Circular Motion of Charges in Magnetic Fields Practice Problems

A beam of singly charged particles of ⁷Li moves through a uniform magnetic field at a speed of 4.0 km/s. A ⁷Li particle has a mass of 1.16 × 10^-26 kg. The magnetic field deflects the particles' beam, and the beam leaves the magnetic field region perpendicular to the initial direction of incidence. Determine the strength of the magnetic field if the distance traveled by the ⁷Li beam is 3.24 cm.

In your laboratory, you set up an experiment with a mass spectrometer used to separate ionized particles. A single ionized particle of mass 3.27 × 10^-25 kg and with an energy of 1.00 MeV moves in a circle of radius 18.00 cm in a uniform magnetic field. What would be the orbital radius of a doubly ionized particle of mass 1.59 × 10^-25 kg entering the same magnetic field with an energy of 1.00 MeV?

In a nuclear experiment, a proton of mass 1.67 × 10^-27 kg and a charge of +1.607 × 10^-19 C moves in a circular trajectory in a uniform magnetic field. The radius of the circular trajectory is 8.70 mm, and the magnitude of the magnetic field is 1.25 T. Calculate (i) the speed, (ii) the period of revolution of the proton, and (iii) the difference in potential that the proton would need to overcome to reach this speed.

A mass spectrometer's ion source generates electrically charged particles with masses of 3.2 × 10^-25 kg and charges of +2 e. The charged particles are accelerated through an electrical field and acquire an energy of 0.40 MeV. Then, the particles pass through a uniform magnetic field of magnitude 1.00 T. Find i) the speed and ii) the radius of the circular path of the charged particles.

In a simplified model, a particle accelerator employs a circular trajectory to confine high-energy electrons, inducing them to undergo cyclotron motion. It is required to contain all the electrons with speeds up to five times the rms speed at a temperature of 1.5 × 10⁸ K. The main inquiry revolves around calculating the magnetic field strength necessary to ensure that the fastest electrons remain confined within cyclotron orbits of a diameter measuring 2 meters. Although actual magnetic fields in practice may be more intricate, your response will provide a reasonable estimate of the required field magnitude.

Given an electron moving in a uniform magnetic field of B = 0.75k̂T and an electric field of E = 4000k̂ V/m, the electron initially follows a circular path in the xy-plane with a radius of 2.5 mm. Determine the number of revolutions completed by the electron at a time when it will have a speed of 6.2 × 10⁸ m/s.

Consider a proton traveling within a particle accelerator. This proton encounters a parallel-plate capacitor that is 3.0 cm in length and has a separation of 4.0 mm between the plates. Perpendicular to the electric field generated by this capacitor, there is a magnetic field. The magnetic field has a strength of 3.5 mT and spans a width of 3.0 cm. The proton is able to pass through this setup without any deflection, but only when the potential difference across the capacitor plates is set to 800 V. Given these conditions, determine the speed at which the proton is moving.

Consider a nuclear experiment involving an alpha particle. This alpha particle is set to traverse a region between two parallel-plate electrodes. These electrodes are 1.5 cm long and are separated by a distance of 2.0 mm. At a right angle to the electric field produced by these electrodes, there exists a magnetic field. This magnetic field has a strength of 1.0 mT and extends across a width of 1.5 cm. The alpha particle is able to navigate through the electrodes without experiencing any deflection, but this is only possible when the potential difference across the electrodes is maintained at 400 V. Given this setup, (i) calculate the speed at which the alpha particle is moving, and (ii) if the potential difference between the plates is set to zero, determine the alpha particle's radius of curvature in the magnetic field.

It has been demonstrated that when charged particles move in circular paths, they emit electromagnetic waves known as cyclotron radiation. Consequently, particles engaged in cyclotron motion with a velocity v experience a gradual depletion of their kinetic energy at a specific rate given by dk/dt = -(µ₀•q⁴•B²•v²)/(8π•m²c). Considering a magnetic field strength of 3.0 T, determine the time it takes for (i) an electron and (ii) a proton to emit one-third of their energy while undergoing spiral motion due to the radiation.

Determine the required magnetic field strength within each of the 42 identical bending magnets in a cyclotron, akin to the one used in particle acceleration. The electrons travel along a circular arc with a length of 0.8 m around each magnet, while maintaining a velocity of 3.2 × 10⁷ m/s. It is assumed that there is a uniform magnetic field inside the magnet and zero magnetic field outside.

What would be the frequency of the electromagnetic waves produced by electrons orbiting around a magnetic field found within a household appliance, if the electrons orbit at a magnetic field strength of 100 mT?

Consider a hydrogen atom with an electron moving at a velocity of 3.60 × 10⁵ m/s. Determine the radius of this hydrogen atom based on the electron's velocity.

Considering that a deuteron and a positron enter a region of a constant magnetic field that also has the same kinetic energy, what will be the ratio of the radii for their circular paths?

An electron is moving in a circular path with a radius of 0.05 m in a magnetic field of 0.2 T directed perpendicular to the plane of the circle. Calculate the magnitude of the electron's linear momentum.

A proton is moving in a circle in a uniform magnetic field B̅. If its kinetic energy stays the same while the magnitude of the magnetic field is tripled, find the relationship between the magnitudes of its initial and final angular momentum.

Given a uniform magnetic field B, derive the expression for the kinetic energy K of a particle with mass m and charge q moving in a circular path in terms of the radius r, charge q, magnetic field B, and mass m. Also, derive the expression for the angular momentum L about the center of the circle in terms of the same variables.

Given a magnetic field of magnitude 0.30 × 10^-4 T pointing northward, determine the speed of a proton with mass 1.67 × 10^-27 kg and charge +e as it moves in a circular path 4.0 km above the surface of Mars. Decide if gravity can be ignored in this situation. (Hint: Ignore relativistic effects. The acceleration due to gravity on Mars is 3.71 m/s² and its radius is 3.39 × 10⁶ m)

A cyclotron is used to accelerate α particles (charge = 2e, mass = 6.64 × 10^-27 kg) at a frequency of 1.2 × 10⁷ revolutions per second. Determine the magnetic field strength required for this operation.