Problem 15 - University Physics with Modern Physics

Question

Protons are accelerated from rest by a potential difference of 4.00 \(\mathrm{kV}\) and strike a metal target. If a proton produces one photon on impact, what is the minimum wavelength of the resulting \(x\) rays? How does your answer compare to the minimum wave-length if \(4.00-\mathrm{keV}\) electrons are used instead? Why do x-ray tubes use electrons rather than protons to produce x rays?

Step-by-Step Solution

Answer

Protons produce longer wavelength x-rays than electrons. Electrons are used because they produce shorter wavelength, higher energy x-rays.

Step 1: Understand the Problem

Protons are accelerated by a potential difference, and we need to find the minimum wavelength of x-rays produced when these protons hit a target. Additionally, we compare this to the wavelength produced by electrons of similar energy and explain the preference for using electrons.

Step 2: Calculate Proton Kinetic Energy

The kinetic energy of the protons after being accelerated from rest by a potential difference of 4.00 kV is given by:\[E_{p} = eV = 4.00 \times 10^{3} \text{ eV}\]where \(e\) is the elementary charge, and \(V\) is the potential difference.

Step 3: Determine the Minimum Wavelength

Use the energy-wavelength relationship for photons, given by the equation:\[\lambda = \frac{hc}{E_{p}}\]Insert the values for Planck's constant \(h = 6.63 \times 10^{-34} \text{ J s}\), speed of light \(c = 3.00 \times 10^{8} \text{ m/s}\), and calculated energy \(E_{p}\) converted to Joules (1 eV = \(1.602 \times 10^{-19} \text{ J}\)).Minimum wavelength \(\lambda_p\) for protons can be determined.

Step 4: Compare with Electrons

For electrons with the same energy, the process is repeated using the same formula to derive the electron minimum wavelength \(\lambda_e\). Since the mass of electrons is much less than protons, the resulting wavelength is shorter for electrons.

Step 5: Reason for Electrons over Protons

Electrons are used in x-ray tubes because they are much lighter than protons, allowing for easier acceleration and resulting in shorter wavelengths. Shorter wavelength x-rays have higher energy, which is valuable in x-ray production.

Key Concepts

Proton Acceleration

In the world of physics, protons are subatomic particles with a positive charge. They are found within an atom's nucleus and are pivotal to atomic structure. To accelerate a proton, we often use an electric field generated by a potential difference. When a proton from rest is accelerated by a potential difference of 4.00 kV, it gains kinetic energy. This energy is calculated using the formula:\[E_p = eV\]where \(e\) represents the elementary charge, approximately \(1.602 \times 10^{-19}\) Coulombs, and \(V\) is the potential difference in volts. For a potential difference of 4.00 kV, the proton gains 4,000 electron volts (eV) of energy.

• This energy increase allows protons to strike a target.
• The impact creates photons, which are integral in producing x-rays.

Understanding proton acceleration is key to grasping x-ray production mechanisms.

Electron Acceleration

Electrons, negatively charged particles, play a critical role in x-ray creation, particularly due to their light mass compared to protons. When electrons are accelerated by a potential difference, similar to protons, their energy transformation is governed by the same formula:\[E = eV\]However, electrons are much lighter, which profoundly impacts their behavior:

• They achieve higher velocities quickly with the same amount of energy input.
• Their impact on a target is more efficient for photon production.

For a potential difference of 4.00 kV, electrons also gain energy equal to 4,000 eV. This energy enables the generation of photons with shorter wavelengths than those produced by heavier protons, making electrons particularly suitable for x-ray tube applications. This is due to their capacity to yield higher-energy x-rays, which are crucial in medical imaging and other applications.

Wavelength Comparison

When both protons and electrons collide with a target, they release energy as photons with specific wavelengths. The relationship between the particle's energy and the resultant photon wavelength is expressed by:\[\lambda = \frac{hc}{E}\]where \(h\) is Planck's constant \(6.63 \times 10^{-34}\) Js, and \(c\) is the speed of light \(3.00 \times 10^{8}\) m/s. By inserting the same energy value (4,000 eV converted to Joules), different minimum wavelengths for electrons and protons can be produced.

• Electrons, being lighter, result in shorter wavelengths.
• Shorter wavelengths equate to higher energy photons.

This reveals why electrons are favored in x-ray equipment: they are efficient at producing high-energy x-rays at wavelengths more suited to medical diagnostics.

Kinetic Energy Calculation

Calculating the kinetic energy acquired by particles like protons and electrons during acceleration is foundational in understanding their role in producing x-rays. The basic formula of kinetic energy acquired through acceleration by electric fields is:\[E = eV\]In both protons and electrons, the energy gained (in eV) is equivalent to the product of the charge and the potential difference. However, when we perform practical application calculations:

• For protons, the greater mass means slower speeds at a given energy input.
• Electrons, given their small mass, achieve much higher speeds, impacting targets more effectively.

In transforming potential energy to kinetic energy, measuring in electron volts provides insights into the energy efficiency of particles. This aids in selecting the right particle for specific purposes, such as choosing electrons over protons for x-ray imaging, due to their efficiency at shorter wavelengths and higher photon energies.