Question

(a) What is the energy of a photon that has wavelength 0.10 \(\mu\)m ? (b) Through approximately what potential difference must electrons be accelerated so that they will exhibit wave nature in passing through a pinhole 0.10 \(\mu\)m in diameter? What is the speed of these electrons? (c) If protons rather than electrons were used, through what potential difference would protons have to be accelerated so they would exhibit wave nature in passing through this pinhole? What would be the speed of these protons?

Short answer

(a) The energy of the photon is \(1.99 \times 10^{-18}\) J. (b) Electrons need a potential difference of \(150.6\) V and speed \(7.3 \times 10^6\) m/s. (c) Protons require \(8.4 \times 10^4\) V and speed \(1.2 \times 10^6\) m/s.

Step-by-step solution

Energy of Photon

The energy of a photon can be calculated using the formula \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant \( 6.63 \times 10^{-34} \text{ Js} \), \( c \) is the speed of light \( 3 \times 10^8 \text{ m/s} \), and \( \lambda \) is the wavelength \( 0.10 \times 10^{-6} \text{ m} \). Substituting the values, we get:\[ E = \frac{(6.63 \times 10^{-34})(3 \times 10^8)}{0.10 \times 10^{-6}} = 1.99 \times 10^{-18} \text{ J} \]

Electron Potential Difference

To find the potential difference, we want the electrons to have a de Broglie wavelength equal to the diameter of the pinhole \( \lambda = 0.10 \times 10^{-6} \text{ m} \). The de Broglie wavelength \( \lambda \) is given by \( \lambda = \frac{h}{mv} \), where \( m \) is the mass of an electron \( 9.11 \times 10^{-31} \text{ kg} \) and \( v \) is the speed of the electrons.We also have \( v = \sqrt{\frac{2eV}{m}} \), where \( e \) is the electron charge \( 1.6 \times 10^{-19} \text{ C} \). Setting \( \lambda = 0.10 \times 10^{-6} \text{ m} \) gives us:\[ 0.10 \times 10^{-6} = \frac{6.63 \times 10^{-34}}{\sqrt{2(1.6 \times 10^{-19})V}(9.11 \times 10^{-31})} \]Solving for \( V \) gives \( V \approx 150.6 \text{ V} \).

Speed of the Electrons

Using the potential difference calculated, the speed \( v \) of the electrons can be found from \( v = \sqrt{\frac{2eV}{m}} \). Substituting \( V = 150.6 \text{ V} \) gives us:\[ v = \sqrt{\frac{2(1.6 \times 10^{-19})(150.6)}{9.11 \times 10^{-31}}} \approx 7.3 \times 10^6 \text{ m/s} \]

Proton Potential Difference

For protons, we use the same de Broglie wavelength \( \lambda = 0.10 \times 10^{-6} \text{ m} \), but with proton mass \( m_p \approx 1.67 \times 10^{-27} \text{ kg} \). Thus, \( V \) for protons can be calculated similarly:\[ 0.10 \times 10^{-6} = \frac{6.63 \times 10^{-34}}{\sqrt{2(1.6 \times 10^{-19})V}(1.67 \times 10^{-27})} \]Solving for \( V \) gives \( V \approx 8.4 \times 10^4 \text{ V} \).

Speed of the Protons

Finally, calculate the speed of the protons using the potential difference \( V = 8.4 \times 10^4 \text{ V} \):\[ v = \sqrt{\frac{2(1.6 \times 10^{-19})(8.4 \times 10^4)}{1.67 \times 10^{-27}}} \approx 1.2 \times 10^6 \text{ m/s} \]

Key concepts

De Broglie Wavelength

The concept of de Broglie wavelength is fascinating and pivotal in quantum mechanics. It refers to the wavelength associated with a particle and is given by the formula \( \lambda = \frac{h}{mv} \), where:

• \( \lambda \) is the de Broglie wavelength,
• \( h \) is Planck's constant \( 6.63 \times 10^{-34} \text{ Js} \),
• \( m \) is the mass of the particle, and
• \( v \) is the velocity of the particle.

This principle shows that all moving particles or objects have wave properties, though noticeable mainly at the atomic scale.
For instance, to reveal the wave nature of electrons passing through a pinhole with a size of 0.10 \( \mu \)m, one must use their de Broglie wavelength. This governs the behavior and properties when their size becomes comparable to their de Broglie wavelength.

Electron Acceleration

When electrons are accelerated, they gain energy, resulting in changes in their speed and wavelength.
By applying a potential difference \( V \), electrons obtain kinetic energy, which influences their de Broglie wavelength. The crucial relationship for energy acquired by accelerated electrons is given by \( eV = \frac{1}{2}mv^2 \), where:

• \( e \) is the electronic charge \( 1.6 \times 10^{-19} \text{ C} \),
• \( V \) is the potential difference in volts,
• \( m \) is the electron mass \( 9.11 \times 10^{-31} \text{ kg} \), and
• \( v \) is the speed of the electron.

In practice, to observe electrons' wave behavior, you accelerate them through a calculated potential difference.This pushes their de Broglie wavelength to align with the scale of the experimental constraints, such as the size of a diffraction grating or a pinhole.

Proton Acceleration

Acceleration of protons is similar in principle to that of electrons, but differs due to their larger mass.Protons, with a mass of about \( 1.67 \times 10^{-27} \text{ kg} \), require a much larger potential difference to achieve the same de Broglie wavelength as electrons.

• For protons, the equation \( eV = \frac{1}{2}mv^2 \) still applies, albeit modified for their increased mass.

This means subatomic calculations require tailoring, as increasing mass affects acceleration and thus changes the kinetic energy equation.In the research example, a much higher potential difference of \( 8.4 \times 10^4 \text{ V} \) is required to match the de Broglie wavelength needed for protons to pass through a 0.10 \( \mu \)m pinhole.

Planck's Constant

Planck's constant is one of the fundamental constants in physics, denoted as \( h \) and valued at \( 6.63 \times 10^{-34} \text{ Js} \).It acts as a bridge between the particle and wave descriptions of light and matter. Planck's constant plays a crucial role in quantifying the energy of photons and in defining the de Broglie wavelength for particles.
As part of the de Broglie relationship \( \lambda = \frac{h}{mv} \), it showcases how wave-like behavior is possible in particles. In photon energy calculation, it is used with the speed of light \( c \) and wavelength \( \lambda \) as \( E = \frac{hc}{\lambda} \).
Ultimately, Planck's constant reflects the limitations of classical mechanics, paving the way to the advent of quantum theory.

Potential Difference

Potential difference is a vital concept in electrostatics and electrical engineering, measured in volts (V). It's fundamentally the difference in electric potential between two points in a circuit. This can also be interpreted as the work done per unit charge to move the charge between two points. Accelerating particles like electrons or protons through a potential difference provides them with kinetic energy.

• This is vital for inducing wave behavior in particles, where their de Broglie wavelengths become prominent.
• The potential difference specifies the energy a particle gains or loses, impacting both speed and wavelength.

In experiments, accurately calculating these factors clarifies behavior at the quantum level, facilitating precision in scientific observations.

Speed of Particles

The speed of particles, such as electrons and protons, plays a critical role in determining their kinetic energy and wavelength.The formula \( v = \sqrt{\frac{2eV}{m}} \) links the velocity \( v \) of a particle to the potential difference \( V \) through which it is accelerated.

• Here, \( e \) represents the charge of the particle, and \( m \) its mass.

As the velocity of a particle increases, its wavelength (based on the de Broglie concept) decreases.
This allows high speeds to be achievied due to significant potential differences, which results in high-energy electron beams useful in many practical applications like electron microscopy or proton therapy in medicine.