Chapter 11: Magnetic Forces and FieldsContent check — University Physics 2, Chapter 11: Magnetic Forces and Fields

Repeat the previous problem with the magnetic field in the x-direction rather than in the z-direction. Check your answers with RHR-1.

A uniform magnetic field of magnitude 1.5 T is directed horizontally from west to east. (a) What is the magnetic force on a proton at the instant when it is moving vertically downward in the field with a speed of $4\times {10}^7\text{m/s?}$ (b) Compare this force with the weight w of a proton.

A straight, flexible length of copper wire is immersed in a magnetic field that is directed into the page. (a) If the wire’s current runs in the +x-direction, which way will the wire bend? (b) Which way will the wire bend if the current runs in the –x-direction?

Check Your Understanding

In what orientation would a magnetic dipole have to be to produce (a) a maximum torque in a magnetic field? (b) A maximum energy of the dipole?

A Hall probe consists of a copper strip, $n=8.5\times {10}^{28}$ electrons per cubic meter, which is 2.0 cm wide and 0.10 cm thick. What is the magnetic field when I = 50 A and the Hall potential is (a) $4.0\mu \text{V}$ and (b) $6.0\mu \text{V}?$

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?

Discuss the similarities and differences between the electrical force on a charge and the magnetic force on a charge.

At a given instant, an electron and a proton are moving with the same velocity in a constant magnetic field. Compare the magnetic forces on these particles. Compare their accelerations.

An electron passes through a magnetic field without being deflected. What do you conclude about the magnetic field?

How could you determine which pole of an electromagnet is north and which pole is south?

Considering the magnetic force law, are the velocity and magnetic field always perpendicular? Are the force and velocity always perpendicular? What about the force and magnetic field?

A magnetic field exerts a force on the moving electrons in a current carrying wire. What exerts the force on a wire?

Hall potentials are much larger for poor conductors than for good conductors with the same current. Why?

What is the direction of the magnetic force on a positive charge that moves as shown in each of the six cases?

What is the direction of the velocity of a negative charge that experiences the magnetic force shown in each of the three cases, assuming it moves perpendicular to B?

What is the direction of the magnetic field that produces the magnetic force on a positive charge as shown in each of the three cases, assuming $\overrightarrow{B}$ is perpendicular to $\overrightarrow{v}$?

(a) Aircraft sometimes acquire small static charges. Suppose a supersonic jet has a 0.500-μC charge and flies horizontally at a speed of 660. m/s over Earth’s south magnetic pole (near the north geographic pole), where the $8.00\times {10}^{\mathrm{-5}}-\text{T}$ magnetic field points straight down into the ground. What are the direction and the magnitude of the magnetic force on the plane? (b) Discuss whether the value obtained in part (a) implies this is a significant or negligible effect.

An electron moving at $4.00\times {10}^3\text{m/s}$ in a 1.25-T magnetic field experiences a magnetic force of $1.40\times {10}^{\mathrm{-16}}\text{N}.$ What angle does the velocity of the electron make with the magnetic field? There are two answers.

A cosmic-ray electron moves at $7.5\times {10}^6\text{m/s}$ perpendicular to Earth’s magnetic field at an altitude where the field strength is $1.0\times {10}^{\mathrm{-5}}\text{T}.$ What is the radius of the circular path the electron follows?

(a) An oxygen-16 ion with a mass of $2.66\times {10}^{\mathrm{-26}}\text{kg}$ travels at $5.0\times {10}^6\text{m/s}$ perpendicular to a 1.20-T magnetic field, which makes it move in a circular arc with a 0.231-m radius. What positive charge is on the ion? (b) What is the ratio of this charge to the charge of an electron? (c) Discuss why the ratio found in (b) should be an integer.

(a) At what speed will a proton move in a circular path of the same radius as the electron in the previous exercise? (b) What would the radius of the path be if the proton had the same speed as the electron? (c) What would the radius be if the proton had the same kinetic energy as the electron? (d) The same momentum?

An alpha-particle $(m=6.64\times {10}^{\mathrm{-27}}\text{kg,}$ $q=$ $3.2\times {10}^{\mathrm{-19}}\text{C})$ travels in a circular path of radius 25 cm in a uniform magnetic field of magnitude 1.5 T. (a) What is the speed of the particle? (b) What is the kinetic energy in electron-volts? (c) Through what potential difference must the particle be accelerated in order to give it this kinetic energy?

What is the direction of the magnetic force on the current in each of the six cases?

What is the direction of the magnetic field that produces the magnetic force shown on the currents in each of the three cases, assuming $\overrightarrow{B}$ is perpendicular to I?

(a) A dc power line for a light-rail system carries 1000 A at an angle of 30.0º to Earth’s $5.0\times {10}^{\mathrm{-5}}\text{T}$ field. What is the force on a 100-m section of this line? (b) Discuss practical concerns this presents, if any.

(a) By how many percent is the torque of a motor decreased if its permanent magnets lose 5.0% of their strength? (b) How many percent would the current need to be increased to return the torque to original values?

Find the current through a loop needed to create a maximum torque of $9.0\text{N}·\text{m}.$ The loop has 50 square turns that are 15.0 cm on a side and is in a uniform 0.800-T magnetic field.

Since the equation for torque on a current-carrying loop is τ = NIAB sin θ, the units of N ⋅ m must equal units of A ⋅ m² T. Verify this.

A proton has a magnetic field due to its spin. The field is similar to that created by a circular current loop $0.65\times {10}^{\mathrm{-15}}\text{m}$ in radius with a current of $1.05\times {10}^4\text{A}.$ Find the maximum torque on a proton in a 2.50-T field. (This is a significant torque on a small particle.)

Repeat the previous problem, but with the loop lying flat on the ground with its current circulating counterclockwise (when viewed from above) in a location where Earth’s field is north, but at an angle 45.0° below the horizontal and with a strength of $6.0\times {10}^{\mathrm{-5}}\text{T}.$

The cross-sectional dimensions of the copper strip shown are 2.0 cm by 2.0 mm. The strip carries a current of 100 A, and it is placed in a magnetic field of magnitude B = 1.5 T. What are the value and polarity of the Hall potential in the copper strip?

A charged particle moves through a velocity selector at constant velocity. In the selector, E = $1.0\times {10}^4\text{N/C}$ and B = 0.250 T. When the electric field is turned off, the charged particle travels in a circular path of radius 3.33 mm. Determine the charge-to-mass ratio of the particle.

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 accelerate the protons? (e) Repeat the calculations for alpha-particles.

(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}^{\mathrm{-25}}\text{kg}$ and $3.95\times {10}^{\mathrm{-25}}\text{kg},$ respectively, and they travel at $3.0\times {10}^5\text{m/s}$ 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 be big enough to be practical in the separation of uranium-235 from uranium-238.

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

The mass and charge of a water droplet are $1.0\times {10}^{\mathrm{-4}}\text{g}$ and $2.0\times {10}^{\mathrm{-8}}\text{C},$ respectively. If the droplet is given an initial horizontal velocity of $5.0\times {10}^5\widehat{i}\text{m/s},$ 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 $\overrightarrow{v}=$ $(4.0\widehat{i}+3.0\widehat{j}+2.0\widehat{k})\times {10}^6\text{m/s}$ enters a region where there is a uniform electric field and a uniform magnetic field. The magnetic field is given by $\overrightarrow{B}=$ $(1.0\widehat{i}-2.0\widehat{j}+4.0\widehat{k})\times {10}^{\mathrm{-2}}\text{T}.$ 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 $2.0\times {10}^{\mathrm{-3}}\text{s}$ to complete eight revolutions in a uniform magnetic field of magnitude $2.0\times {10}^{\mathrm{-2}}\text{T}.$ 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 $m_d=2m_p$ and $m_{\alpha }=4m_p.$

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 $R_A\text{/}{R}_B$ of the radii of their circular orbits?

An electron moving along the +x -axis at $5.0\times {10}^6\text{m/s}$ 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 $8.0\times {10}^{\mathrm{-3}}\text{T}.$ 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°.

The current loop shown in the accompanying figure lies in the plane of the page, as does the magnetic field. Determine the net force and the net torque on the loop if I = 10 A and B = 1.5 T.

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 $30\text{°}$ 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:

$$\frac{\mu }{L}=\frac{e}{2m}.$$

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 $6.64\times {10}^{\mathrm{-27}}\text{kg}$ (that of a helium atom) moving at $8.70\times {10}^5\text{m/s}$ 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 = $6\times {10}^5\text{m/s}$ enters a region of uniform magnetic field of B = 0.5 T at an angle of $q=30\text{°}$ 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 $B_x=$ $0,$ $B_y=$ $0,$ $and{B}_z=$ $ax,$ 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.

The accompanied figure shows an arrangement for measuring mass of ions by an instrument called the mass spectrometer. An ion of mass m and charge +q is produced essentially at rest in source S, a chamber in which a gas discharge is taking place. The ion is accelerated by a potential difference $V_{\text{acc}}$ and allowed to enter a region of constant magnetic field ${\overrightarrow{B}}_0.$ In the uniform magnetic field region, the ion moves in a semicircular path striking a photographic plate at a distance x from the entry point. Derive a formula for mass m in terms of $B_0,$ q, $V_{\text{acc}},$ and x.

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 $\overrightarrow{B}=$ $2.0\widehat{i}+$ $5.0x^2\widehat{j},$ 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.