| | The Divergence Operator |
|---|
| 2.1.1* | In Cartesian coordinates, A = (Ao /d2)(x2ix + y2 iy +
z2 iz), where Ao and d are constants. Show that div A
= 2Ao (x + y + z)/d2.
|
| 2.1.2* | In Cartesian coordinates, three vector functions are
where Ao, k, and d are constants.
| (a) | Show that the divergence of each is zero.
|
| (b) | Devise three vector functions that have a finite
divergence and evaluate their divergences.
|
|
| 2.1.3 | In cylindrical coordinates, the divergence operator is given in
Table I at the end of the text. Evaluate the divergence of the
following vector functions.
|
| 2.1.4* | In cylindrical coordinates, unit vectors are as defined in
Fig.~P2.1.4a. An incremental volume element having sides
( r, r , z)
is as shown in Fig.~P2.1.4b. Determine the divergence operator by
evaluating (2), using steps analogous to those leading from (3)
to (5). Show that the result is as given in Table I at the end of
the text. (Hint: In carrying out the integrations over the surface
elements in Fig.~P2.1.4b having normals  ir, note that not
only is Ar evaluated at r = r  r, but so also is
r. For this
reason, it is most convenient to group Ar and r together in
manipulating the contributions from this surface.)
|
| 2.1.5 | The divergence operator is given in spherical coordinates in Table I
at the end of the text. Use that operator to evaluate the divergence
of the following vector functions.
|
| 2.1.6* | In spherical coordinates, an incremental volume element has sides
r, r\Delta , r sin \Delta. Using
steps analogous to those
leading from (3) to (5), determine the divergence operator by evaluating
(2.1.2). Show that the result is as given in Table I at the end of
the text.
|
| | Gauss' Integral Theorem |
|---|
| 2.2.1* | Given a well-behaved vector function A, Gauss' theorem shows
that the same result will be obtained by integrating its divergence
over a volume V or by integrating its normal component over the
surface S that encloses that volume. The following steps exemplify
this fact. Consider the particular vector function A = (Ao/d)(x
ix + y iy) and a cubical volume having surfaces in the planes
x = d, y = d, and z = d.
|
| 2.2.2 | With A = (Ao/d3)(xy2 ix + x2 y iy), carry out the
steps in Prob. 2.2.1.
|
| | Continuity, and Charge Conservation
|
|---|
| 2.3.1* | For a line charge along the z axis of Prob. 1.3.1, E was
written in Cartesian coordinates as (a).
| (a) | Use Gauss' differential law in Cartesian coordinates to show that the charge density is indeed
zero everywhere except along the z axis.
|
| (b) | Obtain the same result by
evaluating Gauss' law using E as given by (1.3.13) and the
divergence operator from Table I in cylindrical coordinates.
|
|
| 2.3.2* | Show that at each point r < a, E and  as given respectively
by (b) and (a) of Prob. 1.3.3 are consistent with Gauss'
differential law.
|
| 2.3.3* | For the flux linkage  f to be independent of S, (2) must
hold. Return to Prob. 1.6.6 and check to see that this condition was
indeed satisfied by the magnetic flux density.
|
| 2.3.4* | Using H expressed in cylindrical coordinates
by (1.4.10), show that the magnetic flux density of a line current
is indeed solenoidal (has no divergence) everywhere except at r = 0.
|
| 2.3.5 | Use the differential law of magnetic flux continuity, (2), to
answer Prob. 1.7.2.
|
| 2.3.6* | In Prob. 1.3.5, E and are found for a one-dimensional
configuration using the integral charge conservation law. Show that
the differential form of this law is satisfied at each position
- s < z < s.
|
| 2.3.7 | For J and as found in Prob. 1.5.1, show that the
differential form of charge conservation, (3), is satisfied.
|
| | The Curl Operator |
|---|
| 2.4.1* | Show that the curls of the three vector functions given in Prob.
2.1.2 are zero. Devise three such functions that have finite curls
(are rotational) and give their curls.
|
| 2.4.2 | Vector functions are given in cylindrical coordinates in Prob.
2.1.3. Using the curl operator as given in cylindrical coordinates by
Table I at the end of the text, show that all of these functions are
irrotational. Devise three functions that are rotational and give
their curls.
|
| 2.4.3* | In cylindrical coordinates, define incremental surface elements
having normals in the r, and z directions, respectively, as
shown in Fig.~P2.4.3. Determine the r, , and z components
of the curl operator. Show that the result is as given in Table I at
the end of the text. (Hint: In integrating in the
directions on the outer and inner incremental contours of
Fig.~P2.4.3c, note that not only is A evaluated at r = r
r, respectively, but so also is r. It is
therefore convenient to treat A r as a single function.)
|
| 2.4.4 | In spherical coordinates, incremental surface elements have normals
in the r, , and directions, respectively, as described in
Appendix 1. Determine the r, , and components of the
curl operator and compare to the result given in Table I at the end of
the text.
|
| 2.4.5 | The following is an identity.
This can be shown in two ways.
| (a) | Apply Stokes' theorem to an arbitrary but closed surface S (one having no edge, so C = 0) and
then Gauss' theorem to argue the identity.
|
| (b) | Write out the the divergence of the curl in Cartesian
coordinates and show that it is indeed identically zero.
|
|
| | Stokes' Integral Theorem |
|---|
| 2.5.1* | To exemplify Stokes' integral theorem, consider the evaluation of
(4) for the vector function A = (Ao /d2) x2 iy
and a rectangular contour consisting of the segments at x = g +
, y = h, x = g, and y = 0. The direction of the
contour is such that da = iz dxdy.
| (a) | Show that the left-hand side of (4) is h Ao [(g + )2 - g2]d2.
|
| (b) | Verify (4) by obtaining the same
result integrating curl A over the area enclosed by C.
|
|
| 2.5.2 | For the vector function A = (Ao /d)(-ix y + iy x),
evaluate the contour and surface integrals of (4) on C and S as
prescribed in Prob. 2.5.1 and show that they are equal.
|
| | Differential Laws of Ampère and Faraday |
|---|
| 2.6.1* | In Prob. 1.4.2, H is given in Cartesian coordinates by (c).
With   o E / t = 0, show that Ampère's differential
law is satisfied at each point r < a.
|
| 2.6.2* | For the H and J given in Prob. 1.4.1, show that Ampère's
differential law, (2), is satisfied with o E / t = 0.
|
| | Visualization of fields and the Divergence and Curl
|
|---|
| 2.7.1 | Using the conventions exemplified in Fig. 2.7.3,
| (a) | Sketch the distributions of charge density and
electric field intensity E for Prob. 1.3.5 and with Eo = 0 and
 o = 0.
|
| (b) | Verify that E is irrotational.
|
| (c) | From observation of the field sketch, why would you
suspect that E is indeed irrotational?
|
|
| 2.7.2 | Using Fig. 2.7.4 as a model, sketch J and H
| (a) | For Prob. 1.4.1.
|
| (b) | For Prob. 1.4.4.
|
| (c) | Verify that in each case, H is solenoidal.
|
| (d) | From observation of these field sketches, why would
you suspect that H is indeed solenoidal?
|
|
| 2.7.3 | Three two-dimensional vector fields are shown in Fig.~P2.7.3.
| (a) | Which of these is irrotational?
|
| (b) | Which are solenoidal?
|
|
| 2.7.4 | For the fields of Prob. 1.6.7, sketch E just above and just
below the plane y = 0 and s in the surface y = 0.
Assume that E1 = E2 = o /o > 0 and adhere to the
convention that the field intensity is represented by the spacing of
the field lines.
|
| 2.7.5 | For the fields of Prob. 1.7.3, sketch H just above and just
below the plane y = 0 and K in the surface y = 0. Assume that
H1 = H2 = Ko > 0 and represent the intensity of H by the
spacing of the field lines.
|
| 2.7.6 | Field lines in the vicinity of the surface y = 0 are shown in
Fig.~P2.7.6.
charge density s on the surface. Is s positive or
negative?
| (a) | If the field lines represent E, there is a surface
|
| (b) | If the field lines represent H, there is a surface
current density K = Kz iz on the surface. Is Kz
positive or negative?
|
|
