more than just a number—it's a bridge linking classical physics and quantum mechanics, pivotal in the discussion of energy and the behavior of particles on microscopic scales.