Chapter 26, Magnetic Fields Ix Video Solutions, Electromagnetic Fields and Waves: Including Electric Circuits

Problem 1

Show that the energy stored in a magnetic circuit is $\Phi^{2} \mathscr{R} / 2$, where $\Phi$ is the magnetic flux and $\mathscr{R}$ is the reluctance.

Ameer Said

Numerade Educator

06:32

(a) This theorem follows from Tellegen's theorem (Sec. 8.6). Suppose one has a passive circuit comprising resistances, self-inductances, and capacitances. One applies an alternating voltage $V_{p}$ to an input port.
Show that
$$
V_{p} I_{p}^{*}=P+2 j \omega\left(\mathscr{E}_{\text {mag.av }}-\mathscr{E}_{\text {el.av }}\right)
$$
where the left-hand side is the input complex power, $P$ is the power dissipated in the circuit, $\mathscr{E}_{\text {mag.av }}$ is the average magnetic stored energy, and $\mathscr{E}_{\text {el.av }}$ is the average electric stored energy. This is the energy theorem.
(b) It is shown in Prob. $25-4$ that the star and the delta of Fig. $25-19$ are equivalent. Now, if one applies an alternating voltage $V(\mathrm{rms})$ between terminals $B$ and $C$ of the star, then $V I^{*}$ is real and equal to $V^{2} / 2000$. According to the energy theorem, the average energy stored in the capacitors of the delta must be equal to the average energy stored in the inductor, at any frequency. Show that this is correct.

Ze-Han Lee

Numerade Educator

10:53

Problem 3

The average stored energies in capacitors and in inductors are $C V^{2} / 2$ and $L I^{2} / 2$, respectively, where $V$ and $I$ are rms values.
Show that
$$
\mathscr{E}_{C, \mathrm{av}}=\frac{I^{2}}{2 \omega^{2} C}, \quad \mathscr{E}_{L, \mathrm{av}}=\frac{V^{2}}{2 \omega^{2} L}
$$

Khoobchandra Agrawal

Numerade Educator

05:08

Problem 4

We saw that
$$
\mathscr{E}_{m}=\frac{1}{2} \int_{v} \boldsymbol{J} \cdot \boldsymbol{A} d v
$$
where $v^{\prime}$ is any volume that encloses all the conductors.
Show that any $\boldsymbol{A}$ such that $\boldsymbol{B}=\boldsymbol{\nabla} \times \boldsymbol{A}$ is satisfactory here,

Keshav Singh

Numerade Educator

01:10

Problem 5

High-frequency currents do not penetrate a conductor as do lowfrequency currents. This is the skin effect (Sec. 29.1). Does the selfinductance of a coaxial line increase or decrease with frequency?

Narayan Hari

Numerade Educator

04:12

Problem 6

The suspension and the propulsion of tracked vehicles become major problems at speeds of several hundred kilometers per hour. Wheels are then impractical because vehicle vibration, track damage, and power loss become excessive. The tractive force also deteriorates with increasing speed.
An air cushion provides a satisfactory suspension at high speeds, but it consumes a large amount of power. Propulsion then requires either a propeller or a linear electric motor, with the stator in the track.
It is also possible to support a vehicle by means of magnetic forces, and several methods have been developed. In one of these, superconducting coils in the vehicle generate a magnetic field that extends down into the track, which is a sheet of aluminum. At rest and at low speeds, the vehicle uses wheels. As the speed increases, the eddy currents induced in the track by the traveling magnetic field exert a force of repulsion on the currents in the vehicle coils, and the vehicle flies about 10 centimeters above the track. There are, of course, problems of stability. Also, the suspension is not lossless because there are Joule losses in the track.
Let us consider a simplified form of levitation. A pair of parallel and coaxial coils of radius $R$ and $N$ turns are separated by a distance $D .$ The lower coil simulates the track. For $D \approx 0.1 R$, the mutual inductance is given by $N^{2}\{2.154-12.04[(D / R)-0.1]\} R$ microhenrys.
(a) Calculate the number of ampere-turns required in each coil to support a mass of 1 metric ton when $R=1$ meter.
(b) Draw a sketch showing the two coils and lines of $\boldsymbol{B}$. Can you explain the force of repulsion qualitatively?

Amit Srivastava

Numerade Educator

01:39

Problem 7

Two parallel bus bars have equal circular cross sections and carry equal currents $I$. The currents are equally distributed over the cross sections,
Show, without any calculation, that the force is the same as if the bus bars were thin wires.

Averell Hause

Carnegie Mellon University

09:55

Problem 8

A superconducting dc power transmission line has been proposed that would carry 100 gigawatts of power at 200 kilovolts over 1000 kilometers. The conductors would have a diameter of 25 millimeters and be separated by a center-to-center distance of 50 millimeters.
(a) Calculate the magnetic force per meter. See the previous problem. It is clearly preferable to use a coaxial line.
(b) Calculate the stored energy in kilowatthours. The self-inductance per meter is $\left[\mu_{0} /(4 \pi)\right][1+4 \ln (D / R)]$.

Keshav Singh

Numerade Educator

02:33

Problem 9

Much work has been done on the large-scale storage of energy in inductors, for public utilities. One author proposes a huge, underground, cryogenized inductor that would operate at a field of 14 teslas.
(a) Calculate the energy density in kilowatthours/meter $^{3}$.
(b) Calculate the magnetic pressure in atmospheres.
(c) It seems more reasonable to store energy in a capacitor, because a capacitor need not be cryogenized and because the force points inward, not outward as in an inductor. Calculate the energy density in kilowatthours/ meter $^{3}$ if $\epsilon_{r}=3$ and the dielectric strength is $1.5 \times 10^{8}$ volts/meter.
Gasoline can store over 100 kilowatthours/meter $^{3}$, and flywheels over 200 .

Shoukat Ali

Other Schools

02:28

Problem 10

We showed in Sec. $26.5$ that, if one active circuit moves with respect to another, the mechanical work performed by the sources is equal to the increase in magnetic energy if the currents are maintained constant. Hence the force between two active circuits is given by the rate of increase of magnetic energy.
Show that, similarly, if the geometry of an isolated active circuit changes, the energy supplied by the sources divides in the same way. Assume again that the current is constant. It follows that, on this assumption, the force on an element of an active circuit is equal to the rate of increase of magnetic energy.

Supratim Pal

Numerade Educator

03:21

Problem 11

(a) Show qualitatively, in two different ways, that the turns of a solenoid tend to squeeze together.
(b) Calculate the axial compression force on a long solenoid.

Shahab Ullah

Numerade Educator

01:13

Problem 12

Magnetic fields can perform mechanical tasks that require a high power level for a very short time. For example, magnetic pressure can crush a light aluminum tube that acts as a shutter to turn off a beam of light or of soft $x$-rays. The tube is placed inside a coil, parallel to the axis. When the coil is suddenly connected to a large capacitor, the change in flux induces a large current in the tube, which collapses under the magnetic pressure.
Let us calculate the pressure. If the current $I$ in the solenoid increases gradually from zero to some large value, the induced current is small and the magnetic pressure is negligible. Let us assume that $d I / d t$ in the coil is so large that the induced current in the tube maintains zero magnetic field inside it. Then there is a magnetic field $B$ only in the annular region between the solenoid and the conducting tube.
(a) Calculate the pressure on the tube in atmospheres at 1 tesla.
(b) What would be the pressure if the conducting tube were parallel to the axis but off the axis?

Keshav Singh

Numerade Educator

01:13

Problem 13

Flux compression is one method of obtaining large magnetic fields. For example, one can insert a light conducting tube in the field $B_{0}$ of a solenoid and then implode the tube by means of an annular explosive charge situated between the tube and the solenoid. Currents flow in the tube, and the magnetic pressure builds up until it is equal to the external gas pressure. The solenoid is fed by a constant-current source.
(a) Show that, if the radius of the tube shrinks very rapidly, the $B$ inside is about $B_{0}\left(R_{0}^{2} / R^{2}\right)$ at the instant when the radius is equal to $R$. For example, if $B_{0}$ is 10 teslas and if $R_{0} / R=10$, then $B=10^{3}$ teslas.
(b) Calculate the surface current density in the tube in amperes/meter.
(c) Calculate the change in magnetic energy, the energy absorbed by the constant-current source feeding the solenoid, and the explosive energy required to compress the field. Assume that the tube is 200 millimeters long, $R_{0}=50$ millimeters, and neglect end effects.

Keshav Singh

Numerade Educator

01:05

Problem 14

(a) Show that a current-carrying coil tends to orient itself in a magnetic field in such a way that the total magnetic flux linking the coil is maximum.
(b) Show that the torque on the coil is $\boldsymbol{m} \times \boldsymbol{B}$, where $\boldsymbol{m}$ is the magnetic moment of the coil and $\boldsymbol{B}$ is the magnetic flux density when the current in the coil is zero.

Ankur S

Numerade Educator

01:18

Problem 15

Show that the torque exerted on a small, cylindrical permanent magnet of dipole moment $\boldsymbol{m}$ situated in a magnetic field is $\boldsymbol{m} \times \boldsymbol{B}$. See the preceding problem.

Ankur S

Numerade Educator

14:52

Problem 16

It is possible to separate magnetic particles in suspension in a fluid by passing the mixture through steel wool subjected to a strong magnetic field. The magnetic particles cling to the steel wires where the field gradient is large. Arrays of fine steel wires normal to $\boldsymbol{B}$ are also used.
With a field of the order of several teslas supplied by superconducting coils, the separation occurs even with materials that are only slightly magnetic. The method is also applicable in air for removing magnetic particles, say in pulverized coal.
Let us see how a small magnetic dipole behaves in a nonuniform $\boldsymbol{B}$. The dipole first orients itself. Then, as we shall see, it tends to move in the direction in which the applied $B$ increases.
Figure $26-11$ shows a small current loop of radius $R$ that is already oriented in a field $\boldsymbol{B}$ that increases symmetrically about the positive direction of the $z$-axis.
(a) Show, without any calculation, that the magnetic force points to the right. Note that this force tends to increase the linking flux.
(b) Show that $F=2 \pi R I B_{\rho}$, where $B_{\rho}$ is the component of $\boldsymbol{B}$ that is normal to the $z$-axis.
(c) Now consider a small volume of thickness $\Delta z$, as in the figure. Use the fact that the net outward flux of $\boldsymbol{B}$ is zero to find $B_{\rho}$ and $F$.
(d) Calculate the force from the rate of increase of magnetic energy.

Amit Srivastava

Numerade Educator

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16