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A beam of \(16 \mathrm{MeV}\) deuterons from a cyclotron strikes a copper block. The beam is equivalent to current of \(15 \mu \mathrm{A}\). (a) At what rate do deuterons strike the block? (b) At what rate is thermal energy produced in the block?

Short Answer

Expert verified
(a) \(9.36 \times 10^{13}\) deuterons/s, (b) 239.88 W

Step by step solution

01

Find the Charge of One Deuteron

A deuteron is the nucleus of deuterium, consisting of one proton and one neutron. The charge of one deuteron is the same as that of a proton because the neutron does not contribute to the charge. The elementary charge, or the charge of a proton, is approximately \(1.602\times10^{-19} \, \mathrm{C}\). Thus, the charge of one deuteron is also \(1.602\times10^{-19} \, \mathrm{C}\).
02

Calculate the Number of Deuterons per Second

Current is the rate of flow of charge. Given that the beam current is \(15 \, \mu\mathrm{A} = 15 \times 10^{-6} \, \mathrm{A}\). The current can also be expressed in terms of the number of deuterons striking per second: \(I = n \cdot e\), where \(I\) is the current, \(n\) is the number of deuterons striking per second, and \(e\) is the charge of a deuteron. Rearrange the formula to find \(n\): \[ n = \frac{I}{e} = \frac{15 \times 10^{-6}}{1.602 \times 10^{-19}} \approx 9.36 \times 10^{13} \, \mathrm{deuterons/s}\]
03

Calculate the Energy per Deuteron in Joules

The energy of the deuterons is given as \(16 \, \mathrm{MeV}\). To convert this to joules, use the conversion factor \(1 \mathrm{eV} = 1.602 \times 10^{-19} \, \mathrm{J}\). Therefore, \[ 16 \times 10^6 \, \mathrm{eV} = 16 \times 10^6 \times 1.602 \times 10^{-19} \, \mathrm{J} = 2.5632 \times 10^{-12} \, \mathrm{J}\]
04

Calculate the Rate of Thermal Energy Production in the Block

The rate of thermal energy production is given by the product of the energy per deuteron and the number of deuterons hitting the block per second:\[ \text{Power} = n \cdot \text{Energy per deuteron} = 9.36 \times 10^{13} \times 2.5632 \times 10^{-12} \, \mathrm{J/s} = 239.88 \, \mathrm{W}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Deuterons
A deuteron is the nucleus of a type of hydrogen known as deuterium. It is a simple atomic particle consisting of one proton and one neutron. Because the neutron is neutral, the charge of a deuteron is the same as that of a proton. This charge is critical for calculations involving interactions with electric and magnetic fields.
The charge of a deuteron is approximately \(1.602 \times 10^{-19}\) coulombs, just like a proton. This charge can allow deuterons to be manipulated in devices like the cyclotron, where they are accelerated to high energies. Understanding the fundamental properties of deuterons helps in explaining nuclear reactions and their applications in both scientific and medical fields. They are fundamental in nuclear fusion processes and serve as important constituents in various nuclear experiments.
Cyclotron
A cyclotron is a type of particle accelerator. Using both a magnetic field and electric fields, it serves to accelerate charged particles, like protons, ions, or in this case, deuterons. The process boots the particles to high kinetic energies as they spiral outward from the center of the device in a circular path.
Cyclotrons are essential tools in nuclear physics for probing atomic nuclei or generating particle beams for research and medical applications. The ability of the cyclotron to increase the energy of particles makes it valuable for experiments where high-energy collisions are necessary. Cyclotrons are also pivotal in treating certain types of cancer and in producing radioactive isotopes used in medical imaging. The device relies on the clever interplay of electric and magnetic fields to sustain particle acceleration without losing precision or depth.
Thermal Energy
When high-energy particles like deuterons strike a material, they transfer their kinetic energy to the atoms in the material. This energy results in the increase of thermal energy, which is essentially the disordered kinetic energy of particles within the material.
In our context, the thermal energy produced when deuterons from a cyclotron strike a copper block can be calculated using the rate at which energy is transferred. Specifically, the power, defined here as the rate of thermal energy production, can be found by multiplying the energy of each deuteron with the number of deuterons striking per second. The calculation involves converting energy units and using formulas such as \( \text{Power} = n \cdot \text{Energy per deuteron} \) to obtain the rate of energy conversion to heat, measured in watts.
Understanding how particles interact to produce thermal energy is critical in applications ranging from energy management in nuclear reactors to the design of radiation shielding materials.

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Most popular questions from this chapter

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