The Divergence Operator | |||||||||
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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.
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2.1.2* | In Cartesian coordinates, three vector functions are
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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.
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2.1.4* | In cylindrical coordinates, unit vectors are as defined in
Fig.~P2.1.4a. An incremental volume element having sides
(![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||||||
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.
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2.1.6* | In spherical coordinates, an incremental volume element has sides
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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 = ![]() ![]() ![]()
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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).
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2.3.2* | Show that at each point r < a, E and ![]()
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2.3.3* | For the flux linkage ![]()
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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.
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2.3.5 | Use the differential law of magnetic flux continuity, (2), to
answer Prob. 1.7.2.
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2.3.6* | In Prob. 1.3.5, E and ![]() ![]() ![]()
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2.3.7 | For J and ![]()
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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.
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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.
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2.4.3* | In cylindrical coordinates, define incremental surface elements
having normals in the r, ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||||||||
2.4.4 | In spherical coordinates, incremental surface elements have normals
in the r, ![]() ![]() ![]() ![]()
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2.4.5 | The following is an identity.
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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 +
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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.
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Differential Laws of Ampère and Faraday | |||||||||
2.6.1* | In Prob. 1.4.2, H is given in Cartesian coordinates by (c).
With ![]() ![]() ![]()
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2.6.2* | For the H and J given in Prob. 1.4.1, show that Ampère's
differential law, (2), is satisfied with ![]() ![]() ![]()
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Visualization of fields and the Divergence and Curl | |||||||||
2.7.1 | Using the conventions exemplified in Fig. 2.7.3,
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2.7.2 | Using Fig. 2.7.4 as a model, sketch J and H
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2.7.3 | Three two-dimensional vector fields are shown in Fig.~P2.7.3.
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2.7.4 | For the fields of Prob. 1.6.7, sketch E just above and just
below the plane y = 0 and ![]() ![]() ![]()
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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.
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2.7.6 | Field lines in the vicinity of the surface y = 0 are shown in
Fig.~P2.7.6.
charge density ![]() ![]()
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