Poisson's and Laplace's Equations 5.1.1In Problem 4.7.1, the potential of a point charge over a perfectly conducting plane (where z > 0) was found to be Eq. (a) of that problem. Identify particular and homogeneous parts of this solution.

5.1.2A solution for the potential in the region -a < y < a, where there is a charge density , satisfies the boundary conditions = 0 in the planes y = +a and y = -a.

(a)	What is in this region?
(b)	Identify _p and _h. What boundary conditions are satisfied by _h at y = +a and y = -a?
(c)	Illustrate another combination of _p and _h that could just as well be used and give the boundary conditions that apply for _h in that case.

5.1.3^*The charge density between the planes x = 0 and x = d depends only on x.

Boundary conditions are that (x = 0) = 0 and (x = d) = V, so = (x) is independent of y and z. equation GIF #5.172

(a)	Show that Poisson's equation therefore reduces to
(b)	Integrate this expression twice and use the boundary conditions to show that the potential distribution is
(c)	Argue that the first term in (c) can be _p, with the remaining terms then _h.
(d)	Show that in that case, the boundary conditions satisfied by _h are

5.1.4With the charge density given as equation GIF #5.175

carry out the steps in Prob. 5.1.3.

5.1.5^*A frequently used model for a capacitor is shown in Fig. P5.1.5, where two plane parallel electrodes have a spacing that is small compared to either of their planar dimensions. The potential difference between the electrodes is v, and so over most of the region between the electrodes, the electric field is uniform. the region well removed from the edges of the electrodes, the field E = -(v/d) i_z satisfies Laplace's equation and the boundary conditions on the electrode surfaces.

(a)	Show that in
(b)	Show that the surface charge density on the lower surface of the upper electrode is _s = _o v/d.
(c)	For a single pair of electrodes, the capacitance C is defined such that q = Cv (13). Show that for the plane parallel capacitor of Fig. P5.1.5, C = A_o /d, where A is the area of one of the electrodes.
(d)	Use the integral form of charge conservation, (1.5.2), to show that i = dq/dt = Cdv/dt.

5.1.6^*In the three-electrode system of Fig.~P5.1.6, the bottom electrode is taken as having the reference potential. The upper and middle electrodes then have potentials v₁ and v₂, respectively. The spacings between electrodes, 2d and d, are small enough relative to the planar dimensions of the electrodes so that the fields between can be approximated as being uniform.

the figure are then approximately E₁ = v₁/2d, E₂ = v₂/d and E_m = (v₁ - v₂)/d.

(a) Show that the fields denoted in

(b) Show that the net charges q₁ and q₂ on the top and middle electrodes, respectively, are related to the voltages by the capacitance matrix [in the form of (12)] equation GIF #5.176

\noprobsect

Continuity Conditions

5.2.1^*The electric potentials ^a and ^b above and below the plane y = 0 are equation GIF #5.177

0.)

(a) Show that (4) holds. (The potential is continuous at y =

(b) Evaluate E tangential to the surface y = 0 and show that it too is continuous. [Equation (1) is then automatically satisfied at y = 0.]

(c) Use (5) to show that in the plane y = 0, the surface charge density, _s = 2_o V cos x, accounts for the discontinuity in the derivative of normal to the plane y = 0.

5.2.2By way of appreciating how the continuity of guarantees the continuity of tangential E [(4) implies that (1) is satisfied], suppose that the potential is given in the plane y = 0: = (x, 0, z). from this information alone?

(a) Which components of E can be determined

(b) For example, if (x, 0, z) = V sin ( x) sin ( z), what are those components of E?

Coordinates 5.3.1^*A region that extends to in the z direction has the square cross-section of dimensions as shown in Fig. P5.4.1. The walls at x = 0 and y = 0 are at zero potential, while those at x = a and y = a have the linear distributions shown. The interior region is free of charge density.

(a) Show that the potential inside is

(b) Show that plots of and E are as shown in the first quadrant of Fig. 4.1.3.

5.3.2One way to constrain a boundary so that it has a potential distribution that is a linear function of position is shown in Fig. P5.4.2a. A uniformly resistive sheet having a length 2a is driven by a voltage source V. For the coordinate x shown, the resulting potential distribution is the linear function of x shown. The constant C is determined by the definition of where the potential is zero. In the case shown in Fig. 5.4.2a, if is zero at x = 0, then C = 0.
cross-section of length 2a on a side, as shown in Fig. 5.4.2b, is constrained in potential by resistive sheets and voltage sources, as shown. Note that the potential is defined to be zero at the lower right-hand corner, where (x, y) = (a, -a). Inside the cylinder, what must the potential be in the planes x = a and y = a?

(a) Suppose a cylindrical region having a square

(b) Find the linear combination of the potentials from the first column of Table 5.4.1 that satisfies the conditions on the potentials required by the resistive sheets. That is, if takes the form so that it satisfies Laplace's equation inside the cylinder, what are the coefficients A, B, C, and D?

(c) Determine E for this potential.

(d) Sketch and E.

(e) Now the potential on the walls of the square cylinder is constrained as shown in Fig. 5.4.2c. This time the potential is zero at the location (x, y) = (0, 0). Adjust the coefficients in (a) so that the potential satisfies these conditions. Determine E and sketch the equipotentials and field lines.

5.3.3^*Shown in cross-section in Fig. P5.4.3 is a cylindrical system that extends to infinity in the z directions. There is no charge density inside the cylinder, and the potentials on the boundaries are

(a) Show that the potential inside the cylinder is

(b) Show that a plot of and E is as given by the part of Fig. 5.4.1 where - /2 < kx < /2.

5.3.4The square cross-section of a cylindrical region that extends to infinity in the z directions is shown in Fig. P5.4.4. The potentials on the boundaries are as shown. Find .

(a) Inside the cylindrical space, there is no charge density.

(b) What is E in this region?

(c) Sketch and E.

5.3.5^*The cross-section of an electrode structure which is symmetric about the x = 0 plane is shown in Fig. P5.4.5. Above this plane are electrodes that alternately either have the potential v(t) or the potential -v(t). The system has depth d (into the paper) which is very long compared to such dimensions as a or l. So that the current i(t) can be measured, one of the upper electrodes has a segment which is insulated from the rest of the electrode, but driven by the same potential. The geometry of the upper electrodes is specified by giving their altitudes above the x = 0 plane. For example, the upper electrode between y = -b and y = b has the shape where is as shown in Fig. P5.4.5.
electrodes is

(a) Show that the potential in the region between the

(b) Show that E in this region is

(c) Show that plots of and E are as shown in Fig. 5.4.2.

(d) Show that the net charge on the upper electrode segment between y = -l and y = l is (Because the surface S in Gauss' integral law is arbitrary, it can be chosen so that it both encloses this electrode and is convenient for integration.)

(e) Given that v(t) = V_o sin t, where V_o and are constants, show that the current to the electrode segment i(t), as defined in Fig. P5.4.5, is

5.3.6In Prob. 5.4.5, the polarities of all of the voltage sources driving the lower electrodes are reversed. between the electrodes.

(a) Find in the region

(b) Determine E.

(c) Sketch and E.

(d) Find the charge q on the electrode segment in the upper middle electrode.

(e) Given that v(t) = V_o cos t, what is i(t)?

Modal Expansion to Satisfy Boundary Conditions
5.4.1^*The system shown in Fig. P5.5.1a is composed of a pair of perfectly conducting parallel plates in the planes x = 0 and x = a that are shorted in the plane y = b. Along the left edge, the potential is imposed and so has a given distribution _d (x). The plates and short have zero potential.
_d (x), the potential distribution for 0 < y < b, 0 < x < a is
where (At this stage, the coefficients in a modal expansion for the field are left expressed as integrals over the yet to be specified potential distribution.)

(a) Show that, in terms of

(b) In particular, if the imposed potential is as shown in Fig. P5.5.1b, show that A_n is

5.4.2^*The walls of a rectangular cylinder are constrained in potential as shown in Fig. P5.5.2. The walls at x = a and y = b have zero potential, while those at y = 0 and x = 0 have the potential distributions V₁ (x) and V₂ (y), respectively.
In particular, suppose that these distributions of potential are uniform, so that V₁(x) = V_a and V₂(y) = V_b, with V_a and V_b defined to be independent of x and y.
potential distribution there is

(a) The region inside the cylinder is free space. Show that the

(b) Show that the distribution of surface charge density along the wall at x = a is

5.4.3In the configuration described in Prob. 5.5.2, the distributions of potentials on the walls at x = 0 and y = 0 are as shown in Fig. P5.5.3, where the peak voltages V_a and V_b are given functions of time.
cylinder.

(a) Determine the potential in the free space region inside the

(b) Find the surface charge distribution on the wall at y = b.

\problem 4^* The cross-section of a system that extends to ``infinity'' out of the paper is shown in Fig. P5.5.4. An electrode in the plane y = d has the potential V. A second electrode has the shape of an ``L.'' One of its sides is in the plane y = 0, while the other is in the plane x = 0, extending from y = 0 almost to y = d. This electrode is at zero potential.
Show that, far to the left, the potential between the electrodes tends to

(a) The electrodes extend to infinity in the -x direction.

(b) Using this result as a part of the solution, _a, the potential between the plates is written as = _a + _b. Show that the boundary conditions that must be satisfied by _b are

(c) Show that the potential between the electrodes is

(d) Show that a plot of and E appears as shown in Fig. 6.6.9c, turned upside down.

5.4.5In the two-dimensional system shown in cross-section in Fig. P5.5.5, plane parallel plates extend to infinity in the -y direction. The potentials of the upper and lower plates are, respectively, -V_o/2 and V_o/2. The potential over the plane y = 0 terminating the plates at the right is specified to be _d(x).
to the left?

(a) What is the potential distribution between the plates far

(b) If is taken as the potential _a that assumes the correct distribution as y -, plus a potential _b, what boundary conditions must be satisfied by _b?

(c) What is the potential distribution between the plates?

5.4.6As an alternative (and in this case much more complicated) way of expressing the potential in Prob. 5.4.1, use a modal approach to express the potential in the interior region of Fig. P5.4.1.
5.4.7^*Take an approach to finding the potential in the configuration of Fig. 5.5.2 that is an alternative to that used in the text. Let = (Vy/b) + ₁. must be satisfied by ₁ are that ₁ = -Vy/b at x = 0 and at x = a, and ₁ = 0 at y = 0 and y = b.

(a) Show that the boundary conditions that

(b) Show that the potential is where (It is convenient to exploit the symmetry of the configuration about the plane x = a/2.)

Conditions 5.5.1^*The potential distribution is to be determined in a region bounded by the planes y = 0 and y = d and extending to infinity in the x and z directions, as shown in Fig. P5.6.1. In this region, there is a uniform charge density _o. On the upper boundary, the potential is (x, d, z) = V_a sin ( x). On the lower boundary, (x, 0, z) = V_b sin ( x). Show that (x, y, z) throughout the region 0 < y < d is
5.5.2For the configuration of Fig. P5.6.1, the charge is again uniform in the region between the boundaries, with density _o, but the potential at y = d is = _o sin (kx), while that at y = 0 is zero (_o and k are given constants). Find in the region where 0 < y < d, between the boundaries.
5.5.3^*In the region between the boundaries at y = d/2 in Fig. P5.6.3, the charge density is

where _o and are given constants. Electrodes at y = d/2 constrain the tangential electric field there to be The charge density might represent a traveling wave of space charge on a modulated particle beam, and the walls represent the traveling-wave structure which interacts with the beam. Thus, in a practical device, such as a traveling-wave amplifier designed to convert the kinetic energy of the moving charge to ac electrical energy available at the electrodes, the charge and potential distributions move to the right with the same velocity. This does not concern us, because we consider the interaction at one instant in time.

(a) Show that a particular solution is

(b) Show that the total potential is the sum of this solution and that solution to Laplace's equation that makes the total solution satisfy the boundary conditions.

(c) The force density (force per unit volume) acting on the charge is E. Show that the force f_x acting on a section of the charge of length in the x direction = 2 /k spanning the region -d/2 < y < d/2 and unit length in the z direction is

5.5.4In the region 0 < y < d shown in cross-section in Fig. P5.6.4, the charge density is where _o and are constants. Electrodes at y = d constrain the potential there to be (x, d) = V_o cos (kx) (V_o and k given constants), while an electrode at y = 0 makes (x, 0) = 0.
everywhere between the electrodes.

(a) Find a particular solution that satisfies Poisson's equation

(b) What boundary conditions must the homogeneous solution satisfy at y = d and y = 0?

(c) Find in the region 0 < y < d.

(d) The force density (force per unit volume) acting on the charge is E. Find the total force f_x acting on a section of the charge spanning the system from y = 0 to y = d, of unit length in the z direction and of length = 2 /k in the x direction.

5.5.5^*A region that extends to infinity in the z directions has a rectangular cross-section of dimensions 2a and b, as shown in Fig. P5.6.5. The boundaries are at zero potential while the region inside has the distribution of charge density

where _o is a given constant. Show that the potential in this region is 5.5.6The cross-section of a two-dimensional configuration is shown in Fig. P5.6.6. The potential distribution is to be determined inside the boundaries, which are all at zero potential. where V and are given constants, what is the charge density in that region?

(a) Given that a particular solution inside the boundaries is

(b) What is ?

5.5.7The cross-section of a metal box that is very long in the z direction is shown in Fig. P5.6.7. It is filled by the charge density _o x/l. Determine inside the box, given that = 0 on the walls.

5.5.8^*In region (b), where y < 0, the charge density is = _o cos ( x) e^y, where _o, , and are positive constants. In region (a), where 0 < y, = 0.

(a) Show that a particular solution in the region y < 0 is

(b) There is no surface charge density in the plane y = 0. Show that the potential is

5.5.9A sheet of charge having the surface charge density _s = _o sin (x - x_o) is in the plane y = 0, as shown in Fig. 5.6.3. At a distance a above and below the sheet, electrode structures are used to constrain the potential to be = V cos x. The system extends to infinity in the x and z directions. The regions above and below the sheet are designated (a) and (b), respectively. , _o, and x_o.

(a) Find _a and _b in terms of the constants V,

(b) Given that the force per unit area acting on the charge sheet is _s E_x (x, 0), what is the force acting on a section of the sheet having length d in the z direction and one wavelength 2 / in the x direction?

(c) Now, the potential on the wall is made a traveling wave having a given angular frequency , (x, a, t) = V cos ( x - t), and the charge moves to the right with a velocity U, so that _s = _o sin (x - Ut - x_o), where U = /. Thus, the wall potentials and surface charge density move in synchronism. Building on the results from parts (a)-(b), what is the potential distribution and hence total force on the section of charged sheet?

(d) What you have developed is a primitive model for an electron beam device used to convert the kinetic energy of the electrons (accelerated to the velocity v by a dc voltage) to high-frequency electrical power output. Because the system is free of dissipation, the electrical power output (through the electrode structure) is equal to the mechanical power input. Based on the force found in part (c), what is the electrical power output produced by one period 2 / of the charge sheet of width w?

(e) For what values of x_o would the device act as a generator of electrical power?

Solutions to Laplace's Equation in Polar Coordinates
5.6.1^*A circular cylindrical surface r = a has the potential = V sin 5. The regions r < a and a < r are free of charge density. Show that the potential is 5.6.2The x - z plane is one of zero potential. Thus, the y axis is perpendicular to a zero potential plane. With measured relative to the x axis and z the third coordinate axis, the potential on the surface at r = R is constrained by segmented electrodes there to be = V sin . region?

(a) If = 0 in the region r < R, what is in that

(b) Over the range r < R, what is the surface charge density on the surface at y = 0?

5.6.3^*An annular region b < r < a where = 0 is bounded from outside at r = a by a surface having the potential = V_a cos 3 and from the inside at r = b by a surface having the potential = V_b sin . Show that in the annulus can be written as the sum of two terms, each a combination of solutions to Laplace's equation designed to have the correct value at one radius while being zero at the other.

5.6.4In the region b < r < a, 0 < < , = 0. On the boundaries of this region at r = a, at = 0 and = , = 0. At r = b, = V_b sin ( / ). Determine in this region.
5.6.5^*In the region b < r < a, 0 < < , = 0. On the boundaries of this region at r = a, r = b and at = 0, = 0. At = , the potential is = V sin [3 ln(r/a)/ln(b/a)]. Show that within the region,
5.6.6The plane = 0 is at potential = V, while that at = 3 /2 is at zero potential. The system extends to infinity in the z and r directions. Determine and sketch and E in the range 0 < < 3 /2.

Examples in Polar Coordinates
5.7.1^*Show that and E as given by (4) and (5), respectively, describe the potential and electric field intensity around a perfectly conducting half-cylinder at r = R on a perfectly conducting plane at x = 0 with a uniform field E_a i_x applied at x . Show that the maximum field intensity is twice that of the applied field, regardless of the radius of the half-cylinder.
5.7.2Coaxial circular cylindrical surfaces bound an annular region of free space where b < r < a. On the inner surface, where r = b, = V_b > 0. On the outer surface, where r = a, = V_a > 0.

(a) What is in the annular region?

(b) How large must V_b be to insure that all lines of E are outward directed from the inner cylinder?

(c) What is the net charge per unit length on the inner cylinder under the conditions of (b)?

5.7.3^*A device proposed for using the voltage v_o to measure the angular velocity \Omega of a shaft is shown in Fig. P5.8.3a. A cylindrical grounded electrode has radius R. (The resistance R_o is ``small.'') Outside and concentric at r = a is a rotating shell supporting the surface charge density distribution shown in Fig. P5.8.3b. (b), respectively, outside and inside the rotating shell,

(a) Given _o and _o, show that in regions (a) and

(b) Show that the charge on the segment of the inner electrode attached to the resistor is where w is the length in the z direction.

(c) Given that _o = \Omega t, show that the output voltage is related to \Omega by so that its amplitude can be used to measure \Omega.

5.7.4Complete the steps of Prob. 5.8.3 with the configuration of Fig. P5.8.3 altered so that the rotating shell is inside rather than outside the grounded electrode. Thus, the radius a of the rotating shell is less than the radius R, and region (a) is a < r < R, while region (b) is r < a.
5.7.5^*A pair of perfectly conducting zero potential electrodes form a wedge, one in the plane = 0 and the other in the plane = . They essentially extend to infinity in the z directions. Closing the region between the electrodes at r = R is an electrode having potential V. Show that the potential inside the region bounded by these three surfaces is
5.7.6In a two-dimensional system, the region of interest is bounded in the = 0 plane by a grounded electrode and in the = plane by one that has = V. The region extends to infinity in the r direction. At r = R, = V. Determine . 5.7.7Figure P5.8.7 shows a circular cylindrical wall having potential V_o relative to a grounded fin in the plane = 0 that reaches from the wall to the center. The gaps between the cylinder and the fin are very small. the boun\-dary conditions at = 0 and = 2. Note that you cannot accept solutions for of negative powers in r.

(a) Find all solutions in polar coordinates that satisfy

(b) Match the boundary condition at r = R.

(c) One of the terms in this solution has an electric field intensity that is infinite at the tip of the fin, where r = 0. Sketch and E in the neighborhood of the tip. What is the _s on the fin associated with this term as a function of r? What is the net charge associated with this term?

(d) Sketch the potential and field intensity throughout the region.

5.7.8A two-dimensional system has the same cross-sectional geometry as that shown in Fig. 5.8.6 except that the wall at = 0 has the potential v. The wall at = _o is grounded. Determine the interior potential.
5.7.9Use arguments analogous to those used in going from (5.5.22) to (5.5.26) to show the orthogonality (14) of the radial modes R_n defined by (13). [Note the comment following (14).] Spherical Coordinates 5.8.1On the surface of a spherical shell having radius r = a, the potential is = V cos . or inside this shell, what is for r < a and for r > a?

(a) With no charge density either outside

(b) Sketch and E.

5.8.2^*A spherical shell having radius a supports the surface charge density _o cos .
volume of interest, the potential is

(a) Show that if this is the only charge in the

(b) Show that a plot of and E appears as shown in Fig. 6.3.1.

5.8.3^*A spherical shell having zero potential has radius a. Inside, the charge density is = _o cos . Show that the potential there is 5.8.4The volume of a spherical region is filled with the charge density = _o (r/a)^m cos , where _o and m are given constants. If the potential = 0 at r = a, what is for r < a? Equation 5.9.1^*In the configuration of Fig. 5.10.2, all surfaces have zero potential except those at x = 0 and x = a, which have = v. Show that

and 5.9.2In the configuration of Fig. 5.10.2, all surfaces have zero potential. In the plane y = a/2, there is the surface charge density _s = _o sin ( x/a) sin ( z/w). Find the potentials ^a and ^b above and below this surface, respectively. 5.9.3The configuration is the same as shown in Fig. 5.10.2 except that all of the walls are at zero potential and the volume is filled by the uniform charge density = _o. Write four essentially different expressions for the potential distribution. \globallineheight15pt\globallinedepth12pt \boxtable \verticallines \globallineheight15pt\globallinedepth12pt \boxtable \verticallines \globallineheight14pt\globallinedepth11pt \boxtable \verticallines %%%%%%%%%%%%%%%%%%%%%deleted for editor's version%%%%%%%%%%%%%%% 5.9.6^*Show that in Cartesian coordinates (defined relative to the polar coordinates in Fig. 5.7.1) the potential for the fringing field of the closely spaced capacitor plates in Fig. 5.7.2b is Now, consider the field surrounding a pair of capacitor plates having infinite length in the z direction, width l in the x direction and a spacing that is small compared to dimensions of interest. The cross-section is shown in Fig. P5.7.6. Write the potential in terms of the angles ₁ and ₂ defined in the figure and then convert the representation to Cartesian coordinates to show that

(a)	Show that the fields denoted in
(b)	Show that the net charges q₁ and q₂ on the top and middle electrodes, respectively, are related to the voltages by the capacitance matrix [in the form of (12)]

(a)	Show that (4) holds. (The potential is continuous at y =
(b)	Evaluate E tangential to the surface y = 0 and show that it too is continuous. [Equation (1) is then automatically satisfied at y = 0.]
(c)	Use (5) to show that in the plane y = 0, the surface charge density, _s = 2_o V cos x, accounts for the discontinuity in the derivative of normal to the plane y = 0.

(a)	Which components of E can be determined
(b)	For example, if (x, 0, z) = V sin ( x) sin ( z), what are those components of E?

(a)	Show that the potential inside is
(b)	Show that plots of and E are as shown in the first quadrant of Fig. 4.1.3.

(a)	Suppose a cylindrical region having a square
(b)	Find the linear combination of the potentials from the first column of Table 5.4.1 that satisfies the conditions on the potentials required by the resistive sheets. That is, if takes the form so that it satisfies Laplace's equation inside the cylinder, what are the coefficients A, B, C, and D?
(c)	Determine E for this potential.
(d)	Sketch and E.
(e)	Now the potential on the walls of the square cylinder is constrained as shown in Fig. 5.4.2c. This time the potential is zero at the location (x, y) = (0, 0). Adjust the coefficients in (a) so that the potential satisfies these conditions. Determine E and sketch the equipotentials and field lines.

(a)	Show that the potential inside the cylinder is
(b)	Show that a plot of and E is as given by the part of Fig. 5.4.1 where - /2 < kx < /2.

(a)	Inside the cylindrical space, there is no charge density.
(b)	What is E in this region?
(c)	Sketch and E.

(a)	Show that the potential in the region between the
(b)	Show that E in this region is
(c)	Show that plots of and E are as shown in Fig. 5.4.2.
(d)	Show that the net charge on the upper electrode segment between y = -l and y = l is (Because the surface S in Gauss' integral law is arbitrary, it can be chosen so that it both encloses this electrode and is convenient for integration.)
(e)	Given that v(t) = V_o sin t, where V_o and are constants, show that the current to the electrode segment i(t), as defined in Fig. P5.4.5, is

(a)	Find in the region
(b)	Determine E.
(c)	Sketch and E.
(d)	Find the charge q on the electrode segment in the upper middle electrode.
(e)	Given that v(t) = V_o cos t, what is i(t)?

(a)	Show that, in terms of
(b)	In particular, if the imposed potential is as shown in Fig. P5.5.1b, show that A_n is

(a)	The region inside the cylinder is free space. Show that the
(b)	Show that the distribution of surface charge density along the wall at x = a is

(a)	Determine the potential in the free space region inside the
(b)	Find the surface charge distribution on the wall at y = b.

(a)	The electrodes extend to infinity in the -x direction.
(b)	Using this result as a part of the solution, _a, the potential between the plates is written as = _a + _b. Show that the boundary conditions that must be satisfied by _b are
(c)	Show that the potential between the electrodes is
(d)	Show that a plot of and E appears as shown in Fig. 6.6.9c, turned upside down.

(a)	What is the potential distribution between the plates far
(b)	If is taken as the potential _a that assumes the correct distribution as y -, plus a potential _b, what boundary conditions must be satisfied by _b?
(c)	What is the potential distribution between the plates?

(a)	Show that the boundary conditions that
(b)	Show that the potential is where (It is convenient to exploit the symmetry of the configuration about the plane x = a/2.)

(a)	Show that a particular solution is
(b)	Show that the total potential is the sum of this solution and that solution to Laplace's equation that makes the total solution satisfy the boundary conditions.
(c)	The force density (force per unit volume) acting on the charge is E. Show that the force f_x acting on a section of the charge of length in the x direction = 2 /k spanning the region -d/2 < y < d/2 and unit length in the z direction is

(a)	Find a particular solution that satisfies Poisson's equation
(b)	What boundary conditions must the homogeneous solution satisfy at y = d and y = 0?
(c)	Find in the region 0 < y < d.
(d)	The force density (force per unit volume) acting on the charge is E. Find the total force f_x acting on a section of the charge spanning the system from y = 0 to y = d, of unit length in the z direction and of length = 2 /k in the x direction.

(a)	Given that a particular solution inside the boundaries is
(b)	What is ?

(a)	Show that a particular solution in the region y < 0 is
(b)	There is no surface charge density in the plane y = 0. Show that the potential is

(a)	Find _a and _b in terms of the constants V,
(b)	Given that the force per unit area acting on the charge sheet is _s E_x (x, 0), what is the force acting on a section of the sheet having length d in the z direction and one wavelength 2 / in the x direction?
(c)	Now, the potential on the wall is made a traveling wave having a given angular frequency , (x, a, t) = V cos ( x - t), and the charge moves to the right with a velocity U, so that _s = _o sin (x - Ut - x_o), where U = /. Thus, the wall potentials and surface charge density move in synchronism. Building on the results from parts (a)-(b), what is the potential distribution and hence total force on the section of charged sheet?
(d)	What you have developed is a primitive model for an electron beam device used to convert the kinetic energy of the electrons (accelerated to the velocity v by a dc voltage) to high-frequency electrical power output. Because the system is free of dissipation, the electrical power output (through the electrode structure) is equal to the mechanical power input. Based on the force found in part (c), what is the electrical power output produced by one period 2 / of the charge sheet of width w?
(e)	For what values of x_o would the device act as a generator of electrical power?

(a)	If = 0 in the region r < R, what is in that
(b)	Over the range r < R, what is the surface charge density on the surface at y = 0?

(a)	What is in the annular region?
(b)	How large must V_b be to insure that all lines of E are outward directed from the inner cylinder?
(c)	What is the net charge per unit length on the inner cylinder under the conditions of (b)?

(a)	Given _o and _o, show that in regions (a) and
(b)	Show that the charge on the segment of the inner electrode attached to the resistor is where w is the length in the z direction.
(c)	Given that _o = \Omega t, show that the output voltage is related to \Omega by so that its amplitude can be used to measure \Omega.

(a)	Find all solutions in polar coordinates that satisfy
(b)	Match the boundary condition at r = R.
(c)	One of the terms in this solution has an electric field intensity that is infinite at the tip of the fin, where r = 0. Sketch and E in the neighborhood of the tip. What is the _s on the fin associated with this term as a function of r? What is the net charge associated with this term?
(d)	Sketch the potential and field intensity throughout the region.

(a)	With no charge density either outside
(b)	Sketch and E.

(a)	Show that if this is the only charge in the
(b)	Show that a plot of and E appears as shown in Fig. 6.3.1.