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Engineering Electromagnetics

William H. Hayt, Jr. John A. Buck

Chapter 7

The Steady Magnetic Field - all with Video Answers

Educators

Chapter Questions

01:40

Problem 1

( $a$ ) Find $\mathbf{H}$ in rectangular components at $P(2,3,4)$ if there is a current filament on the $z$ axis carrying $8 \mathrm{~mA}$ in the $\mathbf{a}_{z}$ direction. ( $b$ ) Repeat if the filament is located at $x=-1, y=2$. ( $c$ ) Find $\mathbf{H}$ if both filaments are present.

Penny Riley
Penny Riley
Numerade Educator
05:00

Problem 2

A filamentary conductor is formed into an equilateral triangle with sides of length $\ell$ carrying current $I$. Find the magnetic field intensity at the center of the triangle.

Supratim Pal
Supratim Pal
Numerade Educator
04:46

Problem 3

Two semi-infinite filaments on the $z$ axis lie in the regions $-\infty<z<-a$ and $a<z<\infty .$ Each carries a current $I$ in the $\mathbf{a}_{z}$ direction. $(a)$ Calculate $\mathbf{H}$ as a function of $\rho$ and $\phi$ at $z=0 .(b)$ What value of $a$ will cause the magnitude of $\mathbf{H}$ at $\rho=1, z=0$, to be one-half the value obtained for an infinite filament?

James Kiss
James Kiss
Numerade Educator
13:29

Problem 4

Two circular current loops are centered on the $z$ axis at $z=\pm h .$ Each loop has radius $a$ and carries current $I$ in the $\mathbf{a}_{\phi}$ direction. $(a)$ Find $\mathbf{H}$ on the $z$ axis over the range $-h<z<h .$ Take $I=1 \mathrm{~A}$ and plot $|\mathbf{H}|$ as a function of $z / a$ if (b) $h=a / 4 ;$ (c) $h=a / 2 ;(d) h=a$. Which choice for $h$ gives the most uniform field? These are called Helmholtz coils (of a single turn each in this case), and are used in providing uniform fields.

Sam Stansfield
Sam Stansfield
Numerade Educator
01:59

Problem 5

The parallel filamentary conductors shown in Figure $7.21$ lie in free space. Plot $|\mathbf{H}|$ versus $y,-4<y<4$, along the line $x=0, z=2$.

Adnan Gill
Adnan Gill
Numerade Educator
02:04

Problem 6

A disk of radius $a$ lies in the $x y$ plane, with the $z$ axis through its center. Surface charge of uniform density $\rho_{s}$ lies on the disk, which rotates about the $z$ axis at angular velocity $\Omega \mathrm{rad} / \mathrm{s}$. Find $\mathbf{H}$ at any point on the $z$ axis.

Manish Jain
Manish Jain
Numerade Educator
01:40

Problem 7

A filamentary conductor carrying current $I$ in the $\mathbf{a}_{z}$ direction extends along the entire negative $z$ axis. At $z=0$ it connects to a copper sheet that fills the $x>0, y>0$ quadrant of the $x y$ plane. $(a)$ Set up the Biot-Savart law and find $\mathrm{H}$ everywhere on the $z$ axis; $(b)$ repeat part $(a)$, but with the copper sheet occupying the entire $x y$ plane (Hint: express $\mathbf{a}_{\phi}$ in terms of $\mathbf{a}_{x}$ and $\mathbf{a}_{y}$ and angle $\phi$ in the integral).

Penny Riley
Penny Riley
Numerade Educator
03:27

Problem 8

For the finite-length current element on the $z$ axis, as shown in Figure $7.5$, use the Biot-Savart law to derive Eq. (9) of Section 7.1.

Mayukh Banik
Mayukh Banik
Numerade Educator
03:22

Problem 9

A current sheet $\mathbf{K}=8 \mathbf{a}_{x} \mathrm{~A} / \mathrm{m}$ flows in the region $-2<y<2$ in the plane $z=0$. Calculate $H$ at $P(0,0,3)$.

Narayan Hari
Narayan Hari
Numerade Educator
03:47

Problem 10

A hollow spherical conducting shell of radius $a$ has filamentary connections made at the top $(r=a, \theta=0)$ and bottom $(r=a, \theta=\pi) .$ A direct current $I$ flows down the upper filament, down the spherical surface, and out the lower filament. Find $\mathrm{H}$ in spherical coordinates $(a)$ inside and $(b)$ outside the sphere.

Narayan Hari
Narayan Hari
Numerade Educator
01:40

Problem 11

An infinite filament on the $z$ axis carries $20 \pi \mathrm{mA}$ in the $\mathbf{a}_{z}$ direction. Three
$\mathbf{a}_{z}$ -directed uniform cylindrical current sheets are also present: $400 \mathrm{~mA} / \mathrm{m}$ at $\rho=1 \mathrm{~cm},-250 \mathrm{~mA} / \mathrm{m}$ at $\rho=2 \mathrm{~cm}$, and $-300 \mathrm{~mA} / \mathrm{m}$ at $\rho=3 \mathrm{~cm}$. Calculate
$H_{\phi}$ at $\rho=0.5,1.5,2.5$, and $3.5 \mathrm{~cm}$.

Penny Riley
Penny Riley
Numerade Educator
04:47

Problem 12

In Figure $7.22$, let the regions $0<z<0.3 \mathrm{~m}$ and $0.7<z<1.0 \mathrm{~m}$ be conducting slabs carrying uniform current densities of $10 \mathrm{~A} / \mathrm{m}^{2}$ in opposite directions as shown. Find $\mathbf{H}$ at $z=:(a)-0.2 ;(b) 0.2 ;(c) 0.4 ;(d) 0.75$;
(e) $1.2 \mathrm{~m}$.

Linda Winkler
Linda Winkler
Numerade Educator
07:41

Problem 13

A hollow cylindrical shell of radius $a$ is centered on the $z$ axis and carries a uniform surface current density of $K_{a} \mathbf{a}_{\phi} .(a)$ Show that $H$ is not a function of $\phi$ or $z .(b)$ Show that $H_{\phi}$ and $H_{\rho}$ are everywhere zero. $(c)$ Show that $H_{z}=0$ for $\rho>a .(d)$ Show that $H_{z}=K_{a}$ for $\rho<a .(e) \mathrm{A}$ second shell, $\rho=b$, carries a current $K_{b} \mathbf{a}_{\phi} .$ Find $\mathbf{H}$ everywhere.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:14

Problem 14

A toroid having a cross section of rectangular shape is defined by the following surfaces: the cylinders $\rho=2$ and $\rho=3 \mathrm{~cm}$, and the planes $z=1$ and $z=2.5 \mathrm{~cm}$. The toroid carries a surface current density of $-50 \mathbf{a}_{z} \mathrm{~A} / \mathrm{m}$ on the surface $\rho=3 \mathrm{~cm}$. Find $\mathbf{H}$ at the point $P(\rho, \phi, z):(a) P_{A}(1.5 \mathrm{~cm}, 0$, $2 \mathrm{~cm}) ;\left(\right.$ b) $P_{B}(2.1 \mathrm{~cm}, 0,2 \mathrm{~cm}) ;$ (c) $P_{C}(2.7 \mathrm{~cm}, \pi / 2,2 \mathrm{~cm}) ;$ (d) $P_{D}(3.5 \mathrm{~cm},$,
$\pi / 2,2 \mathrm{~cm})$

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
03:04

Problem 15

Assume that there is a region with cylindrical symmetry in which the conductivity is given by $\sigma=1.5 e^{-150 \rho} \mathrm{kS} / \mathrm{m}$. An electric field of $30 \mathbf{a}_{z} \mathrm{~V} / \mathrm{m}$
is present. ( $a$ ) Find $\mathbf{J}$. $(b)$ Find the total current crossing the surface $\rho<\rho_{0}$, $z=0$, all $\phi$. ( $c$ ) Make use of Ampère's circuital law to find $\mathbf{H}$.

Keshav Singh
Keshav Singh
Numerade Educator
01:40

Problem 16

A current filament carrying $I$ in the $-\mathbf{a}_{z}$ direction lies along the entire positive $z$ axis. At the origin, it connects to a conducting sheet that forms the $x y$ plane. (a) Find $\mathbf{K}$ in the conducting sheet. $(b)$ Use Ampere's circuital law to find $\mathbf{H}$ everywhere for $z>0 ;(c)$ find $\mathbf{H}$ for $z<0$.

Penny Riley
Penny Riley
Numerade Educator
01:40

Problem 17

A current filament on the $z$ axis carries a current of $7 \mathrm{~mA}$ in the $\mathbf{a}_{z}$ direction, and current sheets of $0.5 \mathrm{a}_{z} \mathrm{~A} / \mathrm{m}$ and $-0.2 \mathrm{a}_{z} \mathrm{~A} / \mathrm{m}$ are located at $\rho=1 \mathrm{~cm}$
and $\rho=0.5 \mathrm{~cm}$, respectively. Calculate $\mathbf{H}$ at: $($ a $) \rho=0.5 \mathrm{~cm} ;(b) \rho=$ $1.5 \mathrm{~cm} ;(c) \rho=4 \mathrm{~cm} .(d)$ What current sheet should be located at $\rho=4 \mathrm{~cm}$ so that $\mathbf{H}=0$ for all $\rho>4 \mathrm{~cm}$ ?

Penny Riley
Penny Riley
Numerade Educator
02:41

Problem 18

A wire of $3 \mathrm{~mm}$ radius is made up of an inner material $(0<\rho<2 \mathrm{~mm})$ for which $\sigma=10^{7} \mathrm{~S} / \mathrm{m}$, and an outer material ( $2 \mathrm{~mm}<\rho<3 \mathrm{~mm}$ ) for which $\sigma=4 \times 10^{7} \mathrm{~S} / \mathrm{m}$. If the wire carries a total current of $100 \mathrm{~mA}$ dc, determine $\mathbf{H}$ everywhere as a function of $\rho$.

Sunita Kumari
Sunita Kumari
Numerade Educator
03:29

Problem 19

In spherical coordinates, the surface of a solid conducting cone is described by $\theta=\pi / 4$ and a conducting plane by $\theta=\pi / 2 .$ Each carries a total current
I. The current flows as a surface current radially inward on the plane to the vertex of the cone, and then flows radially outward throughout the cross section of the conical conductor. $(a)$ Express the surface current density as a function of $r ;(b)$ express the volume current density inside the cone as a function of $r ;(c)$ determine $\mathbf{H}$ as a function of $r$ and $\theta$ in the region between the cone and the plane; $(d)$ determine $\mathbf{H}$ as a function of $r$ and $\theta$ inside the cone.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:07

Problem 20

A solid conductor of circular cross section with a radius of $5 \mathrm{~mm}$ has a conductivity that varies with radius. The conductor is $20 \mathrm{~m}$ long, and there is a potential difference of $0.1 \mathrm{~V}$ dc between its two ends. Within the conductor, $\mathbf{H}=10^{5} \rho^{2} \mathbf{a}_{\phi} \mathrm{A} / \mathrm{m} .(a)$ Find $\sigma$ as a function of $\rho .(b)$ What is the resistance between the two ends?

Ajay Singhal
Ajay Singhal
Numerade Educator
01:14

Problem 21

A cylindrical wire of radius $a$ is oriented with the $z$ axis down its center line. The wire carries a nonuniform current down its length of density $\mathbf{J}=b \rho \mathbf{a}_{z} \mathrm{~A} / \mathrm{m}^{2}$ where $b$ is a constant. ( $a$ ) What total current flows in the wire? $(b)$ Find $\mathbf{H}_{i n}(0<\rho<a)$, as a function of $\rho ;(c)$ find $\mathbf{H}_{\text {out }}(\rho>a)$, as a function of $\rho ;(d)$ verify your results of parts $(b)$ and $(c)$ by using $\nabla \times \mathbf{H}=\mathbf{J}$.

Dominador Tan
Dominador Tan
Numerade Educator
17:01

Problem 22

A solid cylinder of radius $a$ and length $L$, where $L \gg a$, contains volume charge of uniform density $\rho_{0} \mathrm{C} / \mathrm{m}^{3}$. The cylinder rotates about its axis (the $z$ axis) at angular velocity $\Omega \mathrm{rad} / \mathrm{s}$. (a) Determine the current density $\mathbf{J}$ as a function of position within the rotating cylinder. (b) Determine $\mathbf{H}$ on-axis by applying the results of Problem 7.6. ( $c$ ) Determine the magnetic field intensity $\mathbf{H}$ inside and outside. $(d)$ Check your result of part ( $c$ ) by taking the curl of $\mathbf{H}$.

Linda Winkler
Linda Winkler
Numerade Educator
04:53

Problem 23

Given the field $\mathbf{H}=20 \rho^{2} \mathbf{a}_{\phi} \mathrm{A} / \mathrm{m}:(a)$ Determine the current density $\mathbf{J}$.
(b) Integrate $\mathbf{J}$ over the circular surface $\rho \leq 1,0<\phi<2 \pi, z=0$, to determine the total current passing through that surface in the $\mathbf{a}_{z}$ direction.
(c) Find the total current once more, this time by a line integral around the circular path $\rho=1,0<\phi<2 \pi, z=0 .$

Ahmed Kamel
Ahmed Kamel
Numerade Educator
04:04

Problem 24

Infinitely long filamentary conductors are located in the $y=0$ plane at $x=n$ meters where $n=0, \pm 1, \pm 2, \ldots$ Each carries $1 \mathrm{~A}$ in the $\mathbf{a}_{z}$ direction. (a) Find $\mathbf{H}$ on the $y$ axis. As a help,
$$\sum_{n=1}^{\infty} \frac{y}{y^{2}+n^{2}}=\frac{\pi}{2}-\frac{1}{2 y}+\frac{\pi}{e^{2 \pi y}-1}$$
(b) Compare your result of part $(a)$ to that obtained if the filaments are replaced by a current sheet in the $y=0$ plane that carries surface current density $\mathbf{K}=1 \mathbf{a}_{z} \mathrm{~A} / \mathrm{m}$.

Aja S
Aja S
Numerade Educator
03:31

Problem 25

When $x, y$, and $z$ are positive and less than 5 , a certain magnetic field intensity may be expressed as $\mathbf{H}=\left[x^{2} y z /(y+1)\right] \mathbf{a}_{x}+3 x^{2} z^{2} \mathbf{a}_{y}-$
$\left[x y z^{2} /(y+1)\right] \mathbf{a}_{z} .$ Find the total current in the $\mathbf{a}_{x}$ direction that crosses the strip $x=2,1 \leq y \leq 4,3 \leq z \leq 4$, by a method utilizing: $(a)$ a surface integral; $(b)$ a closed line integral.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:01

Problem 26

Consider a sphere of radius $r=4$ centered at $(0,0,3)$. Let $S_{1}$ be that portion of the spherical surface that lies above the $x y$ plane. Find $\int_{S_{1}}(\nabla \times \mathbf{H}) \cdot d \mathbf{S}$ if $\mathbf{H}=3 \rho \mathbf{a}_{\phi}$ in cylindrical coordinates.

Raj Bala
Raj Bala
Numerade Educator
03:44

Problem 27

The magnetic field intensity is given in a certain region of space as $\mathbf{H}=$ $\left[(x+2 y) / z^{2}\right] \mathbf{a}_{y}+(2 / z) \mathbf{a}_{z} \mathrm{~A} / \mathrm{m} .(a)$ Find $\nabla \times \mathbf{H} .(b)$ Find $\mathbf{J} .(c)$ Use $\mathbf{J}$ to find
the total current passing through the surface $z=4,1 \leq x \leq 2,3 \leq z \leq 5$, in the $\mathbf{a}_{z}$ direction. ( $d$ ) Show that the same result is obtained using the other side of Stokes' theorem.

Laszlo Zalavari
Laszlo Zalavari
Numerade Educator
03:57

Problem 28

Given $\mathbf{H}=\left(3 r^{2} / \sin \theta\right) \mathbf{a}_{\theta}+54 r \cos \theta \mathbf{a}_{\phi} \mathrm{A} / \mathrm{m}$ in free space: $(a)$ Find the total
current in the $\mathbf{a}_{\theta}$ direction through the conical surface $\theta=20^{\circ}, 0 \leq \phi \leq 2 \pi$, $0 \leq r \leq 5$, by whatever side of Stokes' theorem you like the best. $(b)$ Check the result by using the other side of Stokes' theorem.

Vikash Ranjan
Vikash Ranjan
Numerade Educator
02:34

Problem 29

A long, straight, nonmagnetic conductor of $0.2 \mathrm{~mm}$ radius carries a uniformly distributed current of 2 A dc. $(a)$ Find $J$ within the conductor.
(b) Use Ampère's circuital law to find $\mathbf{H}$ and $\mathbf{B}$ within the conductor.
(c) Show that $\nabla \times \mathbf{H}=\mathbf{J}$ within the conductor. $(d)$ Find $\mathbf{H}$ and $\mathbf{B}$ outside the conductor. $(e)$ Show that $\nabla \times \mathbf{H}=\mathbf{J}$ outside the conductor.

- -
- -
Numerade Educator
01:07

Problem 30

(An inversion of Problem 7.20.) A solid, nonmagnetic conductor of circular cross section has a radius of $2 \mathrm{~mm}$. The conductor is inhomogeneous, with $\sigma=10^{6}\left(1+10^{6} \mathrm{\rho}^{2}\right) \mathrm{S} / \mathrm{m}$. If the conductor is $1 \mathrm{~m}$ in length and has a voltage of $1 \mathrm{mV}$ between its ends, find: $(a) \mathbf{H}$ inside; $(b)$ the total magnetic flux inside the conductor.

Narayan Hari
Narayan Hari
Numerade Educator
10:15

Problem 31

The cylindrical shell defined by $1 \mathrm{~cm}<\rho<1.4 \mathrm{~cm}$ consists of a nonmagnetic conducting material and carries a total current of $50 \mathrm{~A}$ in the $\mathbf{a}_{-}$ direction. Find the total magnetic flux crossing the plane $\phi=0,0<z<1$ :
(a) $0<\rho<1.2 \mathrm{~cm}$;
(b) $1.0 \mathrm{~cm}<\rho<1.4 \mathrm{~cm} ;$
(c) $1.4 \mathrm{~cm}<\rho<20 \mathrm{~cm}$.

Vishal Gupta
Vishal Gupta
Numerade Educator
01:31

Problem 32

The free space region defined by $1<z<4 \mathrm{~cm}$ and $2<\rho<3 \mathrm{~cm}$ is a toroid of rectangular cross section. Let the surface at $\rho=3 \mathrm{~cm}$ carry a surface current $\mathbf{K}=2 \mathbf{a}_{z} \mathrm{kA} / \mathrm{m} .(a)$ Specify the current densities on the surfaces at $\rho=2 \mathrm{~cm}, z=1 \mathrm{~cm}$, and $z=4 \mathrm{~cm} .(b)$ Find $\mathrm{H}$ everywhere. $(c)$ Calculate the total flux within the toroid.

Narayan Hari
Narayan Hari
Numerade Educator
07:23

Problem 33

Use an expansion in rectangular coordinates to show that the curl of the gradient of any scalar field $G$ is identically equal to zero.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:40

Problem 34

A filamentary conductor on the $z$ axis carries a current of $16 \mathrm{~A}$ in the $\mathbf{a}_{z}$ direction, a conducting shell at $\rho=6$ carries a total current of $12 \mathrm{~A}$ in the $-\mathbf{a}_{z}$ direction, and another shell at $\rho=10$ carries a total current of $4 \mathrm{~A}$ in the $-\mathbf{a}_{z}$ direction. $(a)$ Find $\mathbf{H}$ for $0<\rho<12 .\left(\right.$ b) Plot $H_{\phi}$ versus $\rho$.
(c) Find the total flux $\Phi$ crossing the surface $1<\rho<7,0<z<1$, at fixed
$\phi .$

Penny Riley
Penny Riley
Numerade Educator
03:31

Problem 35

A current sheet, $\mathbf{K}=20 \mathbf{a}_{z} \mathrm{~A} / \mathrm{m}$, is located at $\rho=2$, and a second sheet, $\mathbf{K}=-10 \mathbf{a}_{z} \mathrm{~A} / \mathrm{m}$, is located at $\rho=4 .(a)$ Let $V_{m}=0$ at $P(\rho=3, \phi=0,$,
$z=5$ ) and place a barrier at $\phi=\pi$. Find $V_{m}(\rho, \phi, z)$ for $-\pi<\phi<\pi$
$(b)$ Let $\mathbf{A}=0$ at $P$ and find $\mathbf{A}(\rho, \phi, z)$ for $2<\rho<4$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:33

Problem 36

Let $\mathbf{A}=(3 y-z) \mathbf{a}_{x}+2 x z \mathbf{a}_{y} \mathrm{~Wb} / \mathrm{m}$ in a certain region of free space.
(a) Show that $\nabla \cdot \mathbf{A}=0 .(b)$ At $P(2,-1,3)$, find $\mathbf{A}, \mathbf{B}, \mathbf{H}$, and $\mathbf{J}$.

Harshita Goel
Harshita Goel
Numerade Educator
02:31

Problem 37

Let $N=1000, I=0.8 \mathrm{~A}, \rho_{0}=2 \mathrm{~cm}$, and $a=0.8 \mathrm{~cm}$ for the toroid shown in Figure $7.12 b$. Find $V_{m}$ in the interior of the toroid if $V_{m}=0$ at $\rho=2.5$ $\mathrm{cm}, \phi=0.3 \pi$. Keep $\phi$ within the range $0<\phi<2 \pi$.

Penny Riley
Penny Riley
Numerade Educator
01:01

Problem 38

A square filamentary differential current loop, $d L$ on a side, is centered at the origin in the $z=0$ plane in free space. The current $I$ flows generally in the $\mathbf{a}_{\phi}$ direction. ( $a$ ) Assuming that $r>>d L$, and following a method similar to that in Section $4.7$, show that
$$d \mathbf{A}=\frac{\mu_{0} I(d L)^{2} \sin \theta}{4 \pi r^{2}} \mathbf{a}_{\phi}$$
(b) Show that
$$d \mathbf{H}=\frac{I(d L)^{2}}{4 \pi r^{3}}\left(2 \cos \theta \mathbf{a}_{r}+\sin \theta \mathbf{a}_{\theta}\right)$$
The square loop is one form of a magnetic dipole.

Narayan Hari
Narayan Hari
Numerade Educator
03:44

Problem 39

Planar current sheets of $\mathbf{K}=30 \mathbf{a}_{z} \mathrm{~A} / \mathrm{m}$ and $-30 \mathbf{a}_{z} \mathrm{~A} / \mathrm{m}$ are located in free space at $x=0.2$ and $x=-0.2$, respectively. For the region $-0.2<x<0.2$
(a) find $\mathbf{H} ;(b)$ obtain an expression for $V_{m}$ if $V_{m}=0$ at $P(0.1,0.2,0.3)$;
(c) find $\mathbf{B} ;(d)$ obtain an expression for $\mathbf{A}$ if $\mathbf{A}=0$ at $P$.

Keshav Singh
Keshav Singh
Numerade Educator
07:18

Problem 40

Show that the line integral of the vector potential A about any closed path is equal to the magnetic flux enclosed by the path, or $\oint \mathbf{A} \cdot d \mathbf{L}=\int \mathbf{B} \cdot d \mathbf{S}$.

Vishal Gupta
Vishal Gupta
Numerade Educator
05:59

Problem 41

Assume that $\mathbf{A}=50 \rho^{2} \mathbf{a}_{z} \mathrm{~Wb} / \mathrm{m}$ in a certain region of free space. $(a)$ Find $\mathbf{H}$ and $\mathbf{B}$. $(b)$ Find $\mathbf{J} .(c)$ Use $\mathbf{J}$ to find the total current crossing the surface $0 \leq \rho \leq 1,0 \leq \phi<2 \pi, z=0 .(d)$ Use the value of $H_{\phi}$ at $\rho=1$ to calculate $\oint \mathbf{H} \cdot d \mathbf{L}$ for $\rho=1, z=0$

Brandy Heflin
Brandy Heflin
Numerade Educator
03:30

Problem 42

Show that $\nabla_{2}\left(1 / R_{12}\right)=-\nabla_{1}\left(1 / R_{12}\right)=\mathbf{R}_{21} / R_{12}^{3}$.

Alex Roush
Alex Roush
Numerade Educator
02:54

Problem 43

Compute the vector magnetic potential within the outer conductor for the coaxial line whose vector magnetic potential is shown in Figure $7.20$ if the outer radius of the outer conductor is $7 a$. Select the proper zero reference and sketch the results on the figure.

Dominador Tan
Dominador Tan
Numerade Educator
00:29

Problem 44

By expanding Eq. (58), Section $7.7$ in rectangular coordinates, show that (59) is correct.

Naomi Motes
Naomi Motes
Numerade Educator