Question

Fully stripped (all electrons removed) sulfur \(\left({ }^{32} \mathrm{~S}\right)\) ions are accelerated in an accelerator from rest using a total voltage of \(1.00 \cdot 10^{9} \mathrm{~V}\). \({ }^{32} \mathrm{~S}\) has 16 protons and 16 neutrons. The accelerator produces a beam consisting of \(6.61 \cdot 10^{12}\) ions per second. This beam of ions is completely stopped in a beam dump. What is the total power the beam dump has to absorb?

Short answer

Answer: The total power the beam dump must absorb is approximately 1.69 x 10^4 W or 16.9 kW.

Step-by-step solution

Calculate the charge of the sulfur ion

Since sulfur has 16 protons and is fully stripped (meaning no electrons are present), the charge of one sulfur ion is the charge of 16 protons. The charge of one proton is \(1.60 \times 10^{-19} \mathrm{C}\). So the charge of the sulfur ion can be calculated as: \(q = 16 \times 1.60 \times 10^{-19} \mathrm{C} = 2.56 \times 10^{-18} \mathrm{C}\)

Calculate the kinetic energy of one ion

The potential energy gained by one ion from the given voltage is equal to the kinetic energy of the ion. We can determine the kinetic energy using the equation: \(KE = qV\), where \(q\) is the charge of the ion and \(V\) is the voltage. Therefore, \(KE = (2.56 \times 10^{-18} \mathrm{C})(1.00 \times 10^{9} \mathrm{V}) = 2.56 \times 10^{-9} \mathrm{J}\)

Calculate the kinetic energy of all ions per second

We know that the accelerator produces \(6.61 \times 10^{12}\) ions per second. So, the total kinetic energy of all ions per second will be: \(KE_{total} = (6.61 \times 10^{12} \text{ ions/s})(2.56 \times 10^{-9} \mathrm{J/ion}) = 1.69 \times 10^{4} \mathrm{J/s}\)

Find the total power that the beam dump has to absorb

Since power is defined as the amount of energy transferred per unit time, the total power that the beam dump has to absorb can be inferred from the total kinetic energy of all ions per second. In this case, the power is equal to: \(P = 1.69 \times 10^{4} \mathrm{J/s}\) So, the total power the beam dump must absorb is approximately \(1.69 \times 10^{4} \mathrm{W}\) or \(16.9 \mathrm{kW}\).

Key concepts

Ion Acceleration

In nuclear physics, ion acceleration is a crucial process where charged particles, or ions, are subjected to an electric field, rapidly increasing their velocity. Here, fully stripped sulfur ions with 16 protons are accelerated from rest across a voltage of 1 billion volts. This process involves transferring energy from the electric field to the ions, making them move faster.

Fully stripped ions mean they lack electrons, resulting in a positive charge equal to the sum of their protons' charges. These ions are directed through an accelerator which uses strong electric fields to push them to higher speeds. This acceleration is key in experiments and practical applications like cancer treatment with ion beams.

Ion accelerators are essential tools in modern science, allowing researchers to probe the structure of matter at a subatomic level and enabling the generation of high-energy beams.

Kinetic Energy Calculation

The concept of kinetic energy refers to the energy a particle possesses due to its motion. Ion acceleration involves converting electrical energy into kinetic energy, which can be calculated using the formula \( KE = qV \), where \( q \) is the charge of the ion and \( V \) is the accelerating voltage.

For a sulfur ion with a charge \( q = 2.56 \times 10^{-18} \mathrm{C} \) and a voltage of \( 1.00 \times 10^{9} \mathrm{V} \), the kinetic energy turns out to be \( 2.56 \times 10^{-9} \mathrm{J} \). This energy represents how much work was done on the ion to accelerate it.

Kinetic energy calculations are fundamental in understanding how ions move and interact, shaping fields such as particle physics and material science.

Beam Dump Power Absorption

When high-energy ions are stopped, as in a beam dump, they transfer their kinetic energy to the material of the beam dump, resulting in energy absorption. This absorbed energy is a form of power, measured in watts.

In this scenario, the accelerator produces \( 6.61 \times 10^{12} \) ions every second, each carrying \( 2.56 \times 10^{-9} \mathrm{J} \) of kinetic energy. Thus, the power absorbed by the beam dump is \( 1.69 \times 10^{4} \mathrm{J/s} \) or \( 16.9 \mathrm{kW} \).

The concept of power absorption is pivotal in ensuring that equipment doesn't overheat and functions safely. It also helps in designing effective thermal management systems to dissipate the absorbed energy.