Problems & Exercises

What is the direction of the magnetic force on a positive charge that moves as shown in each of the six cases shown in Figure 22.56?

Figure 22.56

What is the direction of the velocity of a negative charge that experiences the magnetic force shown in each of the three cases in Figure 22.57, assuming it moves perpendicular to $\mathbf{\text{B}}\mathrm{?}$

Figure 22.57

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 in the figure below, assuming $\mathbf{\text{B}}$ is perpendicular to $\mathbf{\text{v}}$?

Figure 22.58

What is the maximum force on an aluminum rod with a $0\text{.}\text{100}\text{-μC}$ charge that you pass between the poles of a 1.50-T permanent magnet at a speed of 5.00 m/s? In what direction is the force?

(a) A cosmic ray proton moving toward the Earth at $\text{5.00}\times {\text{10}}^7\text{m/s}$ experiences a magnetic force of $1\text{.}\text{70}\times {\text{10}}^{-\text{16}}\text{N}$. What is the strength of the magnetic field if there is a $\text{45º}$ angle between it and the proton’s velocity? (b) Is the value obtained in part (a) consistent with the known strength of the Earth’s magnetic field on its surface? Discuss.

(a) A physicist performing a sensitive measurement wants to limit the magnetic force on a moving charge in her equipment to less than $1\text{.}\text{00}\times {\text{10}}^{-\text{12}}\mathrm{N}$. What is the greatest the charge can be if it moves at a maximum speed of 30.0 m/s in the Earth’s field? (b) Discuss whether it would be difficult to limit the charge to less than the value found in (a) by comparing it with typical static electricity and noting that static is often absent.

A cosmic ray electron moves at $7\text{.}\text{50}\times {\text{10}}^6\text{m/s}$ perpendicular to the Earth’s magnetic field at an altitude where field strength is $1\text{.}\text{00}\times {\text{10}}^{-5}\mathrm{T}$. What is the radius of the circular path the electron follows?

(a) Viewers of Star Trek hear of an antimatter drive on the Starship Enterprise. One possibility for such a futuristic energy source is to store antimatter charged particles in a vacuum chamber, circulating in a magnetic field, and then extract them as needed. Antimatter annihilates with normal matter, producing pure energy. What strength magnetic field is needed to hold antiprotons, moving at $5\text{.}\text{00}\times {\text{10}}^7\text{m/s}$ in a circular path 2.00 m in radius? Antiprotons have the same mass as protons but the opposite (negative) charge. (b) Is this field strength obtainable with today’s technology or is it a futuristic possibility?

What radius circular path does an electron travel if it moves at the same speed and in the same magnetic field as the proton in Exercise 22.13?

An electron in a TV CRT moves with a speed of $6\text{.}\text{00}\times {\text{10}}^7\text{m/s}$, in a direction perpendicular to the Earth’s field, which has a strength of $5\text{.}\text{00}\times {\text{10}}^{-5}\mathrm{T}$. (a) What strength electric field must be applied perpendicular to the Earth’s field to make the electron moves in a straight line? (b) If this is done between plates separated by 1.00 cm, what is the voltage applied? (Note that TVs are usually surrounded by a ferromagnetic material to shield against external magnetic fields and avoid the need for such a correction.)

A mass spectrometer is being used to separate common oxygen-16 from the much rarer oxygen-18, taken from a sample of old glacial ice. (The relative abundance of these oxygen isotopes is related to climatic temperature at the time the ice was deposited.) The ratio of the masses of these two ions is 16 to 18, the mass of oxygen-16 is $2\text{.}\text{66}\times {\text{10}}^{-\text{26}}\text{kg},$ and they are singly charged and travel at $5\text{.}\text{00}\times {\text{10}}^6\text{m/s}$ in a 1.20-T magnetic field. What is the separation between their paths when they hit a target after traversing a semicircle?

A large water main is 2.50 m in diameter and the average water velocity is 6.00 m/s. Find the Hall voltage produced if the pipe runs perpendicular to the Earth’s $5\text{.}\text{00}\times {\text{10}}^{-5}\text{-T}$ field.

(a) What is the speed of a supersonic aircraft with a 17.0-m wingspan, if it experiences a 1.60-V Hall voltage between its wing tips when in level flight over the north magnetic pole, where the Earth’s field strength is $8\text{.}\text{00}\times {\text{10}}^{-5}\text{T?}$ (b) Explain why very little current flows as a result of this Hall voltage.

Calculate the Hall voltage induced on a patient’s heart while being scanned by an MRI unit. Approximate the conducting path on the heart wall by a wire 7.50 cm long that moves at 10.0 cm/s perpendicular to a 1.50-T magnetic field.

Using information in Example 20.6, what would the Hall voltage be if a 2.00-T field is applied across a 10-gauge copper wire (2.588 mm in diameter) carrying a 20.0-A current?

A patient with a pacemaker is mistakenly being scanned for an MRI image. A 10.0-cm-long section of pacemaker wire moves at a speed of 10.0 cm/s perpendicular to the MRI unit’s magnetic field and a 20.0-mV Hall voltage is induced. What is the magnetic field strength?

What is the direction of the magnetic force on the current in each of the six cases in Figure 22.59?

Figure 22.59

What is the direction of the magnetic field that produces the magnetic force shown on the currents in each of the three cases in Figure 22.61, assuming $\mathbf{\text{B}}$ is perpendicular to $\mathbf{\text{I}}$?

Figure 22.61

(a) A DC power line for a light-rail system carries 1000 A at an angle of $\text{30.0º}$ to the Earth’s $\text{5.00}\times {\text{10}}^{-5}\text{-T}$ field. What is the force on a 100-m section of this line? (b) Discuss practical concerns this presents, if any.

A wire carrying a 30.0-A current passes between the poles of a strong magnet that is perpendicular to its field and experiences a 2.16-N force on the 4.00 cm of wire in the field. What is the average field strength?

(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 $\text{90º}$ with the field?

(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 $9\text{.}\text{00 N}\cdot \text{m.}$ 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 $\tau =\text{NIAB}\text{sin}\theta$, the units of $\mathrm{N}\cdot \mathrm{m}$ must equal units of $\mathrm{A}\cdot {\mathrm{m}}^2\mathrm{T}$. Verify this.

A proton has a magnetic field due to its spin on its axis. The field is similar to that created by a circular current loop $0\text{.}\text{650}\times {\text{10}}^{-\text{15}}\mathrm{m}$ in radius with a current of $1\text{.}\text{05}\times {\text{10}}^4\mathrm{A}$ (no kidding). Find the maximum torque on a proton in a 2.50-T field. (This is a significant torque on a small particle.)

Repeat Exercise 22.41, but with the loop lying flat on the ground with its current circulating counterclockwise (when viewed from above) in a location where the Earth’s field is north, but at an angle $\text{45}\text{.}\mathrm{0º}$ below the horizontal and with a strength of $\text{6.}\text{00}\times {\text{10}}^{-5}\mathrm{T}$.

(a) The hot and neutral wires supplying DC power to a light-rail commuter train carry 800 A and are separated by 75.0 cm. What is the magnitude and direction of the force between 50.0 m of these wires? (b) Discuss the practical consequences of this force, if any.

A 2.50-m segment of wire supplying current to the motor of a submerged submarine carries 1000 A and feels a 4.00-N repulsive force from a parallel wire 5.00 cm away. What is the direction and magnitude of the current in the other wire?

An AC appliance cord has its hot and neutral wires separated by 3.00 mm and carries a 5.00-A current. (a) What is the average force per meter between the wires in the cord? (b) What is the maximum force per meter between the wires? (c) Are the forces attractive or repulsive? (d) Do appliance cords need any special design features to compensate for these forces?

Find the direction and magnitude of the force that each wire experiences in Figure 22.64(a) by, using vector addition.

Figure 22.64

Indicate whether the magnetic field created in each of the three situations shown in Figure 22.65 is into or out of the page on the left and right of the current.

Figure 22.65

What are the directions of the currents in the loop and coils shown in Figure 22.67?

Figure 22.67

To see why an MRI utilizes iron to increase the magnetic field created by a coil, calculate the current needed in a 400-loop-per-meter circular coil 0.660 m in radius to create a 1.20-T field (typical of an MRI instrument) at its center with no iron present. The magnetic field of a proton is approximately like that of a circular current loop $0\text{.}\text{650}\times {\text{10}}^{-\text{15}}\mathrm{m}$ in radius carrying $1\text{.}\text{05}\times {\text{10}}^4\mathrm{A}$. What is the field at the center of such a loop?

Nonnuclear submarines use batteries for power when submerged. (a) Find the magnetic field 50.0 cm from a straight wire carrying 1200 A from the batteries to the drive mechanism of a submarine. (b) What is the field if the wires to and from the drive mechanism are side by side? (c) Discuss the effects this could have for a compass on the submarine that is not shielded.

What current is needed in the solenoid described in Exercise 22.58 to produce a magnetic field ${\text{10}}^4$ times the Earth’s magnetic field of $5\text{.}\text{00}\times {\text{10}}^{-5}\mathrm{T}$?

Measurements affect the system being measured, such as the current loop in Figure 22.62. (a) Estimate the field the loop creates by calculating the field at the center of a circular loop 20.0 cm in diameter carrying 5.00 A. (b) What is the smallest field strength this loop can be used to measure, if its field must alter the measured field by less than 0.0100%?

Find the magnitude and direction of the magnetic field at the point equidistant from the wires in Figure 22.64(a), using the rules of vector addition to sum the contributions from each wire.

What current is needed in the top wire in Figure 22.64(a) to produce a field of zero at the point equidistant from the wires, if the currents in the bottom two wires are both 10.0 A into the page?

Integrated Concepts

(a) A pendulum is set up so that its bob (a thin copper disk) swings between the poles of a permanent magnet as shown in Figure 22.69. What is the magnitude and direction of the magnetic force on the bob at the lowest point in its path, if it has a positive $0\text{.}\text{250 μC}$ charge and is released from a height of 30.0 cm above its lowest point? The magnetic field strength is 1.50 T. (b) What is the acceleration of the bob at the bottom of its swing if its mass is 30.0 grams and it is hung from a flexible string? Be certain to include a free-body diagram as part of your analysis.

Figure 22.69

Integrated Concepts

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.

Integrated Concepts

(a) Using the values given for an MHD drive in Exercise 22.59, and assuming the force is uniformly applied to the fluid, calculate the pressure created in ${\text{N/m}}^2\text{.}$ (b) Is this a significant fraction of an atmosphere?

Integrated Concepts

A current balance used to define the ampere is designed so that the current through it is constant, as is the distance between wires. Even so, if the wires change length with temperature, the force between them will change. What percent change in force per degree will occur if the wires are copper?

Integrated Concepts

A cyclotron accelerates charged particles as shown in Figure 22.70. Using the results of the previous problem, calculate the frequency of the accelerating voltage needed for a proton in a 1.20-T field.

Integrated Concepts

(a) What is the direction of the force on a wire carrying a current due east in a location where the Earth’s field is due north? Both are parallel to the ground. (b) Calculate the force per meter if the wire carries 20.0 A and the field strength is $3\text{.}\text{00}\times {\text{10}}^{-5}\mathrm{T}$. (c) What diameter copper wire would have its weight supported by this force? (d) Calculate the resistance per meter and the voltage per meter needed.

Unreasonable Results

(a) Find the charge on a baseball, thrown at 35.0 m/s perpendicular to the Earth’s $5\text{.}\text{00}\times {\text{10}}^{-5}\mathrm{T}$ field, that experiences a 1.00-N magnetic force. (b) What is unreasonable about this result? (c) Which assumption or premise is responsible?

Unreasonable Results

An inventor wants to generate 120-V power by moving a 1.00-m-long wire perpendicular to the Earth’s $5\text{.}\text{00}\times {\text{10}}^{-5}\mathrm{T}$ field. (a) Find the speed with which the wire must move. (b) What is unreasonable about this result? (c) Which assumption is responsible?

Unreasonable Results

A surveyor 100 m from a long straight 200-kV DC power line suspects that its magnetic field may equal that of the Earth and affect compass readings. (a) Calculate the current in the wire needed to create a $5\text{.}\text{00}\times {\text{10}}^{-5}\mathrm{T}$ field at this distance. (b) What is unreasonable about this result? (c) Which assumption or premise is responsible?