Question

Given that the work function of tungsten is \(4.55 \mathrm{eV}\) what is the stopping potential in an experiment using tungsten cathodes at \(360 \mathrm{nm} ?\)

Short answer

Answer: To find the stopping potential, follow these steps: 1. Convert the wavelength into energy (E) using Planck's equation: \(E = \dfrac{h c}{\lambda}\) 2. Convert the energy obtained from step 1 into electron volts (eV): \(E (\mathrm{eV}) = \dfrac{E (\mathrm{J})}{1.6 \times 10^{-19} \mathrm{J/eV}}\) 3. Calculate the maximum kinetic energy of the ejected electrons (K_max) using the photoelectric effect equation: \(K_\text{max} = E - ϕ\) 4. Find the stopping potential (V_stop) using the relationship between the maximum kinetic energy and the stopping potential: \(V_\text{stop} = \dfrac{K_\text{max}}{e}\) Applying these steps will give you the stopping potential for the given experiment.

Step-by-step solution

Convert the wavelength into energy (E)

Using the Planck's equation, we can find the energy (E) of the incident photons: \(E = \dfrac{h c}{\lambda}\) Where: - h = Planck's constant (6.63 x \(10^{-34} \mathrm{Js}\)) - c = Speed of light (3 x \(10^8 \mathrm{ms^{-1}}\)) - λ = Wavelength (360 nm) First, we need to convert the wavelength from nanometers into meters: \(λ = 360 \mathrm{nm} \times \dfrac{1 \mathrm{m}}{10^9 \mathrm{nm}} = 3.6 \times 10^{-7} \mathrm{m}\) Now, we can calculate the energy of the incident photons: \(E = \dfrac{6.63 \times 10^{-34} \mathrm{Js} \times 3 \times 10^8 \mathrm{ms^{-1}}}{3.6 \times 10^{-7} \mathrm{m}}\)

Convert the energy into electron volts (eV)

We need to convert the energy (E) obtained in the previous step from Joules to electron volts (eV) using the conversion: \(1 \mathrm{eV} = 1.6 \times 10^{-19} \mathrm{J}\) \(E (\mathrm{eV}) = \dfrac{E (\mathrm{J})}{1.6 \times 10^{-19} \mathrm{J/eV}}\)

Calculate the maximum kinetic energy of the ejected electrons (K_max)

Now that we have the energy (E) in electron volts, we can calculate the maximum kinetic energy of the ejected electrons (K_max) using the photoelectric effect equation: \(K_\text{max} = E - ϕ\) Where: - K_max = Maximum kinetic energy of the ejected electrons - E = Energy of the incident photons (in eV) - ϕ = Work function (4.55 eV)

Find the stopping potential (V_stop)

Finally, we can find the stopping potential (V_stop) using the relationship between the maximum kinetic energy and the stopping potential: \(K_\text{max} = e * V_\text{stop}\) Where: - e = Elementary charge (1.6 x \(10^{-19} \mathrm{C}\)) We can now solve for the stopping potential (V_stop): \(V_\text{stop} = \dfrac{K_\text{max}}{e}\)

Key concepts

The Photoelectric Effect and Work Function

Understanding the photoelectric effect is crucial when studying how metals emit electrons under the influence of light. This phenomenon was explained by Einstein, building on the work of Max Planck.

At the heart of this process is a property unique to each material known as the 'work function.' The work function is the minimum energy required to eject an electron from the surface of a material. Expressed in electron volts (eV), it is a threshold value: if the energy of incident photons is less than the work function, no electrons are emitted regardless of the light's intensity.

For tungsten, this work function is given as 4.55 eV, meaning electrons can be emitted only if the photon energy exceeds this value. This is a key point for students to grasp, as it illustrates the quantum nature of the interaction between light and matter.

Calculating Photon Energy with Planck's Equation

Planck's equation is a fundamental tool in quantum mechanics and is used to calculate the energy of photons, the basic units of light. This equation connects the seemingly unrelated properties of light — its energy and its wavelength.

The equation is written as:
\[ E = \frac{hc}{\lambda} \]
where E is the photon energy, h is Planck's constant (6.63 x 10⁻³⁴ J.s), c is the speed of light (3 x 10⁸ m/s), and \lambda (lambda) is the wavelength of the light.

By inserting the wavelength into Planck's equation, students can compute the exact energy in joules (J) of the photons hitting the material. It's important to convert this energy into electron volts (eV) since the work function is typically given in this unit. With this calculated value, one can determine if the photoelectric effect will occur based on the material's work function.

Stopping Potential in Photoelectric Effect Experiments

Once photons with sufficient energy strike a surface and eject electrons, the electrons will move away from the material with certain kinetic energy. If we apply an electric potential that opposes the motion of these electrons, we can determine the 'stopping potential.' This is the minimum potential difference needed to reduce the kinetic energy of the photoelectrons to zero, thus stopping them entirely.

This stopping potential provides valuable information. It helps to determine the maximum kinetic energy of the ejected electrons, which is directly related to the photon energy and the work function of the material:
\[ K_{\text{max}} = E - \phi \]
where K_max is the maximum kinetic energy, E is the energy of incident photons, and \phi (phi) is the work function. Stopping potential is measured in volts (V) and is crucial for assessing the energy conversion from photons to kinetic energy in ejected electrons.

Understanding how to measure and apply stopping potential aids students in grasping the physics underlying the photoelectric effect and the conservation of energy within this process.

Photon Energy Calculation Step by Step

Calculating photon energy is elemental for predicting the photoelectric effect in various materials. This energy can be determined using the wavelength of the incident light and Planck's constant, as indicated in Planck's equation.

To compute the energy in a step-by-step manner: first convert the wavelength to meters, plug the values into Planck's equation, and then convert the resultant energy to electron volts. Once you have the photon energy in eV, you can determine if it's greater than the work function of the material in question.

If the photon energy surpasses the work function, electrons will be emitted and acquire kinetic energy that can be quantified as a difference in potential – the stopping potential. The steps one takes in these calculations are crucial in understanding the fundamental principles of how light interacts with matter at the atomic level.