Chapter 39: Problem 14
(a) In an electron microscope, what accelerating voltage is needed to produce electrons with wavelength 0.0600 nm? (b) If protons are used instead of electrons, what accelerating voltage is needed to produce protons with wavelength 0.0600 nm? (\(Hint\): In each case the initial kinetic energy is negligible.)
Short Answer
Step by step solution
Understanding the De Broglie Wavelength
Relating Wavelength to Voltage for Electrons
Calculate Accelerating Voltage for Electrons
Relating Wavelength to Voltage for Protons
Calculate Accelerating Voltage for Protons
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Electron Microscope
Electron microscopes work by accelerating electrons towards the sample we are examining. Once the electrons hit the sample, they interact with it, providing a detailed image of its structure. The shorter the wavelength of the electrons, the more detailed the image. This is why the concept of de Broglie wavelength is critical—because it relates to how we control the wavelength of these electrons using voltage.
- Shorter wavelengths yield higher resolution images.
- Electrons are accelerated using high voltages.
- De Broglie wavelength helps us understand how to calculate the needed voltage for clarity.
Accelerating Voltage
The relationship between voltage and electron speed is founded on the conversion of electric potential energy into kinetic energy. As electrons are released from a cathode, they are accelerated across a voltage difference. This process converts the potential energy (due to the voltage) directly into the electrons' kinetic energy.
- Higher accelerating voltage reduces the effective wavelength of electrons.
- This lower wavelength aids in achieving greater resolution in electron microscopes.
Kinetic Energy
In our equations, the kinetic energy is related to the voltage by the simple relationship: \[ KE = eV \] where \( KE \) is the kinetic energy, \( e \) is the elementary charge, and \( V \) is the voltage. This makes kinetic energy directly proportional to the accelerating voltage.
- As voltage increases, so does the kinetic energy of electrons.
- Higher kinetic energy leads to shorter wavelengths and higher resolution images.
Planck's Constant
Planck's constant allows us to calculate a particle's wavelength from its momentum. In the context of electron microscopes, this is crucial for determining how we can influence the wavelength through momentum changes driven by kinetic energy and voltage.
- Planck's constant \( (h) \) is approximately \( 6.626 \times 10^{-34} \, \text{J} \cdot \text{s} \).
- A key part of de Broglie's equation: \( \lambda = \frac{h}{p} \).
- It helps in understanding the wave-particle duality of matter.