Controlled fusion is a possible future energy source that would harness the same nuclear fusion reactions that power the sun. The simplest fusion reaction is ²H⁺ + ²H⁺ → ³He⁺⁺ + n + energy, in which nuclei of two deuterium atoms fuse into a nucleus while ejecting a neutron and releasing a substantial amount of energy. Deuterium is not an element but is the name given to 'heavy hydrogen,' in which the nucleus is not simply a proton but a proton and a neutron, with atomic mass 2 u. Two positive deuterium nuclei, which repel each other, can get close enough to fuse only if they have very high speeds. This can be achieved by creating a plasma of ionized deuterium gas at a temperature of 1.0 x 10⁸ K. No material substance can contain a plasma at this temperature, so the idea is to contain the plasma with magnetic fields. Consider the simplest model of using a solenoid to confine the ions to cyclotron motion around the field lines. The plasma ions have a range of speeds, and it's necessary to contain all the ions with speeds up to three times the rms speed at the plasma temperature. What magnetic field strength is needed to keep the fastest ions in 20-cm-diameter cyclotron orbits? The actual magnetic fields are considerably more complex, but your answer is a reasonable estimate of the required field strengths.

Verified step by step guidance

1. Step 1: Start by understanding the problem. The goal is to calculate the magnetic field strength required to confine the fastest ions in a plasma to cyclotron orbits of a given diameter. The fastest ions have speeds up to three times the root-mean-square (rms) speed at the plasma temperature.
2. Step 2: Recall the formula for the rms speed of particles in a gas: \( v_{\text{rms}} = \sqrt{\frac{3k_B T}{m}} \), where \( k_B \) is the Boltzmann constant, \( T \) is the temperature, and \( m \) is the mass of a single ion. For deuterium, the mass \( m \) is approximately \( 2u \), where \( u \) is the atomic mass unit. Use this formula to calculate the rms speed, and then multiply it by 3 to find the maximum speed \( v_{\text{max}} \).
3. Step 3: The radius of the cyclotron orbit is given by \( r = \frac{mv}{qB} \), where \( m \) is the mass of the ion, \( v \) is the speed of the ion, \( q \) is the charge of the ion (equal to the elementary charge \( e \) for deuterium), and \( B \) is the magnetic field strength. Rearrange this formula to solve for \( B \): \( B = \frac{mv}{qr} \).
4. Step 4: Substitute the values into the formula for \( B \). Use \( v = v_{\text{max}} \), \( r = \frac{d}{2} \) (where \( d \) is the diameter of the orbit, given as 20 cm or 0.2 m), \( q = e \), and \( m = 2u \). Ensure all units are consistent (e.g., convert \( u \) to kilograms).
5. Step 5: Perform the calculation step-by-step to find the magnetic field strength \( B \). This will give you an estimate of the required field strength to confine the fastest ions in the plasma to the specified cyclotron orbits.

Key Concepts

Nuclear Fusion

Nuclear fusion is the process where two light atomic nuclei combine to form a heavier nucleus, releasing energy in the process. This reaction powers stars, including the sun, and is characterized by the fusion of isotopes, such as deuterium. The energy released during fusion is due to the conversion of mass into energy, as described by Einstein's equation E=mc². Understanding fusion is crucial for exploring potential energy sources that could provide clean and sustainable power.

Plasma Physics

Plasma is a state of matter where gas is ionized, meaning it contains free electrons and ions. In fusion research, creating and maintaining a plasma at extremely high temperatures (like 1.0 x 10⁸ K) is essential for overcoming the electrostatic repulsion between positively charged nuclei. Plasma physics involves understanding the behavior of charged particles in electromagnetic fields, which is critical for confining plasma using magnetic fields in fusion reactors.

Cyclotron Motion

Cyclotron motion refers to the circular motion of charged particles in a magnetic field, where the radius of the orbit depends on the particle's speed and the strength of the magnetic field. In the context of fusion, maintaining deuterium ions in cyclotron orbits allows for effective confinement of the plasma. The relationship between the magnetic field strength and the radius of the orbit is essential for calculating the required field strength to keep the fastest ions within a specified diameter, ensuring they remain in the desired confinement area.