Stopping Potential Formula Explained

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Hello there! I'm here to help you understand the formula for stopping potential. I'll provide a clear, detailed, and correct answer to make sure you grasp this important concept.

Correct Answer

The formula for stopping potential (V₀) is given by: V₀ = (KE) / e, where KE is the maximum kinetic energy of the emitted photoelectrons, and e is the elementary charge.

Detailed Explanation

Let's delve deeper into the formula for stopping potential and the underlying principles of the photoelectric effect. This will help you understand not just what the formula is, but why it works.

The Photoelectric Effect: A Brief Overview

The photoelectric effect is the phenomenon where electrons are emitted from a material when light of a certain frequency shines on it. This effect is crucial to understanding many modern technologies, including solar panels and light sensors. The key is that light, composed of photons, interacts with the material's electrons, potentially giving them enough energy to escape.

Key Concepts

Before we jump into the formula, let's define some core concepts:

  • Photon: A quantum of light. It behaves as a particle and carries energy that is proportional to its frequency (E = hf, where 'h' is Planck's constant and 'f' is the frequency).
  • Work Function (Φ): The minimum amount of energy required to remove an electron from the surface of a material. It's like the 'entrance fee' an electron has to pay to escape.
  • Kinetic Energy (KE): The energy of motion. Photoelectrons emitted due to the photoelectric effect possess kinetic energy.
  • Stopping Potential (V₀): The negative voltage applied to a material that is just sufficient to stop the most energetic photoelectrons from reaching the other electrode in a photoelectric setup. It's the 'brakes' we apply to the electrons.
  • Elementary Charge (e): The magnitude of the electric charge carried by a single electron (approximately 1.602 x 10⁻¹⁹ Coulombs).

Understanding the Stopping Potential

The stopping potential is a critical concept in understanding the photoelectric effect. It is a measure of the maximum kinetic energy of the photoelectrons emitted when light strikes a material. When light shines on a metal, electrons can be ejected if the light's energy (photon energy) is greater than the metal's work function. The extra energy gets converted into the kinetic energy of the emitted electrons. It's like pushing a swing: the harder you push (higher energy light), the faster the swing (higher kinetic energy electrons) will go. We use the stopping potential to measure this maximum kinetic energy.

Derivation of the Formula

Here's how the formula for stopping potential is derived:

  1. Energy Conservation: When a photon strikes a metal surface, its energy (hf) is used for two main things:

    • Overcoming the work function (Φ) of the metal.
    • Providing kinetic energy (KE) to the emitted electron.

    This relationship can be represented by Einstein's photoelectric equation:

    hf = Φ + KE

  2. Stopping Potential and Kinetic Energy: The stopping potential (V₀) is the potential difference required to stop the most energetic photoelectrons. This potential difference creates an electric field that does work on the electrons, slowing them down until they have zero kinetic energy. The work done by the electric field is equal to the charge of the electron times the stopping potential (e * V₀).

    At the stopping potential, the maximum kinetic energy (KE) of the electrons is completely converted into potential energy to stop them from reaching the other electrode. Thus,

    KE = e * V₀

  3. Rearranging for Stopping Potential: To find the formula for the stopping potential (V₀), rearrange the equation above:

    V₀ = KE / e

    This formula tells us that the stopping potential is directly proportional to the maximum kinetic energy of the emitted photoelectrons and inversely proportional to the elementary charge. The higher the maximum KE of the photoelectrons, the higher the stopping potential needed to stop them. Conversely, a larger elementary charge (though it's a constant for electrons) will result in a smaller stopping potential for a given KE.

Example and Application

Let's illustrate with an example:

  • Scenario: Suppose light of a certain frequency strikes a metal, and the emitted photoelectrons have a maximum kinetic energy (KE) of 3.2 x 10⁻¹⁹ Joules. What is the stopping potential (V₀)?

  • Solution:

    1. We know that e = 1.602 x 10⁻¹⁹ Coulombs.
    2. Using the formula: V₀ = KE / e
    3. Substitute the values: V₀ = (3.2 x 10⁻¹⁹ J) / (1.602 x 10⁻¹⁹ C)
    4. Calculate: V₀ ≈ 2 Volts

    So, the stopping potential is approximately 2 Volts. This means you would need to apply a negative voltage of 2 Volts to stop the most energetic photoelectrons from reaching the other electrode.

Practical Significance

The formula for stopping potential is more than just a mathematical equation; it's a tool that helps us understand and measure the energy of light and its interaction with matter. Applications include:

  • Determining Planck's Constant: By measuring the stopping potential for different frequencies of light, scientists can determine Planck's constant (h) experimentally.
  • Photomultiplier Tubes: These devices use the photoelectric effect to detect extremely faint light signals. Understanding stopping potential is crucial to their design and operation.
  • Solar Cells: While not directly related to the stopping potential, the photoelectric effect is the core principle behind solar cells. Understanding the energy of the emitted electrons is vital.

Factors Affecting Stopping Potential

Several factors influence the stopping potential:

  • Frequency of Incident Light: The stopping potential increases linearly with the frequency of the incident light. Higher frequency light means more energetic photons, resulting in photoelectrons with higher kinetic energy and thus a higher stopping potential.
  • Intensity of Incident Light: The intensity of light (the number of photons hitting the surface per second) does not affect the stopping potential. It does affect the number of photoelectrons emitted, but not their individual energy.
  • Material of the Cathode: Different materials have different work functions. A material with a higher work function will require higher frequency light to emit photoelectrons and will result in a lower stopping potential for a given frequency of light.

Common Misconceptions

  • Intensity Affects Stopping Potential: Many people incorrectly believe that increasing the light's intensity will increase the stopping potential. While increasing the intensity increases the number of photoelectrons emitted, it does not affect the maximum kinetic energy of the photoelectrons, and therefore, does not affect the stopping potential.
  • Stopping Potential and the Work Function are the Same: The stopping potential is related to the maximum kinetic energy of the photoelectrons, while the work function is the minimum energy required to remove an electron. They are related through the photoelectric equation, but they are not the same thing.

Key Takeaways

  • The stopping potential (V₀) is the negative voltage required to stop the most energetic photoelectrons.
  • The formula is: V₀ = KE / e, where KE is the maximum kinetic energy and e is the elementary charge.
  • The stopping potential is directly proportional to the maximum kinetic energy of the photoelectrons.
  • The stopping potential is independent of the intensity of the incident light.
  • The stopping potential is a crucial concept in understanding the photoelectric effect, used in determining Planck's constant and in various technologies.