Stopping Potential Formula Explained

by Wholesomestory Johnson 37 views

Hello there! 👋 Are you curious about the stopping potential and its formula? You've come to the right place! In this article, we will break down the formula for stopping potential, explain its significance, and provide a detailed explanation to ensure you understand it thoroughly. Let's dive in!

Correct Answer

The formula for stopping potential (V₀) is given by: V₀ = KE_max / e, where KE_max is the maximum kinetic energy of the emitted photoelectrons, and e is the elementary charge (1.602 × 10⁻¹⁹ Coulombs).

Detailed Explanation

The stopping potential is a crucial concept in understanding the photoelectric effect, a phenomenon where electrons are emitted from a material when light shines on it. The stopping potential, denoted as V₀, is the reverse voltage required to stop the most energetic photoelectrons from reaching the collector plate in an experimental setup. This concept helps us relate the energy of the incident photons to the kinetic energy of the emitted electrons.

Key Concepts

Before we delve deeper into the formula, let's define some key concepts:

  • Photoelectric Effect: The phenomenon where electrons are emitted from a material (usually a metal) when light of sufficient frequency shines on it.
  • Photoelectrons: The electrons emitted from the material due to the photoelectric effect.
  • Kinetic Energy (KE): The energy possessed by an object due to its motion. In this context, we are concerned with the kinetic energy of the photoelectrons.
  • Maximum Kinetic Energy (KE_max): The highest kinetic energy possessed by any of the emitted photoelectrons.
  • Elementary Charge (e): The magnitude of the electric charge carried by a single proton or electron, approximately 1.602 × 10⁻¹⁹ Coulombs.
  • Stopping Potential (V₀): The reverse voltage required to stop the most energetic photoelectrons from reaching the collector, thus reducing the photocurrent to zero.

Understanding the Formula

The formula for stopping potential is derived from the principle of energy conservation. When a photon strikes a metal surface, it transfers its energy to an electron. This energy is used in two ways:

  1. To overcome the work function (Φ) of the metal, which is the minimum energy required to remove an electron from the metal's surface.
  2. The remaining energy is converted into the kinetic energy (KE) of the emitted electron.

According to Einstein's photoelectric equation:

E = Φ + KE_max

Where:

  • E is the energy of the incident photon (E = hf, where h is Planck's constant and f is the frequency of the light).
  • Φ is the work function of the metal.
  • KE_max is the maximum kinetic energy of the emitted photoelectrons.

To determine the stopping potential, we need to relate it to the maximum kinetic energy of the photoelectrons. When a reverse voltage (V₀) is applied, the electric potential energy gained by an electron (eV₀) must be equal to the maximum kinetic energy of the photoelectrons for them to be stopped. Therefore:

eV₀ = KE_max

Where:

  • e is the elementary charge (1.602 × 10⁻¹⁹ Coulombs).
  • V₀ is the stopping potential.

Now, rearranging the equation to solve for V₀, we get:

V₀ = KE_max / e

This is the formula for the stopping potential. It shows that the stopping potential is directly proportional to the maximum kinetic energy of the emitted photoelectrons and inversely proportional to the elementary charge.

Steps to Calculate Stopping Potential

To calculate the stopping potential, follow these steps:

  1. Determine the Maximum Kinetic Energy (KE_max):

    • If the energy of the incident photon (E) and the work function (Φ) of the metal are given, use Einstein's photoelectric equation: KE_max = E - Φ.
    • The energy of the incident photon can be calculated using E = hf, where h is Planck's constant (6.626 × 10⁻³⁴ J·s) and f is the frequency of the light.
    • Alternatively, if the wavelength (λ) of the light is given, use E = hc/λ, where c is the speed of light (3.00 × 10⁸ m/s).

  2. Use the Stopping Potential Formula:

    • Once you have the maximum kinetic energy (KE_max), divide it by the elementary charge (e = 1.602 × 10⁻¹⁹ Coulombs) to find the stopping potential (V₀): V₀ = KE_max / e.

Example Calculation

Let's go through an example to illustrate how to calculate the stopping potential.

Example:

Suppose light with a wavelength of 200 nm is incident on a metal surface with a work function of 4.0 eV. Calculate the stopping potential.

Solution:

  1. Calculate the Energy of the Incident Photon (E):

    • Using E = hc/λ:
    • E = (6.626 × 10⁻³⁴ J·s × 3.00 × 10⁸ m/s) / (200 × 10⁻⁹ m)
    • E = 9.939 × 10⁻¹⁹ J
    • Convert Joules to electron volts (eV): E (eV) = E (J) / (1.602 × 10⁻¹⁹ J/eV)
    • E = (9.939 × 10⁻¹⁹ J) / (1.602 × 10⁻¹⁹ J/eV) ≈ 6.2 eV

  2. Calculate the Maximum Kinetic Energy (KE_max):

    • Using Einstein's photoelectric equation: KE_max = E - Φ
    • KE_max = 6.2 eV - 4.0 eV = 2.2 eV

  3. Calculate the Stopping Potential (V₀):

    • Using V₀ = KE_max / e:
    • Since KE_max is already in electron volts, we can directly relate it to the stopping potential in volts:
    • V₀ = 2.2 V

Therefore, the stopping potential for this scenario is 2.2 volts.

Significance of Stopping Potential

The stopping potential provides valuable insights into the photoelectric effect and the properties of materials. Here are some key points regarding its significance:

  • Direct Measurement of Maximum Kinetic Energy: The stopping potential gives a direct measure of the maximum kinetic energy of the emitted photoelectrons. This is crucial for understanding the energy distribution of photoelectrons.
  • Verification of Einstein's Photoelectric Equation: Experimental measurements of the stopping potential can be used to verify Einstein's photoelectric equation, which is a cornerstone of quantum mechanics.
  • Determination of Work Function: By measuring the stopping potential for different frequencies of light, the work function of a material can be determined. This is important for material characterization.
  • Applications in Photomultiplier Tubes and Solar Cells: The principle of stopping potential is utilized in devices like photomultiplier tubes and solar cells, where understanding and controlling the kinetic energy of photoelectrons are essential for efficient operation.

Factors Affecting Stopping Potential

Several factors can influence the stopping potential in a photoelectric experiment:

  • Frequency of Incident Light: The stopping potential increases with the frequency of the incident light. Higher frequency photons have more energy, leading to photoelectrons with higher kinetic energy.
  • Work Function of the Metal: The stopping potential decreases with an increase in the work function of the metal. Metals with higher work functions require more energy to emit electrons, resulting in lower kinetic energy for the photoelectrons.
  • Intensity of Incident Light: The intensity of the light does not affect the stopping potential. Intensity affects the number of photoelectrons emitted but not their maximum kinetic energy.

Common Misconceptions

There are some common misconceptions regarding the stopping potential that are worth clarifying:

  • Misconception 1: Stopping potential depends on the intensity of light.

    • Clarification: The stopping potential is independent of the intensity of light. It only depends on the frequency of the incident light and the work function of the metal.

  • Misconception 2: Higher frequency light will always result in a higher photocurrent.

    • Clarification: While higher frequency light can lead to photoelectrons with higher kinetic energy, the photocurrent (number of electrons emitted per unit time) depends on the intensity of the light, not its frequency.

  • Misconception 3: Stopping potential is the same as the work function.

    • Clarification: The stopping potential is the voltage required to stop the most energetic photoelectrons, while the work function is the minimum energy required to remove an electron from the metal surface. They are related but not the same.

Real-World Applications

The concepts related to stopping potential and the photoelectric effect have several real-world applications:

  • Photomultiplier Tubes (PMTs): PMTs use the photoelectric effect to detect weak light signals. Photoelectrons emitted from a photocathode are amplified through a series of dynodes, and the stopping potential concept helps in optimizing the electron collection efficiency.
  • Solar Cells: Solar cells convert sunlight into electricity using the photoelectric effect. Understanding the stopping potential and the factors affecting it is crucial for designing efficient solar cells.
  • Light Sensors: Many light sensors, such as those used in cameras and other electronic devices, utilize the photoelectric effect to detect light. The stopping potential concept aids in calibrating and optimizing these sensors.
  • Medical Imaging: Techniques like X-ray imaging and PET scans involve the photoelectric effect. Understanding the interactions of photons with matter and the resulting electron emissions is essential for accurate imaging.

Key Takeaways

To summarize, here are the key takeaways about the formula for stopping potential:

  • The formula for stopping potential (V₀) is V₀ = KE_max / e, where KE_max is the maximum kinetic energy of the emitted photoelectrons, and e is the elementary charge.
  • The stopping potential is the reverse voltage required to stop the most energetic photoelectrons from reaching the collector in a photoelectric experiment.
  • The stopping potential is directly proportional to the maximum kinetic energy of the photoelectrons and inversely proportional to the elementary charge.
  • Factors affecting the stopping potential include the frequency of incident light and the work function of the metal.
  • The stopping potential concept has various real-world applications, including photomultiplier tubes, solar cells, and light sensors.

I hope this detailed explanation has clarified the formula for stopping potential and its significance. If you have any more questions, feel free to ask! Happy learning! 😊