Stopping Potential: Formula And Explanation

Hey there! 👋 Today, we're diving into the fascinating world of physics to understand the "stopping potential formula." You might be wondering, "What is stopping potential, and how do I calculate it?" Don't worry; I'm here to break it down for you in a simple, step-by-step manner. We’ll explore the formula, the concepts behind it, and some real-world applications. Let’s get started!

Correct Answer:

The stopping potential (${V_s}$) is given by the formula: ${V_s = \frac{KE_{max}}{e} = \frac{h \cdot f - \phi}{e}}$, where ${KE_{max}}$ is the maximum kinetic energy of the emitted electrons, ${e}$ is the elementary charge, ${h}$ is Planck's constant, ${f}$ is the frequency of the incident light, and ${\phi}$ is the work function of the metal.

Detailed Explanation:

Let's delve deeper into the stopping potential formula and understand each component involved. 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. To fully grasp this, we need to cover a few key concepts.

Key Concepts:

• Photoelectric Effect: The photoelectric effect is the emission of electrons when electromagnetic radiation, such as light, hits a material. These electrons are called photoelectrons.
• Work Function (${\phi}$): The work function is the minimum amount of energy required to remove an electron from the surface of a solid, usually a metal.
• Kinetic Energy (${KE_{max}}$): When photons (light particles) hit the metal surface, they transfer their energy to the electrons. If the energy of the photon is greater than the work function, electrons are emitted with a certain kinetic energy. ${KE_{max}}$ represents the maximum kinetic energy of these emitted electrons.
• Stopping Potential (${V_s}$): The stopping potential is the voltage required to stop the most energetic photoelectrons from reaching the collector in a photoelectric effect experiment. In simpler terms, it's the voltage needed to completely halt the flow of photoelectrons.
• Planck's Constant (${h}$): Planck's constant is a fundamental constant in quantum mechanics, approximately equal to ${6.626 \times 10^{-34} \text{ Js}}$.
• Frequency (${f}$): The frequency of the incident light refers to the number of oscillations per unit time of the electromagnetic wave. It is measured in Hertz (Hz).
• Elementary Charge (${e}$): The elementary charge is the electric charge carried by a single proton or electron, approximately equal to ${1.602 \times 10^{-19} \text{ C}}$.

The Stopping Potential Formula Explained:

The stopping potential formula is derived from the principle of energy conservation. When a photon strikes the metal, its energy (${h \cdot f}$) is used to overcome the work function (${\phi}$) and provide the electron with kinetic energy (${KE_{max}}$). This is expressed as:

${h \cdot f = \phi + KE_{max}}$

To determine the stopping potential (${V_s}$), we need to relate it to the maximum kinetic energy of the emitted electrons. The work done by the stopping potential in halting the electrons is equal to the maximum kinetic energy of the electrons. Mathematically, this is represented as:

${KE_{max} = e \cdot V_s}$

Where:

• ${KE_{max}}$ is the maximum kinetic energy of the emitted electrons.
• ${e}$ is the elementary charge.
• ${V_s}$ is the stopping potential.

From these equations, we can derive the formula for the stopping potential:

${V_s = \frac{KE_{max}}{e}}$

Substituting ${KE_{max} = h \cdot f - \phi}$ into the equation, we get:

${V_s = \frac{h \cdot f - \phi}{e}}$

So, the stopping potential (${V_s}$) is given by:

${V_s = \frac{KE_{max}}{e} = \frac{h \cdot f - \phi}{e}}$

Breaking Down the Formula:

Let's break down the formula step by step to ensure clarity:

1. ${h \cdot f}$ (Photon Energy):

   • This term represents the energy of the incident photon. Here, ${h}$ is Planck's constant, and ${f}$ is the frequency of the light. The product gives the energy of a single photon.
   • Example: If light with a frequency of ${7.5 \times 10^{14} \text{ Hz}}$ is incident on a metal, the photon energy is: ${h \cdot f = (6.626 \times 10^{-34} \text{ Js}) \cdot (7.5 \times 10^{14} \text{ Hz}) = 4.9695 \times 10^{-19} \text{ J}}$

2. ${\phi}$ (Work Function):

   • This is the minimum energy required to eject an electron from the metal surface. It's a property of the metal itself.
   • Example: The work function for sodium is approximately ${2.75 \text{ eV}}$. Converting this to Joules: ${\phi = 2.75 \text{ eV} = 2.75 \times 1.602 \times 10^{-19} \text{ J} = 4.4055 \times 10^{-19} \text{ J}}$

3. ${KE_{max} = h \cdot f - \phi}$ (Maximum Kinetic Energy):

   • This is the maximum kinetic energy of the emitted electrons. It's the difference between the photon energy and the work function.
   • Example: Using the values from above: ${KE_{max} = 4.9695 \times 10^{-19} \text{ J} - 4.4055 \times 10^{-19} \text{ J} = 5.64 \times 10^{-20} \text{ J}}$

4. ${e}$ (Elementary Charge):

   • This is the charge of a single electron, approximately ${1.602 \times 10^{-19} \text{ C}}$.

5. ${V_s = \frac{KE_{max}}{e}}$ (Stopping Potential):

   • Finally, the stopping potential is the maximum kinetic energy divided by the elementary charge.
   • Example: Using the values from above: ${V_s = \frac{5.64 \times 10^{-20} \text{ J}}{1.602 \times 10^{-19} \text{ C}} = 0.352 \text{ V}}$

Practical Applications and Examples:

Understanding the stopping potential formula has numerous practical applications, especially in technologies that rely on the photoelectric effect.

1. Photomultipliers:

   • Photomultipliers are extremely sensitive light detectors used in various scientific instruments. They utilize the photoelectric effect to amplify weak light signals. The stopping potential helps in controlling the electron flow within the multiplier tubes.

2. Solar Cells:

   • Solar cells convert light energy into electrical energy through the photoelectric effect. Understanding and optimizing the stopping potential is crucial for improving the efficiency of solar cells.

3. Light Sensors:

   • Many light sensors, such as those used in digital cameras and light meters, employ the photoelectric effect. The stopping potential is used to measure the intensity of light.

Example Problem:

Let's go through an example problem to solidify our understanding:

Problem: Light with a frequency of ${8.0 \times 10^{14} \text{ Hz}}$ is incident on a metal with a work function of ${3.0 \text{ eV}}$. Calculate the stopping potential.

Solution:

1. Calculate Photon Energy: ${h \cdot f = (6.626 \times 10^{-34} \text{ Js}) \cdot (8.0 \times 10^{14} \text{ Hz}) = 5.3008 \times 10^{-19} \text{ J}}$

2. Convert Work Function to Joules: ${\phi = 3.0 \text{ eV} = 3.0 \times 1.602 \times 10^{-19} \text{ J} = 4.806 \times 10^{-19} \text{ J}}$

3. Calculate Maximum Kinetic Energy: ${KE_{max} = 5.3008 \times 10^{-19} \text{ J} - 4.806 \times 10^{-19} \text{ J} = 4.948 \times 10^{-20} \text{ J}}$

4. Calculate Stopping Potential: ${V_s = \frac{4.948 \times 10^{-20} \text{ J}}{1.602 \times 10^{-19} \text{ C}} = 0.309 \text{ V}}$

Therefore, the stopping potential is approximately ${0.309 \text{ V}}$.

Key Takeaways:

• The stopping potential is the voltage required to stop the emission of photoelectrons in the photoelectric effect.
• The stopping potential formula is ${V_s = \frac{KE_{max}}{e} = \frac{h \cdot f - \phi}{e}}$.
• Understanding the stopping potential helps in analyzing and optimizing devices such as photomultipliers and solar cells.
• The photoelectric effect demonstrates the quantum nature of light, where light behaves as particles (photons) with discrete energy.

I hope this explanation clarifies the stopping potential formula for you! Feel free to ask if you have more questions! Happy learning! 🚀