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Magazine Chapter 30: Problem 21
In a proton accelerator used in elementary particle physics experiments, the trajectories of protons are controlled by bending magnets that produce a magnetic field of 4.80 T. What is the magnetic-field energy in a 10.0 -cm' volume of space where \(B=4.80 \mathrm{T} ?\)
Short Answer
Expert verified
The magnetic-field energy is approximately 9.16 Joules.
Step by step solution
01
Understanding the Formula
The energy stored in a magnetic field within a given volume can be calculated using the formula \(U_B = \frac{1}{2\mu_0} B^2 V\) where \(U_B\) is the magnetic energy, \(B\) is the magnetic field strength, \(V\) is the volume, and \(\mu_0\) is the permeability of free space, approximately \(4\pi \times 10^{-7} \, \text{T} \cdot \text{m/A}\).
02
Converting Units
The volume provided is 10.0 cm鲁. First, convert this volume into cubic meters because the SI unit for volume in this context is meters. Therefore, \(10.0 \, \text{cm}^3 = 10.0 \times 10^{-6} \, \text{m}^3\).
03
Calculating Magnetic Energy
Using the formula \(U_B = \frac{1}{2\mu_0} B^2 V\), substitute the values \(B = 4.80 \, \text{T}\), \(\mu_0 = 4\pi \times 10^{-7} \, \text{T} \cdot \text{m/A} \), and \(V = 10.0 \times 10^{-6} \, \text{m}^3\). This gives: \(U_B = \frac{1}{2 \times 4\pi \times 10^{-7}} (4.80)^2 (10.0 \times 10^{-6})\).
04
Performing the Calculation
Calculate \(B^2 = (4.80)^2 = 23.04\). Then, \(\frac{1}{2 \times 4\pi \times 10^{-7}} = \frac{1}{8\pi \times 10^{-7}}\). This value is approximately \(3.9785 \times 10^{5}\). Finally, calculate \(U_B \approx 3.9785 \times 10^{5} \times 23.04 \times 10^{-6}\).
05
Conclusion
The magnetic field energy is calculated as approximately \(9.164 \, \text{Joule}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Proton Accelerator
A proton accelerator is a device that uses electromagnetic fields to propel charged particles, like protons, to high speeds. These machines are fundamentally important in the field of particle physics, as they allow scientists to examine the fundamental components of matter. Proton accelerators are used in a variety of applications, including research, medicine, and industry.
Protons are accelerated using magnetic and electric fields, with the focus on controlling their direction and speed. The intense energy levels achieved allow the particles to overcome barriers and make new reactions possible.
- In research, particularly elementary particle physics, they help in probing the properties of atoms, nuclei, and the particles that constitute them, by smashing particles at nearly the speed of light.
- In medicine, accelerators are used in cancer treatment through a method known as proton therapy, which targets tumors with high precision.
- In industry, they can be used to produce materials with specific properties.
Protons are accelerated using magnetic and electric fields, with the focus on controlling their direction and speed. The intense energy levels achieved allow the particles to overcome barriers and make new reactions possible.
Bending Magnets
Bending magnets are a critical component in the function of a proton accelerator. These magnets generate magnetic fields that can alter the paths of moving charged particles, such as protons, within the accelerator.
By carefully controlling these magnets, accelerators can achieve consistent and repeatable results, allowing for high-purity scientific experiments.
- Function: They are designed to change the trajectory of the particles without altering their speed. By adjusting the magnetic field, scientists can precisely control the pathways, making sure protons follow the intended route within the accelerator.
- Design: Typically composed of magnetized materials wrapped around bends in the accelerator, guiding protons to their specific points of collision.
- Role: Essential in ensuring that high-energy protons are directed accurately, thus maximizing efficiency and effectiveness in creating the desired experimental conditions.
By carefully controlling these magnets, accelerators can achieve consistent and repeatable results, allowing for high-purity scientific experiments.
Magnetic Field Strength
Magnetic field strength, denoted as \(B\), represents the intensity of a magnetic field in a given area. This measure is critical in proton accelerators as it influences how particles like protons are guided and controlled.
The unit of magnetic field strength is Tesla (T), which is a measure of the concentration of magnetic flux. In the context of the exercise, a field strength of 4.80 T is quite strong, indicating a powerful magnetic field capable of significantly altering the path of protons.
Accurately understanding and controlling this parameter ensures that complex experiments can be conducted reliably.
The unit of magnetic field strength is Tesla (T), which is a measure of the concentration of magnetic flux. In the context of the exercise, a field strength of 4.80 T is quite strong, indicating a powerful magnetic field capable of significantly altering the path of protons.
- High magnetic field strengths are necessary in accelerators to create the force needed to bend the trajectory of fast-moving protons.
- These forces help ensure that protons reach the necessary energy and precise location for intended collisions or interactions.
- Magnetic field strength must be carefully monitored and adjusted to maintain performance and achieve experimental objectives.
Accurately understanding and controlling this parameter ensures that complex experiments can be conducted reliably.
Permeability of Free Space
The permeability of free space, represented by the symbol \(\mu_0\), is a fundamental constant that measures the ability of a vacuum to support a magnetic field. It is used in calculations involving magnetic forces and fields, such as those in proton accelerators.
Understanding \(\mu_0\) is crucial for physicists when designing and operating magnetic systems in accelerators, ensuring the necessary conditions for experimental research are met efficiently and safely.
- The value of \(\mu_0\) is approximately \(4\pi \times 10^{-7} \, \text{T}\cdot\text{m/A}\).
- This constant plays a vital role when determining the energy stored in a magnetic field, as shown in the solution formula \(U_B = \frac{1}{2\mu_0} B^2 V\).
- It acts as a bridge between magnetic field strength, volume, and energy, allowing scientists to calculate how much energy is involved in a specific magnetic setup.
Understanding \(\mu_0\) is crucial for physicists when designing and operating magnetic systems in accelerators, ensuring the necessary conditions for experimental research are met efficiently and safely.
