11.7 Applications of Magnetic Forces and Fields

A cyclotron is to be designed to accelerate protons to kinetic energies of 20 MeV using a magnetic field of 2.0 T. What is the required radius of the cyclotron?

A physicist is designing a cyclotron to accelerate protons to one-tenth the speed of light. The magnetic field will have a strength of 1.5 T. Determine (a) the rotational period of the circulating protons and (b) the maximum radius of the protons’ orbit.

The magnetic field in a cyclotron is 1.25 T, and the maximum orbital radius of the circulating protons is 0.40 m. (a) What is the kinetic energy of the protons when they are ejected from the cyclotron? (b) What is this energy in MeV? (c) Through what potential difference would a proton have to be accelerated to acquire this kinetic energy? (d) What is the period of the voltage source used to accel…

(a) Triply charged uranium-235 and uranium-238 ions are being separated in a mass spectrometer. (The much rarer uranium-235 is used as reactor fuel.) The masses of the ions are $$$3.90 \times 10^{−25} \text{kg}$ and $3.95 \times 10^{−25} \text{kg} ,$ respectively, and they travel at $3.0 \times 10^{5} \text{m}/\text{s}$ in a 0.250-T field. What is the separation between their paths when they hit a target after traversing a semicircle? (b) Discuss whether this distance between their paths seems to…

Repeat the previous problem with a new magnetic field of $\left(0.4 \hat{i} + 1.2 \hat{k}\right) \text{T} .$

The mass and charge of a water droplet are $1.0 \times 10^{−4} \text{g}$ and $2.0 \times 10^{−8} \text{C} ,$ respectively. If the droplet is given an initial horizontal velocity of $5.0 \times 10^{5} \hat{i} \text{m}/\text{s} ,$ what magnetic field will keep it moving in this direction? Why must gravity be considered here?

An electron of kinetic energy 2000 eV passes between parallel plates that are 1.0 cm apart and kept at a potential difference of 300 V. What is the strength of the uniform magnetic field B that will allow the electron to travel undeflected through the plates? Assume E and B are perpendicular.

An electron moving with a velocity $\overset{\rightarrow}{v} = \left(4.0 \hat{i} + 3.0 \hat{j} + 2.0 \hat{k}\right) \times 10^{6} \text{m}/\text{s}$ enters a region where there is a uniform electric field and a uniform magnetic field. The magnetic field is given by $\overset{\rightarrow}{B} = \left(1.0 \hat{i} - 2.0 \hat{j} + 4.0 \hat{k}\right) \times 10^{−2} \text{T} .$ If the electron travels through a region without being deflected, what is the electric field?

What is the (a) path of a proton and (b) the magnetic force on the proton that is traveling west to east with a kinetic energy of 10 keV in Earth’s magnetic field that has a horizontal component of 1.8 x 10^–5 T north and a vertical component of 5.0 x 10^–5 T down?

An electron and a proton move with the same speed in a plane perpendicular to a uniform magnetic field. Compare the radii and periods of their orbits.

A singly charged ion takes $2.0 \times 10^{−3} \text{s}$ to complete eight revolutions in a uniform magnetic field of magnitude $2.0 \times 10^{−2} \text{T} .$ What is the mass of the ion?

A proton, deuteron, and an alpha-particle are all accelerated from rest through the same potential difference. They then enter the same magnetic field, moving perpendicular to it. Compute the ratios of the radii of their circular paths. Assume that $m_{d} = 2 m_{p}$ and $m_{\alpha} = 4 m_{p} .$

Two particles have the same linear momentum, but particle A has four times the charge of particle B. If both particles move in a plane perpendicular to a uniform magnetic field, what is the ratio $R_{A} / R_{B}$ of the radii of their circular orbits?

An electron moving along the +x -axis at $5.0 \times 10^{6} \text{m}/\text{s}$ enters a magnetic field that makes a $75^{\text{o}}$ angle with the x-axis of magnitude 0.20 T. Calculate the (a) pitch and (b) radius of the trajectory.

(a) What is the angle between a wire carrying an 8.00-A current and the 1.20-T field it is in if 50.0 cm of the wire experiences a magnetic force of 2.40 N? (b) What is the force on the wire if it is rotated to make an angle of 90º with the field?

A 5.0-m section of a long, straight wire carries a current of 10 A while in a uniform magnetic field of magnitude $8.0 \times 10^{−3} \text{T} .$ Calculate the magnitude of the force on the section if the angle between the field and the direction of the current is (a) 45°; (b) 90°; (c) 0°; or (d) 180°.

A circular coil of wire of radius 5.0 cm has 20 turns and carries a current of 2.0 A. The coil lies in a magnetic field of magnitude 0.50 T that is directed parallel to the plane of the coil. (a) What is the magnetic dipole moment of the coil? (b) What is the torque on the coil?

A 4.0-cm by 6.0-cm rectangular current loop carries a current of 10 A. What is the magnetic dipole moment of the loop?

The current through a circular wire loop of radius 10 cm is 5.0 A. (a) Calculate the magnetic dipole moment of the loop. (b) What is the torque on the loop if it is in a uniform 0.20-T magnetic field such that $\mu$ and B are directed at $30 °$ to each other? (c) For this position, what is the potential energy of the dipole?

Consider an electron rotating in a circular orbit of radius r. Show that the magnitudes of the magnetic dipole moment μ and the angular momentum L of the electron are related by:

The density of charge carriers for copper is $8.47 \times 10^{28}$ electrons per cubic meter. What will be the Hall voltage reading from a probe made up of $3 \text{cm} \times \text{2 cm} \times \text{1 cm} \left(\text{L} \times \text{W} \times \text{T}\right)$ copper plate when a current of 1.5 A is passed through it in a magnetic field of 2.5 T perpendicular to the $3 \text{cm} \times \text{2 cm} .$

Show that the Hall voltage across wires made of the same material, carrying identical currents, and subjected to the same magnetic field is inversely proportional to their diameters. (Hint: Consider how drift velocity depends on wire diameter.)

Find the radius of curvature of the path of a 25.0-MeV proton moving perpendicularly to the 1.20-T field of a cyclotron.

Unreasonable results A charged particle having mass $6.64 \times 10^{−27} \text{kg}$ (that of a helium atom) moving at $8.70 \times 10^{5} \text{m}/\text{s}$ perpendicular to a 1.50-T magnetic field travels in a circular path of radius 16.0 mm. (a) What is the charge of the particle? (b) What is unreasonable about this result? (c) Which assumptions are responsible?

Unreasonable results Frustrated by the small Hall voltage obtained in blood flow measurements, a medical physicist decides to increase the applied magnetic field strength to get a 0.500-V output for blood moving at 30.0 cm/s in a 1.50-cm-diameter vessel. (a) What magnetic field strength is needed? (b) What is unreasonable about this result? (c) Which premise is responsible?

A proton of speed v = $6 \times 10^{5} \text{m}/\text{s}$ enters a region of uniform magnetic field of B = 0.5 T at an angle of $q = 30 °$ to the magnetic field. In the region of magnetic field proton describes a helical path with radius R and pitch p (distance between loops). Find R and p.

In a region a non-uniform magnetic field exists such that $B_{x} = 0 , B_{y} = 0 , a n d B_{z} = a x ,$ where a is a constant. At some time t, a wire of length L is carrying a current I is located along the x-axis from origin to x = L. Find the magnetic force on the wire at this instant in time.

A long-rigid wire lies along the x-axis and carries a current of 2.5 A in the positive x-direction. Around the wire is the magnetic field $\overset{\rightarrow}{B} = 2.0 \hat{i} + 5.0 x^{2} \hat{j} ,$ with x in meters and B in millitesla. Calculate the magnetic force on the segment of wire between x = 2.0 m and x = 4.0 m.