Chapter 23, Magnetic Fields Vi Video Solutions, Electromagnetic Fields and Waves: Including Electric Circuits

Problem 1

The thought experiment of Fig. $23-3$
Show that there is conservation of energy in the thought experiment of Fig. 23-3(a),

David Collins

Numerade Educator

01:44

Tides and the magnetic ficld of the carth
Discuss how tides affect the magnetic field of the earth by considering the case of a river flowing into the sea in the east-to-west direction in the northern hemisphere. Remcmber that the magnetic pole situated at the north geographic pole is a south magnetic pole. The vector $B$ points downward in the northern hemisphere.

Pankaj Jain

Numerade Educator

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Problem 3

The magnetic braking force on a satellite,
A natural satellite whose diameter is $10^{4}$ meters moves at a velocity of 1 kilometer/second in the direction normal to the magnetic field of a planet in a region where $B=10^{-7}$ tesla. The satellite has an appreciable conductivity.
(a) The satellite moves in a perfect vacuum. What happens?
(b) The ambient gas has a density of the order of $10^{\text {in }}$ particles per cubic meter, the particles being either electrons or singly charged ions. Each half of the satellite collects particles of the correct sign in swecping through space. Calculate the order of magnitude of the current.
(c) Calculate the order of magnitude of the braking force.
(d) Someone suggests that this current could provide power for an artificial satellite traveling in the same ficld at the same velocity. Inversely, a current in the opposite direction could serve to propel the satellite. What is your opinion?
Artificial satellite velocities range from about 4 to 8 kilometers/sccond, and $v \times B$ in the ionosphere and magnetosphere ranges from about 100 microvolts/meter to 320 millivolts/meter.

Lainey Roebuck

Numerade Educator

01:28

Problem 4

Eddy-current damping
Figure $23-9$ shows one common type of eddy-current damper. Motion of the copper plate in the field of the permanent magnet induces currents that tend to oppose the motion, according to Lenz's law. Joule losses in the plate dissipate its kinetic energy.
Dampers of this general type are used mostly, but not exclusively, in low-power devices such as watt-hour meters and balances. As you will see, the braking force is proportional to the velocity, as in a viscous fluid.
(a) Explain qualitatively, but in greater detail, the origin of the braking force.
(b) How could you design an automobile speedometer that uses eddy currents?
(c) Say $B$ is uniform over the pole face. The path followed by the current is complex; set $R=3 a /$ obs. This quantity is of the order 3 . The plate has a thickness $s$ and a conductivity $\sigma .$ Calculate the current.
(d) Calculate the braking force $F$. This is proportional to the conductivity. So the plate should be either copper or aluminum. An even better solution is to use an iron plate faced with copper.

Penny Riley

Numerade Educator

03:20

Problem 5

Detecting flaws in metal tubing
Figure 23-10 shows the principle of operation of a device for detecting flaws in metal tubing, or rod. The coils a provide a large gradient of magnetic field along the axis, as in Prob. 18-10. Coil $b$ is connected to a monitor. The tubing $T$ moves at a constant velocity $\boldsymbol{v}$ along the axis of symmetry. A voltage appears across coil $b$ when a flaw passes through. Explain.

Km Neeraj

Numerade Educator

02:15

Problem 6

The flux-gate magnetometer
A magnetometer measures $B$. One common type is the fux-gate magnetometer, which puts to use the hysteresis curve. There exist many forms, one of which is shown in Fig. $23-11(a)$. The two rods are made of a ferromagnetic material such as a ferrite, whose hysteresis curve is shown in Fig. 23-11(b). The twin coils are in series and are wound as in the figure so as to magnetize the rods in opposite directions. The current through these coils is sufficient to carry the material through a complete hysteresis loop.
In the absence of an external field $H_{e x}$ the magnetic fluxes through the rods cancel, and $V=0$.
(a) Sketch $\Phi(t)$ and $V(t)$ for each rod when $H_{e x}=0$.
(b) Sketch the same quantities for $H_{e s} \neq 0$. You will notice that if the oscillator operates at a frequency $f$, the fundamental frequency of $V$ is $2 f$.
This facilitates the measurement because the detector can be made to reject, the frequency $f$.
Flux-gate magnetometers can measure fields down to a few nanoteslas.

Ajay Singhal

Numerade Educator

06:58

Problem 7

The peaking strip.
A peaking strip serves to measure $B$. It consists of a fine wire of permalloy (see below) oricnted in the direction of $B$ with a small pickup coil of a few thousand turns near the center, on the axis of a solenoid, as in Fig. 23-12.
To measure the ambient $B$, the solenoid carries a direct current that just cancels $B$, plus a small alternating current. Then the $H$ on the axis of the solenoid is that of the alternating current, and the strip goes through a hysteresis loop at every cycle.
With molybdenum permalloy the loop is approximately rectangular, and the voltage induced in the small coil has two sharp peaks, one positive and one negative, which can be observed on an oscilloscope.
When the oscilloscope swecp is synchronized with the alternating current in the solenoid, the two peaks are symmetric if the time-averaged $H$ on the axis of the solenoid is zero. Then the steady field of the solenoid exactly
cancels the ambient $B$ and the current in the solenoid is then a measure of $B$

The peaking strip has a rather limited range of applications. (1) The solenoid has to be at least about 10 centimeters long because it must be at least a few times longer than the strip, to avoid excessive end effects. But the length of the strip must be much larger than its diameter, again to reduce end effects, and a decrease in the strip cross section decreases the signal proportionately. (2) The ambient $B$ cannot be larger than a few hundredths of a tesla, for otherwise the power dissipated in the solenoid becomes excessive. (3) If one measures $B$ in the neighborhood of a pole-picce, the field of the solenoid alters the permeability of the iron locally.
Calculate the peak voltage induced in the pickup coil under the following. conditions: strip diameter, 25 micrometers; number of turns in the pickup coil, $1000 ;$ maximum value of $\mu, 75,000$; frequency, 60 hertz; amplitude of the altemating $H, 7$ ampere-turns/meter.

Amit Srivastava

Numerade Educator

04:18

Problem 8

Measuring a resistivity without contacts
It is useful to be able to measure the resistivity of a sample without having to cement contacts to it. One method involves placing a disk of the material inside a solenoid carrying an alternating current, with the two axes parallel, and measuring the power absorbed by the disk. The disk has a radius a, a thickness $s$, and ai conductivity $\sigma$. The magnetic field is uniform. and $B=B_{-m}$ cos ar. We neglect the magnetic ficld of the induced currents. We therefore restrict ourselves to low-conductivity materials.
Find the relation between $\sigma$ and the average dissipated power $P$.

Eduard Sanchez

Numerade Educator

04:02

Problem 9

The induction linear accelerator
Figure 23-13 shows a schematic diagram of an induction linear accelerator. It consists of a series of ferrite toroids linked by the ion beam and by one-turn loops that carry large pulsed currents.
One such accelerator comprises 200 toroids and accelerates a 10kiluampere pulsed electron beam to 50 million electronvolts. Its total length is 80 meten, and the pulses are 70 nanometers wide.
Fxplain its operation qualitatively.

Zachary Warner

Numerade Educator

02:15

Problem 10

A magnetometer that uses eddy currents
Figure 23-14 shows the principle of operation of a magnetometer that can measure magnetic fields as small as $10^{-4}$ tesla and up to $10^{-2}$ tesla. The aluminum plate $P$ turns on the axis $A A$ in the ambient field $B_{0}$ that we wish to measure. The fluctuating eddy currents induced in $P$ produce a fluctuating magnetic flux through the fixed coil $C$, which has $N$ turns, and the voltage $V$ is a measure of $B_{0}$ -
The plate is 10 millimeters square and is cemented inside the plastic rotor of a small air turbine that operates at 1000 revolutions/second. The only metallic parts are the plate and the coil.
An exact calculation of $V$ as a function of geometry, of $\omega$, and of $B_{0}$ would be difficult. But this is unnecessary because we can calibrate the instrument with Helmholtz coils (Prob. 18-9).
(a) How does $V$ vary with $B_{01}$ and with $\omega$ ? Set $\omega t=0$ when the plate lies in the planc of $C$.
(b) What is the frequency of $V$ ?

Ajay Singhal

Numerade Educator

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