It is shown in more advanced courses that charged particles in circular orbits radiate electromagnetic waves, called cyclotron radiation. As a result, a particle undergoing cyclotron motion with speed v is actually losing kinetic energy at the rate$\frac{dK}{dt}=-\left(\frac{{\mu }_0{q}^4}{6\pi c{m}^2}\right)B^2{v}^2$
How long does it take (a) an electron and (b) a proton to radiate away half its energy while spiraling in a 2.0 T magnetic field?

The microwaves in a microwave oven are produced in a special tube called a magnetron. The electrons orbit the magnetic field at 2.4 GHz, and as they do so they emit 2.4 GHz electromagnetic waves. What is the magnetic field strength?

Particle accelerators, such as the Large Hadron Collider, use magnetic fields to steer charged particles around a ring. Consider a proton ring with 36 identical bending magnets connected by straight segments. The protons move along a 1.0-m-long circular arc as they pass through each magnet. What magnetic field strength is needed in each magnet to steer protons around the ring with a speed of 2.5 x 10⁷ m/s? Assume that the field is uniform inside the magnet, zero outside.

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.

A proton moves in the uniform fields E = 2500 k V/m and B = 0.50 k T. At t = 0 s the proton is moving in a 1.0-cm-diameter circle in the xy-plane. How many revolutions will the proton have made during this time interval?

An electron in a cathode-ray beam passes between 2.5-cm-long parallel-plate electrodes that are 5.0 mm apart. A 2.0 mT, 2.5-cm-wide magnetic field is perpendicular to the electric field between the plates. The electron passes through the electrodes without being deflected if the potential difference between the plates is 600 V. What is the electron's speed?

An electron in a cathode-ray beam passes between 2.5-cm-long parallel-plate electrodes that are 5.0 mm apart. A 2.0 mT, 2.5-cm-wide magnetic field is perpendicular to the electric field between the plates. The electron passes through the electrodes without being deflected if the potential difference between the plates is 600 V. If the potential difference between the plates is set to zero, what is the electron's radius of curvature in the magnetic field?

What is the radius of a hydrogen atom whose electron moves at 7.3×10⁵ m/s?

What strength of magnetic field is used in a cyclotron in which protons make 3.5 x 10⁷ revolutions per second?