Conduction Constitutive Laws

7.1.1

In a metal such as copper, where each atom contributes approximately one conduction electron, typical current densities are the result of electrons moving at a surprisingly low velocity. To estimate this velocity, assume that each atom contributes one conduction electron and that the material is copper, where the molecular weight M_o = 63.5 and the mass density is = 8.9 x 10³ kg/m³. Thus, the density of electrons is approximately (A_o/M_o), where A_o = 6.023 x 10²⁶ molecules/kg-mole is Avogadro's number. Given from Table 7.1.1, what is the mobility of the electrons in copper? What electric field intensity is required to drive a current density of l amp/cm²? What is the electron velocity?

Steady Ohmic Conduction

7.2.1^*

The circular disk of uniformly conducting material shown in Fig. P7.2.1 has a dc voltage v applied to its surfaces at r = a and r = b by means of perfectly conducting electrodes. The other boundaries are interfaces with free space. Show that the resistance R = ln(a/b)/2 d.

Figure P7.2.1

7.2.2

In a spherical version of the resistor shown in Fig. P7.2.1, a uniformly conducting material is connected to a voltage source v through spherical perfectly conducting electrodes at r = a and r = b. What is the resistance?

7.2.3^*

By replacing , resistors are made to have the same geometry as shown in Fig. P6.5.1. In general, the region between the plane parallel perfectly conducting electrodes is filled by a material of conductivity = (x). The boundaries of the conductor that interface with the surrounding free space have normals that are either in the x or the z direction.

(a) Show that even if d is large compared to l and c, E between the plates is (v/d)i_y.

(b) If the conductor is piece-wise uniform, with sections having conductivities _a and _b of width a and b, respectively, as shown in Fig. P6.5.1a, show that the conductance G = c(_b b + _a a)/d.

(c) If = _a(1 + x/l), show that G = 3_a cl/2d.

7.2.4

A pair of uniform conductors form a resistor having the shape of a circular cylindrical half-shell, as shown in Fig. P7.2.4. The boundaries at r = a and r = b, and in planes parallel to the paper, interface with free space. Show that for steady conduction, all boundary conditions are satisfied by a simple piece-wise continuous potential that is an exact solution to Laplace's equation. Determine the resistance. figure GIF #2

Figure P7.2.4

7.2.5^*

The region between the planar electrodes of Fig. 7.2.4 is filled with a material having conductivity = _o/(1 + y/a), where _o and a are constants. The permittivity is uniform.

(a) Show that G = A _o/d(1 + d/2a).

(b) Show that _u = Gv/A_o a.

7.2.6

The region between the planar electrodes of Fig. 7.2.4 is filled with a uniformly conducting material having permittivity = _a/(1 + y/a).

(a) What is G?

(b) What is _u in the conductor?

7.2.7^*

A section of a spherical shell of conducting material with inner radius b and outer radius a is shown in Fig. P7.2.7. Show that if = _o (r/a)², the conductance G = 6 (1 - cos /2 ) ab³ _o/(a³ - b³).

Figure P7.2.7

7.2.8

In a cylindrical version of the geometry shown in Fig. P7.2.7, the material between circular cylindrical outer and inner electrodes of radii a and b, respectively, has conductivity = _o (a/r). The boundaries parallel to the page interface free space and are a distance d apart. Determine the conductance G.

Distributed Current Sources and Associated Fields

7.3.1^*

An infinite half-space of uniformly conducting material in the region y > 0 has an interface with free space in the plane y = 0. There is a point current source of I amps located at (x, y, z) = (0, h, 0) on the y axis. Using an approach analogous to that used in Prob. 6.6.5, show that the potential inside the conductor is equation GIF #7.159

Now that the potential of the interface is known, show that the potential in the free space region outside the conductor, where y < 0, is equation GIF #7.160

7.3.2

The half-space y > 0 is of uniform conductivity while the remaining space is insulating. A uniform line current source of density K_l (A/m) runs parallel to the plane y = 0 along the line x = 0, y = h.

(a)	Determine in the conductor.
(b)	In turn, what is in the insulating half-space?

7.3.3^*

A two-dimensional dipole current source consists of uniform line current sources K_l have the spacing d. The cross-sectional view is as shown in Fig. 7.3.4, with . Show that the associated potential is equation GIF #7.161

in the limit K_l , d 0, K_l d finite.

Superposition and Uniqueness of Steady Conduction Solutions

7.4.1^*

A material of uniform conductivity has a spherical insulating cavity of radius b at its center. It is surrounded by segmented electrodes that are driven by current sources in such a way that at the spherical outer surface r = a, the radial current density is J_r = - J_o cos , where J_o is a given constant.

(a) Show that inside the conducting material, the potential is equation GIF #7.162

(b) Evaluated at r = b, this gives the potential on the surface bounding the insulating cavity. Show that the potential in the cavity is equation GIF #7.163

7.4.2

A uniformly conducting material has a spherical interface at r = a, with a surrounding insulating material and a spherical boundary at r = b (b < a), where the radial current density is J_r = J_o cos , essentially independent of time.

(a) What is in the conductor?

(b) What is in the insulating region surrounding the conductor?

7.4.3

In a system that stretches to infinity in the x and z directions, there is a layer of uniformly conducting material having boundaries in the planes y = 0 and y = -a. The region y > 0 is free space, while a potential = V cos x is imposed on the boundary at y = -a.

(a) Determine in the conducting layer.

(b) What is in the region y > 0?

7.4.4^*

The uniformly conducting material shown in cross-section in Fig. P7.4.4 extends to infinity in the z directions and has the shape of a 90-degree section from a circular cylindrical annulus. At = 0 and = /2, it is in contact with grounded electrodes. The boundary at r = a interfaces free space, while at r = b, an electrode constrains the potential to be v. Show that the potential in the conductor is

Figure P7.4.4

Figure P7.4.5

7.4.5

The cross-section of a uniformly conducting material that extends to infinity in the z directions is shown in Fig. P7.4.5. The boundaries at r = b, at = 0, and at = interface insulating material. At r = a, voltage sources constrain = -v/2 over the range 0 < < /2, and = v/2 over the range /2 < < .

(a) Find an infinite set of solutions for that satisfy the boundary conditions at the three insulating surfaces.

(b) Determine in the conductor.

7.4.6

The system of Fig. P7.4.4 is altered so that there is an electrode on the boundary at r = a. Determine the mutual conductance between this electrode and the one at r = b.

Steady Currents in Piece-Wise Uniform Conductors

7.5.1^*

A sphere having uniform conductivity _b is surrounded by material having the uniform conductivity _a. As shown in Fig. P7.5.1, electrodes at "infinity" to the right and left impose a uniform current density J_o at infinity. Steady conduction prevails. Show that

Figure P7.5.1

7.5.2

Assume at the outset that the sphere of Prob. 7.5.1 is much more highly conducting than its surroundings.

(a) As far as the fields in region (a) are concerned, what is the boundary condition at r = R?

(b) Determine the approximate potential in region (a) and compare to the appropriate limiting potential from Prob. 7.5.1.

(c) Based on this potential in region (a), determine the approximate potential in the sphere and compare to the appropriate limit of as found in Prob. 7.5.1.

(d) Now, assume that the sphere is much more insulating than its surroundings. Repeat the steps of parts (a)-(c).

7.5.3^*

A rectangular box having depth b, length l and width much larger than b has an insulating bottom and metallic ends which serve as electrodes. In Fig. P7.5.3a, the right electrode is extended upward and then back over the box. The box is filled to a depth b with a liquid having uniform conductivity. The region above is air. The voltage source can be regarded as imposing a potential in the plane z = -l between the left and top electrodes that is linear.

(a)	Show that the potential in the conductor is = -vz/l.
(b)	In turn, show that in the region above the conductor, = v(z/l)(x - a)/a.
(c)	What are the distributions of _u and _u?
(d)	Now suppose that the upper electrode is slanted, as shown in Fig. P7.5.3b. Show that in the conductor is unaltered but in the region between the conductor and the slanted plate, = v[(z/l) + (x/a)].

Figure P7.5.3

7.5.4

The structure shown in Fig. P7.5.4 is infinite in the z directions. Each leg has the same uniform conductivity, and conduction is stationary. The walls in the x and in the y planes are perfectly conducting.

(a)	Determine , E, and J in the conductors.
(b)	What are and E in the free space region?
(c)	Sketch and E in this region and in the conductors.

Figure P7.5.4

7.5.5

The system shown in cross-section by Fig. P7.5.6a extends to infinity in the x and z directions. The material of uniform conductivity _a to the right is bounded at y = 0 and y = a by electrodes at zero potential. The material of uniform conductivity _b to the left is bounded in these planes by electrodes each at the potential v. The approach to finding the fields is similar to that used in Example 6.6.3.

(a) What is ^a as x and ^b as x -?

(b) Add to each of these solutions an infinite set such that the boundary conditions are satisfied in the planes y = 0 and y = a and as x .

(c) What two boundary conditions relate ^a to ^b in the plane x = 0?

(d) Use these conditions to determine the coefficients in the infinite series, and hence find throughout the region between the electrodes.

(e) In the limits _b _a and _b = _a, sketch and E. (A numerical evaluation of the expressions for is not required.)

(f) Shown in Fig. P7.5.6b is a similar system but with the conductors bounded from above by free space. Repeat the steps (a) through (e) for the fields in the conducting layer.

Figure P7.5.5

Conduction Analogs

7.6.1^*

In deducing (4) relating the capacitance of electrodes in an insulating material to the conductance of electrodes having the same shape in a conducting material, it is assumed that not only are the ratios of all dimensions in one situation the same as in the other (the systems are geometrically similar), but that the actual size of the two physical situations is the same. Show that if the systems are again geometrically similar but the length scale of the capacitor is l while that of the conduction cell is l , RC = ( / )(l /l ).

Charge Relaxation in Uniform Conductors

7.7.1^*

In the two-dimensional configuration of Prob. 4.1.4, consider the field transient that results if the region within the cylinder of rectangular cross-section is filled by a material having uniform conductivity and permittivity .

(a) With the initial potential given by (a) of Prob. 4.1.4, with _o and _o a given constant, show that _u (x, y, t = 0) is given by (c) of Prob. 4.1.4.

(b) Show that for t > 0, is given by (c) of Prob. 4.1.4 multiplied by exp (-t/ ), where = /.

(c) Show that for t > 0, the potential is given by (a) of Prob. 4.1.4 multiplied by exp (-t/ ).

(d) Show that for t > 0, the current i(t) from the electrode segment is (f) of Prob. 4.1.4

7.7.2

When t = 0, the only net charge in a material having uniform and is the line charge of Prob. 4.5.4. As a function of time for t > 0, determine the

(a)	line charge density,
(b)	charge density elsewhere in the medium, and
(c)	the potential (x, y, z, t).

7.7.3^*

When t = 0, the charged particle of Example 7.7.2 has a charge q = q_o < -q_c.

(a) Show that, as long as q remains less than -q_c, the net current to the particle is i = - / q.

(b) Show that, as long as q < -q_c, q = q_o exp (-t/₁) where ₁ = /.

7.7.4

Relative to the potential at infinity on a plane passing through the equator of the particle in Example 7.7.2, what is the potential of the particle when its charge reaches q = q_c?

Electroquasistatic Conduction Laws for Inhomogeneous Materials

7.8.1^*

Use an approach similar to that illustrated in this section to show uniqueness of the solution to Poisson's equation for a given initial distribution of and a given potential = on the surface S^', and a given current density -( + / t) n = J on S^" where S^' + S^" encloses the volume of interest V.

Charge Relaxation in Uniform and Piece-Wise Uniform Systems

7.9.1^*

We return to the coaxial circular cylindrical electrode configurations of Prob. 6.5.5. Now the material in region (2) of each has not only a uniform permittivity but a uniform conductivity as well. Given that V(t) = Re exp (j t),

(a)	show that E in the first configuration of Fig. P6.5.5 is i_r v/r ln(a/b),
(b)	while in the second configuration, where Det = [ ln(a/R)] + j [_o ln(R/b) + ln(a/R)].
(c)	Show that in the first configuration a length l (into the paper) is equivalent to a conductance G in parallel with a capacitance C where while in the second, it is equivalent to the circuit of Fig. 7.9.5 with

7.9.2

Interpret the configurations shown in Fig. P6.5.5 as spherical. An outer spherically shaped electrode has inside radius a, while an inner electrode positioned on the same center has radius b. Region (1) is free space while (2) has uniform and .

(a) For V = V_o cos ( t), determine E in each region.

(b) What are the elements in the equivalent circuit for each?

7.9.3^*

Show that the hemispherical electrode of Fig. 7.3.3 is equivalent to a circuit having a conductance G = 2 a in parallel with a capacitance C = 2 a.

7.9.4

The circular cylinder of Fig. P7.9.4a has _b and _b and is surrounded by material having _a and _a. The electric field E(t)i_x is applied at x = .

(a) Find the potential in and around the cylinder and the surface charge density that result from applying a step in field to a system that initially is free of charge.

(b) Find these quantities for the sinusoidal steady state response.

(c) Argue that these fields are equally applicable to the description of the configuration shown in Fig. P7.9.4b with the cylinder replaced by a half-cylinder on a perfectly conducting ground plane. In the limit where the exterior region is free space while the half-cylinder is so conducting that its charge relaxation time is short compared to times characterizing the applied field (1/ in the sinusoidal steady state case), what are the approximate fields in the exterior and in the interior regions? (See Prob. 7.9.5 for a direct calculation of these approximate fields.)

Figure P7.9.4

7.9.5^*

The half-cylinder of Fig. P7.9.4b has a relaxation time that is short compared to times characterizing the applied field E(t). The surrounding region is free space (_a = 0).

(a) Show that in the exterior region, the potential is approximately equation GIF #7.170

(b) In turn, show that the field inside the half-cylinder is approximately equation GIF #7.171

7.9.6

An electric dipole having a z-directed moment p(t) is situated at the origin and at the center of a spherical cavity of free space having a radius a in a material having uniform and . When t < 0, p = 0 and there is no charge anywhere. The dipole is a step function of time, instantaneously assuming a moment p_o when t = 0.

(a) An instant after the dipole is established, what is the distribution of inside and outside the cavity?

(b) Long after the electric dipole is turned on and the fields have reached a steady state, what is the distribution of ?

(c) Determine (r, , t).

7.9.7^*

A planar layer of semi-insulating material has thickness d, uniform permittivity , and uniform conductivity , as shown in Fig. P7.9.7. From below it is bounded by contacting electrode segments that impose the potential = V cos x. The system extends to infinity in the x and z directions.

(a)	The potential has been applied for a long time. Show that at y = 0, _su = _o V cos x/cosh d.
(b)	When t = 0, the applied potential is turned off. Show that this unpaired surface charge density decays exponentially from the initial value from part (a) with the time constant = (_o tanh d + )/.

Figure P7.9.7

7.9.8^*

Region (b), where y < 0, has uniform permittivity and conductivity , while region (a), where 0 < y, is free space. Before t = 0 there are no charges. When t = 0, a point charge Q is suddenly "turned on" at the location (x, y, z) = (0, h, 0).

(a) Show that just after t = 0, equation GIF #7.172 equation GIF #7.173 where q_b Q[( /_o ) - 1]/[( /_o ) + 1] and q_a 2Q/[( /_o ) + 1].

(b) Show that as t , q_b Q and the field in region (b) goes to zero.

(c) Show that the transient is described by (a) and (b) with equation GIF #7.174 equation GIF #7.175 where = (_o + )/.

7.9.9^*

The cross-section of a two-dimensional system is shown in Fig. P7.9.9. The parallel plate capacitor to the left of the plane x = 0 extends to x = -, with the lower electrode at potential v(t) and the upper one grounded. This upper electrode extends to the right to the plane x = b, where it is bent downward to y = 0 and inward to the plane x = 0 along the surface y = 0. Region (a) is free space while region (b) to the left of the plane x = 0 has uniform permittivity and conductivity . The applied voltage v(t) is a step function of magnitude V_o.

(a) The voltage has been on for a long-time. What are the field and potential distributions in region (b)? Having determined ^b, what is the potential in region (a)?

(b) Now, is to be found for t > 0. Example 6.6.3 illustrates the approach that can be used. Show that in the limit t , becomes the result of part (a).

(c) In the special case where = _o, sketch the evolution of the field from the time just after the voltage is applied to the long-time limit of part (a).

Figure P7.9.9