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Unformatted text preview: 7 The Electric Field in Various Circumstances
(Continued) 7—1 Methods for ﬁnding the electrostatic ﬁeld This chapter is a continuation of our consideration of the characteristics of
electric ﬁelds in various particular situations. We shall ﬁrst describe some of the
more elaborate methods for solving problems with conductors. It is not expected
that these more advanced methods can be mastered at this time. Yet it may be of
interest to have some idea about the kinds of problems that can be solved, using
techniques that may be learned in more advanced courses. Then we take up two
examples in which the charge distribution is neither ﬁxed nor is carried by a con
ductor, but instead is determined by some other law of physics. As we found in Chapter 6, the problem of the electrostatic ﬁeld 18 fundamen
tally simple when the distribution of charges is speciﬁed; it requires only the evalua
tion of an integral. When there are conductors present, however, complications
arise because the charge distribution on the conductors is not initially known;
the charge must distribute itself on the surface of the conductor in such a way that
the conductor is an equipotential. The solution of such problems is neither direct
nor simple. We have looked at an indirect method of solving such problems, in which we
ﬁnd the equipotentials for some speciﬁed charge distribution and replace one of
them by a conducting surface. In this way we can build up a catalog of special
solutions for conductors in the shapes of spheres, planes, etc. The use of images,
described in Chapter 6, is an example of an indirect method. We shall describe
another in this chapter. If the problem to be solved does not belong to the class of problems for which
we can construct solutions by the indirect method, we are forced to solve the prob
lem by a more direct method. The mathematical problem of the direct method is
the solution of Laplace’s equation, (7.1) subject to the condition that d: 15 a suitable constant on certain boundaries—the
surfaces of the conductors. Problems which involve the solution of a differential
ﬁeld equation subject to certain boundary conditions are called boundaryvalue
problems. They have been the object of considerable mathematical study. In
the case of conductors having complicated shapes, there are no general analytical
methods. Even such a simple problem as that of a charged cylindrical metal can
closed at both ends—a beer can—presents formidable mathematical difﬁculties.
It can be solved only approximately, using numerical methods. The only general
methods of solution are numerical. There are a few problems for which Eq. (7.1) can be solved directly. For
example, the problem of a charged conductor having the shape of an ellipsoid of
revolution can be solved exactly in terms of known special functions. The solution
for a thin disc can be obtained by letting the ellipsoid become inﬁnitely oblate.
In a similar manner, the solution for a needle can be obtained by letting the ellipsord
become inﬁnitely prolate. However, it must be stressed that the only direct methods
of general applicability are the numerical techniques. Boundaryvalue problems can also be solved by measurements of a physical
analog. Laplace’s equation arises in many different physical situations: in steady
state heat ﬂow. in irrotational ﬂuid ﬂow, in current flow in an extended medium, 7—I Methods for ﬁnding the
electrostatic ﬁeld Twodimensional ﬁelds;
functions of the complex
variable Plasma oscillations Colloidal particles in an
electrolyte The electrostatic ﬁeld of a grid and in the deﬂection of an elastic membrane. It is frequently possible to set up a
physical model which is analogous to an electrical problem which we wish to solve.
By the measurement of a suitable analogous quantity on the model, the solution
to the problem of interest can be determined. An example of the analog technique
is the use of the electrolytic tank for the solution of twodimensional problems in
electrostatics. This works because the differential equation for the potential in a
uniform conducting medium is the same as it is for a vacuum. There are many physical situations in which the variations of the physical
ﬁelds in one direction are zero, or can be neglected in comparison with the varia
tions in the other two directions. Such problems are called twodimensional; the
ﬁeld depends on two coordinates only. For example, if we place a long charged
wire along the zaxis, then for points not too far from the wire the electric ﬁeld
depends on x and y, but not on 2; the problem is twodimensional. Since in a two
dimensional problem a/az = 0, the equation for ¢> in free Space is 62¢ 62¢ _
W W _ o. (7.2) Because the twodimensional equation is comparatively simple, there is a wide
range of conditions under which it can be solved analytically. There is, in fact,
a very powerful indirect mathematical technique which depends on a theorem
from the mathematics of functions of a complex variable, and which we will now
describe. 72 Twodimensional ﬁelds; functions of the complex variable The complex variable a is deﬁned as 3=x+iy. (Do not confuse 3 with the zcoordinate, which we ignore in the following dis
cussion because we assume there is no zdependence of the ﬁelds.) Every point in
x and y then corresponds to a complex number 3. We can use a as a single
(complex) variable, and with it write the usual kinds of mathematical functions
F(a). For example, F0) = 32,
or F(3) = 1/33,
or 17(3) = 310g 3, and so forth.
Given any particular F(3) we can substitute 3 = x + iy, and we have a
function of x and y—with real and imaginary parts. For example, 32 = (x + iy)2 = x2 — y2 + 2ixy. (7.3) Any function F (a) can be written as a sum of a pure real part and a pure
imaginary part, each part a function of x and y: 17(3) = U(x,y) + iV(x,y), (74) where U(x, y) and V(x, y) are real functions. Thus from any complex function
F(a) two new functions U(x, y) and V(x, y) can be derived. For example, F (a) = 32
gives us the two functions U(x, y) = x2  yz. (75)
and
V(x, y) = 2xy. (7.6) Now we come to a miraculous mathematical theorem which is so delightful
that we shall leave a proof of it for one of your courses in mathematics. (We
should not reveal all the mysteries of mathematics, or that subject matter would 7—2 become too dull.) It is this. For any “ordinary function” (mathematicians will
deﬁne it better) the functions U and Vautomatz'cally satisfy the relations 6U~6V 5; _ .6}. (7.7)
aV 6U
57 _ ~67. (7.8) It follows immediately that each of the functions U and Vsatisfy Laplace’s equation: a2U 62U
3.72 + 372 = 0’ (79)
62V 62V
'5? + 3;? = 0 (7'10) These equations are clearly true for the functions of (7.5) and (7.6). Thus, starting with any ordinary function, we can arrive at two functions
U(x, y) and V(x, y), which are both solutions of Laplace’s equation in two dimen
sions. Each function represents a possible electrostatic potential. We can pick any
function F(a) and it should represent some electric ﬁeld problem—in fact, two
problems, because U and V each represent solutions. We can write down as many
solutions as we wish—by just making up functions—then we just have to ﬁnd the
problem that goes with each solution. It may sound backwards, but it’s a possible
approach. I I / y \
\
I \
/ / ’ l ' ‘ \
/ ‘4] \ \\
‘6 / _3 \ 6
'5 / l \ \ \
/ / / 2 \ \ ’ \ \ \
/ / I I '3 / / \ \ \ \
_ \ 2 \
— / BIl Bll \ \ \
, / / \
a / / \ \
_ _ , I A o no \
4 3 2 l A=l 4
3:0 _ x
\ \ Bil 8.0 BI’l /
~ \ \ / ,
\ \ AO A30 / /
\ \ \2 I /
\ Al—I /
x \ 3 / ’3/ / /
‘ _ /
~ \\\ \ 4 \ \ 2 I I 4— ’ / /
\ 5 \ \ I _ z
6 \ \ \ ‘3, I /
\ \ _ l
\ \ i 4 I I //
\ \ l l I I I
\ \ \ \ I I / Fig. 7]. Two sets of orthogonal curves which can represent
equipotentials in a twodimensional electrostatic ﬁeld. As an example, let’s see what physics the function F(3) = 32 gives us. From
it we get the two potential functions of (7.5) and (7.6). To see what problem the
function U belongs to, we solve for the equipotential surfaces by setting U = A,
a constant: This is the equation of a rectangular hyperbola. For various values of A, we get
the hyperbolas shown in Fig. 7—1. When A = O, we get the special case of diagonal
straight lines through the origin. Such a set of equipotentials corresponds to several possible physical situations.
First, it represents the ﬁne details of the ﬁeld near the point halfway between two 7—3 Fig. 7—3. lens. Fig. 7—2. The ﬁeld near the point C
is the some as that in Fig. 71. The field in a quadrupole equal point charges. Second, it represents the ﬁeld at an inside rightangle corner
of a conductor. If we have two electrodes shaped like those in Fig. 7—2, which are
held at different potentials, the ﬁeld near the corner marked C will look just like
the ﬁeld above the origin in Fig. 7—1. The solid lines are the equipotentials, and
the broken lines at right angles correspond to lines of E. Whereas at points or
protuberances the electric ﬁeld tends to be high, it tends to be low in dents or
hollows. The solution we have found also corresponds to that for a hyperbolashaped
electrode near a rightangle corner, or for two hyperbolas at suitable potentials.
You will notice that the ﬁeld of Fig. 7—1 has an interesting property. The xcom
ponent of the electric ﬁeld, E1, is given by The electric ﬁeld is proportional to the distance from the axis. This fact is used to
make devices (called quadrupole lenses) that are useful for focusing particle beams
(see Section 29—9). The desired ﬁeld is usually obtained by using four hyperbola
shaped electrodes, as shown in Fig. 7—3. For the electric ﬁeld lines in Fig. 7—3,
we have simply copied from Fig. 7—1 the set of brokenline curves that represent
V = constant. We have a bonus! The curves for V = constant are orthogonal
to the ones for U = constant because of the equations (7.7) and (7.8). Whenever
we choose a function F (a), we get from U and V both the equipotentials and ﬁeld
lines. And you Will remember that we have solved either of two problems, depend
ing on which set of curves we call the equipotentials.
As a second example, consider the function F(6) = x/g. (7.11)
If we write
a=x+iy=pe“’,
where
p = W2 + y2
and
tan 0 = y/x,
then = pllzeW/Z
2 pl/2 (COS g + [Sin 1
from which
2 21/2 1/2 2 2 1/2 _ 1/2
H3) z [9.7” yz) +1] + . (7_12) 7—4 The curves for U(x, y) = A and V(x, y) = B, using U and Vfrom Eq. (7.12),
are plotted in Fig. 7—4. Again, there are many possible situations that could be
described by these ﬁelds. One of the most interesting is the ﬁeld near the edge of a
thin plate. If the line B = 0——to the right of the yaxis—represents a thin charged
plate, the ﬁeld lines near it are given by the curves for various values of A. The
physical situation is shown in Fig. 7—5. Further examples are [7(3) = 23/2,, (7.13)
which yields the ﬁeld outside a rectangular corner 17(3) = log a, (714)
which yields the ﬁeld for a line charge, and F(a) = 1/3, (7.15) which gives the ﬁeld for the twodimensional analog of an electric dipole, i.e.,
two parallel line charges with opposite polarities, very close together. We will not pursue this subject further in this course, but should emphasize
that although the complex variable technique is often powerful, it is limited to
twodimensional problems; and also, it is an indirect method. 7—3 Plasma oscillations We consider now some physical situations in which the ﬁeld is determined
neither by ﬁxed charges nor by charges on conducting surfaces, but by a com
bination of two physical phenomena. In other words, the ﬁeld will be governed
simultaneously by two sets of equations: (1) the equations from electrostatics
relating electric ﬁelds to charge distribution, and (2) an equation from another
part of physics that determines the positions or motions of the charges in the
presence of the ﬁeld. The ﬁrst example that we will discuss is a dynamic one in which the motion
of the charges is governed by Newton’s laws. A simple example of such a situation
occurs in a plasma, which is an ionized gas consisting of ions and free electrons
distributed over a region in space. The ionosphere—an upper layer of the atmos
phere—is an example of such a plasma. The ultraviolet rays from the sun knock 7—5 Fig. 7—4. Curves of constant U(x, y)
and Vix, y) from Eq. (7.12). GROUNDED
PLATE Fig. 7—5. The electric ﬁeld near the
edge of a thin grounded plate. W2
and Fig. 7—6. Motion in a plasma wave.
The electrons at the plane a move to a',
and those at b move to b'. electrons off the molecules of the air, creating free electrons and ions. In such a
plasma the positive ions are very much heavier than the electrons, so we may
neglect the ionic motion, in comparison to that of the electrons. Let no be the density of electrons in the undisturbed, equilibrium state.
This must also be the densrty of positive ions, since the plasma is electrically
neutral (when undisturbed). Now we suppose that the electrons are somehow
moved from equilibrium and ask what happens. If the density of the electrons in
one region is increased, they will repel each other and tend to return to their
equilibrium positions. As the electrons move toward their original positions they
pick up kinetic energy, and instead of coming to rest in their equilibrium conﬁgura
tion, they overshoot the mark. They will oscillate back and forth. The situation
is similar to what occurs in sound waves, in which the restoring force is the gas
pressure. In a plasma, the restoring force is the electrical force on the electrons. To simplify the discussion, we will worry only about a situation in which the
motions are all in one dimensron, say x. Let us suppose that the electrons origi
nally at x are, at the instant I, displaced from their equilibrium positions by a small
amount s(x, 1). Since the electrons have been displaced, their density will, in general,
be changed. The change in density is easily calculated. Referring to Fig. 7—6.
the electrons initially contained between the two planes a and b have moved and
are now contained between the planes 0’ and b’. The number of electrons that
were between a and b is proportional to noAx; the same number are now contained
in the space whose width is Ax + As. The density has changed to noAx n0 _ Ax + As : l—+ (As/Ax). (7'16) It If the change in density is small, we can write [using the binomial expansion for (l + E)‘]]
As
" = "0(1 " We assume that the pOSlllVC ions do not move appreciably (because of the much
larger inertia), so their density remains no. Each electron carries the charge —q,.,
so the average charge density at any point is given by OH) p = ~(rl — new.» 01' ds p = ngqe a; (where we have written the differential form for As/Ax).
The charge density is related to the electric ﬁeld by Maxwell‘s equations. in
particular,
V . E = fl . (7.19)
60
If the problem is indeed onedimensional (and if there are no other ﬁelds but the one due to the displacements of the electrons), the electric ﬁeld E has a single
component [5,. Equation (7.19), together with (7.18), gives 6E1 noqe 0s 3; ~ 60 b}. (7.20)
Integrating Eq. (7.20) gives a=?Ws+K. 0m)
60
Since E1 = 0 when s = 0, the integration constant K is zero.
The force on an electron in the displaced position is
"043
F, = — s, (7.22)
60 7~6 a restoring force proportional to the displacements of the electron. This leads to
a harmonic oscillation of the electrons. The equation of motion of a displaced
electron is dzs _ noqe2 m —— _.
edtz 60 s. (7.23) We ﬁnd that s will vary harmonically. Its time variation will be as cos wt, or—
using the exponential notation of Vol. I——as e‘W‘. (7.24) The frequency of oscillation (0,, is determined from (7 .23): 2
2 _ not):
(up — 60mg, (7.25) and is called the plasma frequency. It is a characteristic number of the plasma.
When dealing with electron charges many people prefer to express their an
SWers in terms of a quantity e2 deﬁned by 2
e2 = 4% = 2.3068 X 1028 newtonmeterz. (726)
W60 Using this convention, Eq. (7.25) becomes 2 41re2no
wp =
m. , (7.27) which is the form you will ﬁnd in most books. Thus we have found that a disturbance of a plasma will set up free oscillations
of the electrons about their equilibrium positions at the natural frequency wp,
which is proportional to the square root of the density of the electrons. The plasma
electrons behave like a resonant system, such as those we described in Chapter
23 of Vol. I. This natural resonance of a plasma has some interesting effects. For example,
if one tries to propagate a radiowave through the ionosphere, one ﬁnds that it
can penetrate only if its frequency is higher than the plasma frequency. Otherwise
the signal is reﬂected back. We must use high frequencies if we wish to communi
cate with a satellite in space. On the other hand, if we wish to communicate with
a radio station beyond the horizon, we must use frequencies lower than the plasma
frequency, so that the signal will be reﬂected back to the earth. Another interesting example of plasma oscillations occurs in metals. In a
metal we have a contained plasma of positive ions, and free electrons. The density
no is very high, so up is also. But it should still be possible to observe the electron
oscillations. Now, according to quantum mechanics, a harmonic oscillator with
a natural frequency to, has energy levels which are separated by the the energy
increment hwp. If, then, one shoots electrons through, say, an aluminum foil, and
makes very careful measurements of the electron energies on the other side, one
might expect to ﬁnd that the electrons sometimes lose the energy ﬂap to the plasma
oscillations. This does indeed happen. It was ﬁrst observed experimentally in
1936 that electrons with energies of a few hundred to a few thousand electron volts
lost energy in jumps when scattering from or going through a thin metal foil. The
cﬂ'ect was not understood until 1953 When Bohm and Pines“ showed that the
observations could be explained in terms of quantum excitations of the plasma
oscillations in the metal. * For some recent work and a bibliography see C. 1. Powell and J. B. Swarm, Phys.
Rev. 115, 869 (1959). 7—7 74 Colloidal particles in an electrolyte We turn to another phenomenon in which the locations of charges is governed
by a potential that arises in part from the same charges. The resulting effects
inﬂuence in an important way the behavror of colloids. A colloid consists of a
suspension in water of small charged partICles which, though microscopic, from
an atomic point of View are still very large. If the colloidal particles were not
charged, they would tend to coagulate into large lumps: but because of their
charge, they repel each other and remain in suspenswn. Now if there is also some salt dissolved in the water, it will be dissociated into
positive and negative ions. (Such a solution of ions is called an electrolyte.) The
negative ions are attracted to the colloid particles (assuming their charge is positive)
and the positive ions are repelled. We will determine how the ions which surround
such a colloidal particle are distributed in space. To keep the ideas simple, we will again solve only a onedimensional case.
If we think of a colloidal particle as a sphere having a very large radius—on an
atomic scale!——we can then treat a small part of its surface as a plane. (Whenever
one is trying to understand a new phenomenon it is a good idea to take a somewhat
oversimpliﬁed model; then, having understood the problem with that model, one
is better able to proceed to tackle the more exact calculation.) We suppose that the distribution of ions generates a charge density p(x), and
an electrical potential ¢, related by the electrostatic law V2¢ = —p/e[, or, for
ﬁelds that vary in only one dimension, by d2¢ _ 9
dx, _ —;B. (7.28) Now supposing there were such a potential p(x), how would the ions dis
tribute themselves in it? This we can determine by the principles of statistical
mechanics. Our problem then is to determine ¢ so that the resulting charge density
from statistical mechanics also satisﬁes (7.28). According to statistical mechanics (see Chapter 40, Vol. I), particles in thermal
equilibrium in a force ﬁeld are distributed in such a way that the density n of
particles at the position x is given by n(x) = noerm’”, (7.29) where U(x) is the potential energy, k is Boltzmann’s constant, and Tis the absolute
temperature. We assume that the ions carry one electronic charge, positive or negative.
At the distance x from the surface of a colloidal particle, a positive ion will have
potential energy qe¢(x), so that U (x) = qe¢(x) The density of positive ions, 11+, is then n+(x) = noe—qmsz. Similarly, the densrty of negative ions is
"*(x) = "OEH19“me
The total charge density is p = qen+ — (ML,
or
p = qenaeWT—Wm”). (7.30) Combining this with Eq. (7.28), we ﬁnd that the potential dz must satisfy 2 . I!
d ¢ qgno (e—qe¢/k1_e+Qe¢/kl)_ (7.31) — 60 This equation is readily solved in general [multiply both sides by 2(d¢/dx), and
integrate with respect to x], but to keep the problem as simple as possible, we will
consider here only the limiting case in which the potentials are small or the tem
perature T is high. The case where «p is small corresponds to a dilute solution. For
these cases the exponent is small, and we can approximate eWI’” = 1 i 1%. (7.32)
Equation (7.31) then gives d 2 .2 as = + .133. «m Notice that this time the sign on the right is positive. The solutions for ¢ are not
oscillatory, but exponential.
The general solution of Eq. (7.33) is ¢ = Art/0 + Be+"/D, (7.34)
with
D2 = (7.35) The constants A and B must be determined from the conditions of the problem.
In our case, B must be zero; otherwise the potential would go to inﬁnity for large x. So we have that
¢ = Art/D, (7.36) in which A is the potential at x = 0, the surface of the colloidal particle. Fig. 7~7. The variation of the po
tential near the surface of a colloidal
particle. D is the Debye length. O D D 30 x The potential decreases by a factor l/e each time the distance increases by D,
as shown in the graph of Fig. 7—7. The number D is called the Debye length, and
is a measure of the thickness of the ion sheath that surrounds a large charged
particle in an electrolyte. Equation (7.36) says that the sheath gets thinner with
increasing concentration of the ions (no) or with decreasing temperature. The constant A in Eq. (7.36) is easily obtained if we know the surface charge a
on the colloid particle. We know that E. = E.(0) = (7.37)
But E is also the gradient of 4n
_ _ 21> _ é
Ex(0) — ax O — + D, (7.38)
from which we get
A = ‘T—Dm (7.39)
6o 7—9 Using this result in (7.36), we ﬁnd (by taking x = 0) that the potential of the
colloidal particle is (ID
6—0. ¢(0) = (7.40)
You will notice that this potential is the same as the potential difference across a
condenser with a plate spacing D and a surface charge density a. We have said that the colloidal particles are kept apart by their electrical
repulsion. But now we see that the ﬁeld a little way from the surface of a particle
18 reduced by the ion sheath that collects around it. If the sheaths get thin enough,
the particles have a good chance of knocking against each other. They will then
stick, and the colloid will coagulate and precipitate out of the liquid. From our
analysis, we understand why adding enough salt to a colloid should cause it to
precipitate out. The process is called “salting out a colloid.” Another interesting example is the effect that a salt solution has on protein
molecules. A protein molecule is a long, complicated, and ﬂexible chain of amino
acids. The molecule has various charges on it, and it sometimes happens that
there is a net charge, say negative, which is distributed along the chain. Because
of mutual repulsion of the negative charges, the protein chain is kept stretched out.
Also, if there are other similar chain molecules present in the solution, they will
be kept apart by the same repulsive effects. We can, therefore, have a suspension
of chain molecules in a liquid. But if we add salt to the liquid we change the proper
ties of the suspension. As salt is added to the solution, decreasing the Debye
distance, the chain molecules can approach one another, and can also coil up.
If enough salt is added to the solution, the chain molecules will precipitate out of
the solution. There are many chemical effects of this kind that can be understood
in terms of electrical forces. 7—5 The electrostatic ﬁeld of a grid As our last example, we would like to describe another interesting property
of electric ﬁelds. It is one which is made use of in the design of electrical instru
ments, in the construction of vacuum tubes, and for other purposes. This is the
character of the electric ﬁeld near a grid of charged wires. To make the problem
as simple as possible, let us consider an array of parallel wires lying in a plane,
the wires being inﬁnitely long and with a uniform spacing between them. If we look at the ﬁeld a large distance above the plane of the wires, we see a
constant electric ﬁeld, just as though the charge were uniformly spread over a
plane. As we approach the grid of wires, the ﬁeld begins to deviate from the
uniform ﬁeld we found at large distances from the grid. We would like to estimate
how close to the grid we have to be in order to see appreciable variations in the
potential. Figure 7—8 shows a rough sketch of the equipotentials at various
distances from the grid. The closer we get to the grid, the larger the variations.
As we travel parallel to the grid, we observe that the ﬁeld ﬂuctuates in a periodic
manner. _——————————_——_ I \ / \ I ‘ \ " / \
:1 :a :i l.) (0‘ .0,
q.
T . *
kO—u'
—.
X Fig. 7—8. Equipotential surfaces
above a uniform grid of charged wires. 7—10 Now we have seen (Chapter 50, Vol. I) that any periodic quantity can be
expressed as a sum of sine waves (Fourier’s theorem). Let’s see if we can ﬁnd a
suitable harmonic function that satisﬁes our ﬁeld equations. If the wires lie in the xyplane and run parallel to the yaxis, then we might
try terms like 21rnx
a ¢(x,z) = Fn(z)cos , (7.41)
where a is the spacing of the wires and n is the harmonic number. (We have as
sumed long wires, so there should be no variation with y.) A complete solution
would be made up of a sum of such terms for n = 1, 2, 3, . . . . If this is to be a valid potential, it must satisfy Laplace’s equation in the
region above the wires (where there are no charges). That is, gas 92.,
6x2 622_' Trying this equation on the ¢ in (7.41), we ﬁnd that 4 2 2 2 d2F,, 2
— 7;,” Fn(z) cos 7:“ + dzg cos 7:“ = o, (7.42) or that F,,(z) must satisfy d251, 4 2 2
(122 = :2" F". (7.43)
So we must have
F" = Aug—“"20, (7.44)
where
a
20 = 2—7m ' We have found that if there is a Fourier component of the ﬁeld of harmonic n,
that component will decrease exponentially with a characteristic distance 20 =
a/27m. For the ﬁrst harmonic (n = 1), the amplitude falls by the factor e‘2’r
(a large decrease) each time we increase 2 by one grid spacing a. The other har
monics fall Off even more rapidly as we move away from the grid. We see that if
we are only a few times the distance a away from the grid, the ﬁeld is very nearly
uniform, i.e., the oscillating terms are small. There would, of course, always
remain the “zero harmonic” ﬁeld ¢o = "E02 to give the uniform ﬁeld at large 2. For a complete solution, we would combine
this term with a sum of terms like (7.41) with Fn from (7.44). The coefficients A”
would be adjusted so that the total sum would, when differentiated, give an electric
ﬁeld that would ﬁt the charge density A of the grid wires. The method we have just developed can be used to explain why electrostatic
shielding by means of a screen is often just as good as with a solid metal sheet.
Except within a distance from the screen a few times the spacing of the screen
wires, the ﬁelds inside a closed screen are zero. We see why copper screen—
lighter and cheaper than copper sheet—is often used to shield sensitive electrical
equipment from external disturbing ﬁelds. 7—11 ...
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 Chaikin
 Physics, Laplace, Electric charge

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