The Divergence Operator

2.1.1^*In Cartesian coordinates, A = (A_o /d²)(x²i_x + y² i_y + z² i_z), where A_o and d are constants. Show that div A = 2A_o (x + y + z)/d².

2.1.2^*In Cartesian coordinates, three vector functions are

where A_o, 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.

2.1.3In 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 i_r, note that not only is A_r evaluated at r = r $fraction GIF #11$ r, but so also is r. For this reason, it is most convenient to group A_r and r together in manipulating the contributions from this surface.) floating figure GIF #13

2.1.5The 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 = (A_o/d)(x i_x + y i_y) and a cubical volume having surfaces in the planes x = d, y = d, and z = d. respectively da = i_x dydz, i_y dxdz, and i_z dydx.

(a) Show that the area elements on these surfaces are

(b) Show that evaluation of the left-hand side of (4) gives equation GIF #2.48

(c) Evaluate the divergence of A and the right-hand side of (4) and show that it gives the same result.

2.2.2With A = (A_o/d³)(xy² i_x + x² y i_y), 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). law in Cartesian coordinates to show that the charge density is indeed zero everywhere except along the z axis.

(a) Use Gauss' differential

(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.

2.3.5Use 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 - $fraction GIF #12$ s < z < 1\over 2 s.

2.3.7For 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.

2.4.2Vector 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 $fraction GIF #13$ r, respectively, but so also is r. It is therefore convenient to treat A r as a single function.) 2.4.4In 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.

2.4.5The following is an identity.

This can be shown in two ways.

arbitrary but closed surface S (one having no edge, so C = 0) and then Gauss' theorem to argue the identity.

(a) Apply Stokes' theorem to an

(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 = (A_o /d²) x² i_y 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 = i_z dxdy.

)² - g²]d².

(a) Show that the left-hand side of (4) is h A_o [(g +

(b) Verify (4) by obtaining the same result integrating curl A over the area enclosed by C.

2.5.2For the vector function A = (A_o /d)(-i_x y + i_y 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.
and Curl 2.7.1Using the conventions exemplified in Fig. 2.7.3, electric field intensity E for Prob. 1.3.5 and with E_o = 0 and _o = 0.

(a) Sketch the distributions of charge density and

(b) Verify that E is irrotational.

(c) From observation of the field sketch, why would you suspect that E is indeed irrotational?

2.7.2Using 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?

2.7.3Three two-dimensional vector fields are shown in Fig.~P2.7.3.

(a) Which of these is irrotational?

(b) Which are solenoidal?

2.7.4For 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.
2.7.5For 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 = K_o > 0 and represent the intensity of H by the spacing of the field lines.
2.7.6Field 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 = K_z i_z on the surface. Is K_z positive or negative?

\globallineheight13pt\globallinedepth12pt \boxtable \verticallines

(a)	Show that the area elements on these surfaces are
(b)	Show that evaluation of the left-hand side of (4) gives
(c)	Evaluate the divergence of A and the right-hand side of (4) and show that it gives the same result.

(a)	Use Gauss' differential
(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.

(a)	Apply Stokes' theorem to an
(b)	Write out the the divergence of the curl in Cartesian coordinates and show that it is indeed identically zero.

(a)	Show that the left-hand side of (4) is h A_o [(g +
(b)	Verify (4) by obtaining the same result integrating curl A over the area enclosed by C.

(a)	Sketch the distributions of charge density and
(b)	Verify that E is irrotational.
(c)	From observation of the field sketch, why would you suspect that E is indeed irrotational?

(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?

(a)	Which of these is irrotational?
(b)	Which are solenoidal?

(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 = K_z i_z on the surface. Is K_z positive or negative?