book
A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 17 of 28
1 January 1881
and the curves intersect at right angles.
If we suppose the potential F= V^ + lca, where k is some con- stant, then Twill satisfy Laplace's equation, and the curves (a) will be equipotential curves. The curves (/3) will be lines of force, and
264 CONJUGATE FUNCTIONS. [184.
the surface-integral of R over unit-length of a cylindrical surface whose projection on the plane of xy is the curve AB will be Jc(ftB — /3^), where ft A and ftB are the values of ft at the extremities of the curve.
If one series of curves corresponding to values of a in arithmetical progression be drawn on the plane, and another series corresponding to a series of values of ft having the same common difference, then the two series of curves will everywhere intersect at right angles, and, if the common difference is small enough, the elements into which the plane is divided will be ultimately little squares, whose sides, in different parts of the field, are in different directions and of different magnitudes, being inversely proportional to R.
If two or more of the equipotential lines (a) are closed curves enclosing a continuous space between them, we may take these for the surfaces of conductors at potentials (^o + ^i)* (^o + ^a2)> &c- respectively. The quantity of electricity upon any one of these be-
Jc tween the lines offeree ft and /32 will be — (ftz—ft)'
The number of equipotential lines between two conductors will therefore indicate their difference of potential, and the number of lines of force which emerge from a conductor will indicate the quantity of electricity upon it.
We must next state some of the most important theorems relating to conjugate functions, and in proving them we may use either the equations (l), containing the differential coefficients, or the original definition, which makes use of imaginary symbols.
184.] THEOKEM I. Ifx and y' are conjugate functions with respect to x and y> and if x" and y" are also conjugate functions with respect to x and y> then the functions x' -f x" and y' +y" will be conjugate functions with respect to x and y.
dx' dy' . dx" dy" For — - = -f- , and -=- = -f- ;
dx dy dx dy
therefore
dx dy
dy' dx" df
Also T- = — -T-J and T-= -- 7-9
dy dx dy dx
dx'
therefore
dy dx
or x + x" and yr --y" are conjugate with respect to x and y.
185.] GKAPHIC METHOD. 265
Graphic Representation of a Function which is the Sum of Two Given Functions.
Let a function (a) of x and y be graphically represented by a series of curves in the plane of xy^ each of these curves corre- sponding to a value of a which belongs to a series of such values increasing by a common difference, 8.
Let any other function, /3, of x and y be represented in the same way by a series of curves corresponding to a series of values of ft having the same common difference as those of a.
Then to represent the function a + ft in the same way, we must draw a series of curves through the intersections of the two former series, from the intersection of the curves (a) and (£) to that of the curves (a + 8) and (ft — 8), then through the intersection of (a + 2 5) and (/3 — 28), and so on. At each of these points the function will have the same value, namely a + /3. The next curve must be drawn through the points of intersection of (a) and ((3 + 8), of (a + 8) and (/3), of (a + 2 8) and (/3 — 8), and so on. The function belonging to this curve will be a -f ft -f 8.
In this way, when the series of curves (a) and the series (/3) are drawn, the series (a + /3) may be constructed. These three series of curves may be drawn on separate pieces of transparent paper, and when the first and second have been properly superposed, the third may be drawn.
The combination of conjugate functions by addition in this way enables us to draw figures of many interesting cases with very little trouble when we know how to draw the simpler cases of which they are compounded. We have, however, a far more powerful method of transformation of solutions, depending on the following theorem.
185.] THEOREM II. If x" and y" are conjugate functions with respect to the variables x' and y\ and if x' and y' are conjugate functions with respect to x and y, then x" and y" will be con- jugate functions with respect to x and y.
dx" dx" dx' dx" dy'
dy" dy dy" dx'
dy dy dx' dy
266 CONJUGATE FUNCTIONS. [185.
dx" _ dx" daf dx" dyf
dy dx' dy dy' dy
_ _
dy' dx dx' dx
dx '
and these are the conditions that x" and y" should be conjugate
functions of x and y.
This may also be shewn from the original definition of conjugate
functions. For x" -\- \/ — \y" is a function of xf + */ — ly, and
#'+ \/— \y is a function of #+ \/— ly. Hence, #"+\/— \y"
is a function of #-f \/ — \y.
In the same way we may shew that if x' and y are conjugate
functions of x and y, then a? and y are conjugate functions of x
and y'.
This theorem may be interpreted graphically as follows : —
Let x ', y' be taken as rectangular coordinates, and let the curves
corresponding to values of x" and of y" taken in regular arithmetical
series be drawn on paper. A double system of curves will thus be
drawn cutting the paper into little squares. Let the paper be also
ruled with horizontal and vertical lines at equal intervals, and let
these lines be marked with the corresponding values of xf and yf.
Next, let another piece of paper be taken in which x and y are
made rectangular coordinates and a double system of curves #', y'
is drawn, each curve being marked with the corresponding value
of x' or y '. This system of curvilinear coordinates will correspond,
point for point, to the rectilinear system of coordinates x', y' on the
first piece of paper.
Hence, if we take any number of points on the curve x" on the
first paper, and note the values of x' and yf at these points, and
mark the corresponding points on the second paper, we shall find
a number of points on the transformed curve x" . If we do the
same for all the curves x" ', y" on the first paper, we shall obtain on
the second paper a double series of curves as", y" of a different form,
but having the same property of cutting the paper into little
squares.
i86.]
THEOREMS.
267
186.] THEOHEM III. If V is any function of x' and /, and if x'
and y' are conjugate functions of x and y, then
dx2 r di
the integration being between the same limits.
For -7— = -=-> -= — | — Y~f ~T- »
dx dx dx dy dx
dx' dy
dx'dy' dx dx dy'2 dx
dx' dx2 dy' dx* '
and
d
-. . daf_dtf_ d2Fdy'*
dx'dy dy dy dy'2 dy
f dx dy2 4 dy'
Adding the last two equations, and remembering- the conditions
of conjugate functions (l), we find
dx
•
Hence
,u- r ,u/~r \ (d® dy' dx' dy'\
= ('dx72 d/*' \dx ~dy "" ~dy ~dx'
r f?2Fx77 rrfd*r d^fdx'dy' dx'dv\
+ -) dxdy =(+(- dfdy>
=(d^ + d> y-
If F is a potential, then, by Poisson's equation
and we may write the result
or the quantity of electricity in corresponding portions of two sys-
tems is the same if the coordinates of one system are conjugate
functions of those of the other.
268 CONJUGATE FUNCTIONS. [187.
Additional Theorems on Conjugate Functions.
187.] THEOREM IV. If xl and ylt and also x.2 and y^ are eon-
jugate functions of x and y, then, if
X=xlx2—yly^ and Y = xly2 + x2yl,
X and Y will be conjugate functions of x and y.
For X+ V^lT = fo + V^T^) (#2 + V^T y2).
THEOREM V. If $ be a solution of the equation
_
dx* d ~
7 '^ n -n ^ fd$ d(h \ , , dx
and if 2 It = log ( -~ + -~ ) > and 0 = — tan-1 — >
\dx dy ' d§
dy
R and 0 will le conjugate functions of x and y.
For H and 0 are conjugate functions of ~- and — , and these
, d. fdf^are conjugate functions of x and y.
EXAMPLE I. — Inversion.
188.] As an example of the general method of transformation
let us take the case of inversion in two dimensions.
If 0 is a fixed point in a plane, and OA a fixed direction, and
if r = OP = aef>, and 0 = AOP, and if #, y are the rectangular
coordinates of P with respect to 0,
tf^tan-1^, ) /5x
x = ae? cos 0, y — ae? sin 0, )
p and 0 are conjugate functions of x and y.
If p'= np and 0'= n0, p' and 6' will be conjugate functions of p
and 0. In the case in which n = — 1 we have
a2
= -, and 0'=-0, (6)
which is the case of ordinary inversion combined with turning the
figure 1 80° round OA.
&_
Inversion in Two Dimensions.
In this case if ;• and / represent the distances of corresponding
points from 0, e and ef the total electrification of a body, S and S'
superficial elements, V and V solid elements, o- and </ surface-
i89.]
ELECTRIC IMAGES IN TWO DIMENSIONS.
2G9
densities, p and p' volume densities, $ and <£' corresponding po-
tentials,
/2
EXAMPLE II. — Electric Images in Two Dimensions.
189.] Let A be the centre of a circle of radius AQ = #, and let
E be a charge at A, then the potential
at any point P is
I
d> = 2JZnog-r7r; (8)
4r*
and if the circle is a section of a hollow
conducting cylinder, the surface- density
at any point Q is — -— y • FiS> V-
27TO
Invert the system with respect to a point Oy making
AO = mb, and a2 = (m2—
then we have a charge at A equal to that at A, where AA'=.
m
The density at Q' ig
AQ'2
and the potential at any point P' within the circle is
4/ = 0 = 2 ^ (log 5 -log AP),
= 2E (log OP'- log AP* - log w). (9)
This is equivalent to a combination of a charge E at ^', and a
charge — E at 0, which is the image of A, with respect to the
circle. The imaginary charge at 0 is equal and opposite to that
at^'.
If the point P' is defined by its polar coordinates referred to the
centre of the circle, and if we put
p = logr— log 3, and p0 = log A A'— logd,
then AP"= be^ AA'= be">t
and the potential at the point (p, 6) is
<J> = E log (e-2<>°— 2 6?~po e? cos 6 -f e2?)
AO =
(10)
(11)
This is the potential at the point (p, 6) due to a charge E, placed
at the point (p0; 0), with the condition that when p = 0, $ = 0.
270 CONJUGATE FUNCTIONS. [190.
In this case p and 6 are the conjugate functions in equations (5) :
p is the logarithm of the ratio of the radius vector of a point to
the radius of the circle, and 6 is an angle.
The centre is the only singular point in this system of coordinates,
and the line-integral of / -=- ds round a closed curve is zero or 2 TT,
according as the closed curve excludes or includes the centre.
EXAMPLE III. — Neumann's Transformation of this Case*.
190.] Now let a and ft be any conjugate functions of x and y^
such that the curves (a) are equipotential curves, and the curves
(/3) are lines of force due to a system consisting of a charge of half
a unit at the origin, and an electrified system disposed in any
manner at a certain distance from the origin.
Let us suppose that the curve for which the potential is a0 is
a closed curve, such that no part of the electrified system except the
half-unit at the origin lies within this curve.
Then all the curves (a) between this curve and the origin will be
closed curves surrounding the origin, and all the curves (/8) will
meet in the origin, and will cut the curves (a) orthogonally.
The coordinates of any point within the curve (a0) will be deter-
mined by the values of a and ft at that point, and if the point
travels round one of the curves (a) in the positive direction, the
value of ft will increase by 2 ir for each complete circuit.
If we now suppose the curve (a0) to be the section of the inner
surface of a hollow cylinder of any form maintained at potential
zero under the influence of a charge of linear density ~E on a line of
which the origin is the projection, then we may leave the external
electrified system out of consideration, and we have for the potential
at any point (a) within the curve
4> = 2^(a-a0), (12)
and for the quantity of electricity on any part of the curve a0
between the points corresponding to ft1 and /32,
If in this way, or in any other, we have determined the dis-
tribution of potential for the case of a given curve of section when
the charge is placed at a given point taken as origin, we may pass
to the case in which the charge is placed at any other point by an
application of the general method of transformation.
* See Crelle's Journal, 1861.
NEUMANN'S TRANSFORMATION. 271
Let the values of a and ft for the point at which the charge is
placed be 04 and 01$ then substituting in equation (ll) a— a0 for p,
and 0—0J for 6, we find for the potential at any point whose co-
ordinates are a and 0,
$ = Slog (I— 2tfa+ai-2aocos(0— 01) + e2(«+ai-2«o))
-Elog(l-2e^co*(0-p1) + e*(*-^)-2E(a1--a0). (14)
This expression for the potential becomes zero when a = a0, and is
finite and continuous within the curve a0 except at the point (al5 0j),
at which point the second term becomes infinite, and in its immediate
neighbourhood is ultimately equal to — 2.#log/, where / is the
distance from that point.
We have therefore obtained the means of deducing the solution
of Green's problem for a charge at any point within a closed curve
when the solution for a charge at any other point is known.
The charge induced upon an element of the curve a0 between the
points 0 and 0-M0 by a charge E placed at the point (al5 0X) is,
with the notation of Art. 183,
n ,
where dsl is measured inwards and a is to be put equal to a0 after
differentiation.
This becomes, by (4) of Art. 183,
_ -go _
" 2^ 1 - 2 <?(«i-o) cos (0-00 + 62(«i-o) **'
From this expression we may find the potential at any point
(al5 0t) within the closed curve, when the value of the potential at
every point of the closed curve is given as a function of 0, and
there is no electrification within the closed curve.
For, by Art. 86, the part of the potential at (a15 0^, due to the
maintenance of the portion d/3 of the closed curve at the potential
V is n V, where n is the charge induced on d@ by unit of electri-
fication at (at, 0!). Hence, if Pis the potential at a point on the
closed curve defined as a function of 0, and (f> the potential at
the point (al5 0X) within the closed curve, there being no electri-
fication within the curve,
1 f" (l-e^-^)Fdft
9~ 27T./0 l-2e'°i-''o>cos/3-/31 + e2(''i—<>>
272 CONJUGATE FUNCTIONS. [191.
EXAMPLE IV. — Distribution of Electricity near an Edge of a
Conductor formed by Two Plane Faces.
191.] In the case of an infinite plane face of a conductor charged
with electricity to the surface-density <70, we find for the potential
at a distance y from the plane
where C is the value of the potential of the conductor itself.
Assume a straight line in the plane as a polar axis, and transform
into polar coordinates, and we find for the potential
7= C—l-no-QaePsinO,
and for the quantity of electricity on a parallelogram of breadth
unity, and length ae? measured from the axis
E = <T^ae?.
Now let us make p = np' and 0 = nb', then, since // and & are
conjugate to p and 0, the equations
V = C— 4 TT <TO aen?' sin n 0'
and E=.
express a possible distribution of electricity and of potential.
If we write r for ae?, r will be the distance from the axis ; we
may also put 6 instead of 6' for the angle. We shall have
rn
V— C— 4 TT <TO -^q sin n 9,
a
V will be equal to C whenever n 0 = 77 or a multiple of IT.
Let the edge be a salient angle of the conductor, the inclination
of the faces being a, then the angle of the dielectric is 2 IT — a, so
that when 6 = 27T— a the point is in the other face of the con-
ductor. We must therefore make
/ \
n(27r — a) = KJ or n =
27T — a
Then F= C— 4ir(rna ( ~] sin
27T — a
2 77— a
The surface-density o- at any distance r from the edge is
a — TT
dE TT
I92-] ELLIPSES AND HYPERBOLAS. 273
When the angle is a salient one a is less than TT, and the surface-
density varies according to some inverse power of the distance
from the edge, so that at the edge itself the density becomes
infinite, although the whole charge reckoned from the edge to any
finite distance from it is always finite.
Thus, when a = 0 the edge is infinitely sharp, like the edge of a
mathematical plane. In this case the density varies inversely as
the square root of the distance from the edge.
When a = - the edge is like that of an equilateral prism, and
o
the density varies inversely as the f power of the distance.
When a = - the edge is a right angle, and the density is in-
2
versely as the cube root of the distance.
o _
When a = -- the edge is like that of a regular hexagonal prism,
3
and the density is inversely as the fourth root of the distance.
When a = TT the edge is obliterated, and the density is constant.
When a = f TT the edge is like that in the inside of the hexagonal
prism, and the density is directly as the square root of the distance
from the edge.
When a = f TT the edge is a re-entrant right angle, and the
density is directly as the distance from the edge.
When a = 1 77 the edge is a re-entrant angle of 60°, and the
density is directly as the square of the distance from the edge.
In reality, in all cases in which the density becomes infinite at
any point, there is a discharge of electricity into the dielectric at
that point, as is explained in Art. 55.
EXAMPLE V. — Ellipses and Hyperbolas. Fig. X.
192.] We have seen that if
ttj = e* cos fa y± — e* sin fa (1)
x and y will be conjugate functions of $ and \j/.
Also> if #2 = e~* cos fa y^ — —e~^ sin fa (2)
#2 and y2 will be conjugate functions. Hence, if
2a> = as1 + a:2 = (e+ + e-+)co8fa 1y = ft+ft = (e+—e~+) sinfa (3)
x and y will also be conjugate functions of <£ and •$.
In this case the points for which </> is constant lie in the ellipse
whose axes are e*-\- er* and e$—e~$.
VOL. I. T
274 CONJUGATE FUNCTIONS. [I93-
The points for which ^ is constant lie in the hyperbola whose
axes are 2 cos \jr and 2 sin ty.
On the axis of a, between x=. — I and #= + 1 ,
$ = 0, \j/ as cos-1^-. (4)
On the axis of #, beyond these limits on either side, we have
# > 1, \jf = 0, 0 = log (#+ \A?2— 1), (5)
X< — 1, \/r = 7T, 0 = log ( >/#2 — 1 — 5?).
Hence, if </> is the potential function, and \j/ the function of flow,
we have the case of electricity flowing from the positive to the
negative side of the axis of x through the space between the points
— 1 and 4- 1 , the parts of the axis beyond these limits being
impervious to electricity.
Since, in this case, the axis of y is a line of flow, we may suppose
it also impervious to electricity.
We may also consider the ellipses to be sections of the equi-
potential surfaces due to an indefinitely long flat conductor of
breadth 2, charged with half a unit of electricity per unit of length.
If we make \j/ the potential function, and <£ the function of flow,
the case becomes that of an infinite plane from which a strip of
breadth 2 has been cut away and the plane on one side charged to
potential IT while the other remains at zero.
These cases may be considered as particular cases of the quadric
surfaces treated of in Chapter X. The forms of the curves are
given in Fig. X.
EXAMPLE VI.— Fig. XI.
193.] Let us next consider of and y as functions of x and ^, where
of— I log Va? +/, / = I tan- l*-, (6)
x
x and y will be also conjugate functions of <£ and \jf.
The curves resulting from the transformation of Fig. X with
respect to these new coordinates are given in Fig. XI.
If x' and y' are rectangular coordinates, then the properties of the
axis of x in the first figure will belong to a series of lines parallel
to x in the second figure for which y ' = bri 'TT, where n' is any
integer.
The positive values of x on these lines will correspond to values
of x greater than unity, for which, as we have already seen,
*' ,~^ v
6+v eb _ iJ. 7
I93-] PARTICULAR CASE OF CONJUGATE FUNCTIONS. 275
The negative values of a?' on the same lines will correspond to
values of x less than unity, for which, as we have seen,
a/
<£ = 0, \l/ = cos"1^ = cos~leb. (8)
The properties of the axis of y in the first figure will belong to a
series of lines in the second figure parallel to #', for which
/= a* (»'+!). (9)
The value of \j/ along these lines is \j/ = * (n + £) for all points
both positive and negative, and
/ ^ tw_ \
* = log(y+ Vy+1) = \og\eb + V eb + i/, (10)
[The curves for which <£ and \j/ are constant may be traced
directly from the equations
As the figure repeats itself for intervals of iib in the values o
it will be sufficient to trace the lines for one such interval.
Now there will be two cases, according as <£ or ^ changes sign
with y'. Let us suppose that 0 so changes sign. Then any curve
for which \fr is constant will be symmetrical about the axis of #',
cutting that axis orthogonally at some point on its negative side.
If we begin with this point for which <£ = 0, and gradually in-
crease <£, the curve will bend round from being at first orthogonal
to being, for large values of $, at length parallel to the axis of of.
The positive side of the axis of ai is one of the system, viz. ty is
there zero, and when/= + ITT^, ^ = JTT. The lines for which \l/
has constant values ranging from 0 to JTT form therefore a system
of curves embracing the positive side of the axes of x'.
The curves for which </> has constant values cut the system \^
orthogonally, the values of </> ranging from +00 to -co. For
any one of the curves 0 drawn above the axis of x the value of $ is
positive, along the negative side of the axis of x the value is zero,
and for any curve below the axis of af the value is negative.
We have seen that the system \js is symmetrical about the axis
of #; let PQR be any curve cutting that system orthogonally and
terminating in P and R in the lines /= + \-nb, the point Q being
in the axis of %'. Then the curve PQR is symmetrical about the axis
of #', but if c be the value of </> along PQ, the value of <£ along QR
will be — c. This discontinuity in the value of $ will be accounted
T a
276 CONJUGATE FUNCTIONS. [194.
for by an electrical distribution in the case which will be discussed
in Art. 195.
If we next suppose that \j/ and not $ changes sign with y', the
values of <p will range from 0 to oo . When <$> = 0 we have the
negative side of the axis of af, and when <£ = oo we have a line
at an infinite distance perpendicular to the axis of af '. Along any
line PQR between these two the value of $ is constant throughout
its entire length and positive.
Any value \jf now experiences an abrupt change at the point
where the curve along which it is constant crosses the negative
side of the axis of #', the sign of ^ changing there. The sig-
nificance of this discontinuity will appear in Art. 197.
The lines we have shewn how to trace are drawn in Fig. XI
if we limit ourselves to two-thirds of that diagram, cutting off the
uppermost third.]
194.] If we consider </> as the potential function, and \js as the
function of flow, we may consider the case to be that of an in-
definitely long strip of metal of breadth 116 with a non-conducting
division extending from the origin indefinitely in the positive
direction, and thus dividing the positive part of the strip into two
separate channels. We may suppose this division to be a narrow
slit in the sheet of metal.
If a current of electricity is made to flow along one of these
divisions and back again along the other, the entrance and exit of
the current being at an indefinite distance on the positive side of
the origin, the distribution of potential and of current will be given
by the functions </> and x//- respectively.
If, on the other hand, we make \js the potential, and $ the
function of flow, then the case will be that of a current in the
general direction of y', flowing through a sheet in which a number
of non-conducting divisions are placed parallel to #', extending from
the axis of yr to an indefinite distance in the negative direction.
195.] We may also apply the results to two important cases in
statical electricity.
(1) Let a conductor in the form of a plane sheet, bounded by a
straight edge but otherwise unlimited, be placed in the plane of xz
on the positive side of the origin, and let two infinite conducting
planes be placed parallel to it and at distances %irb on either side.
Then, if \]/ is the potential function, its value is 0 for the middle
conductor and J TT for the two planes.
Let us consider the quantity of electricity on a part of the middle
EDGE OF AN ELECTRIFIED PLATE. 277
conductor, extending to a distance 1 in the direction of *, and from
the origin to #'= a.
The electricity on the part of this strip extending from x{ to #2'
is — (<£2 — <^j).
Hence from the origin to x'= a the amount is
477
If a is large compared with I, this becomes
1 -
*" =
477
477^
, v
Hence the quantity of electricity on the plane bounded by the
straight edge is greater than it would have been if the electricity
had been uniformly distributed over it with the same density that
it has at a distance from the boundary, and it is equal to the
quantity of electricity having the same uniform surface-density,
but extending to a breadth equal to I \oge 2 beyond the actual
boundary of the plate.
This imaginary uniform distribution is indicated by the dotted
straight lines in Fig. XI. The vertical lines represent lines of
force, and the horizontal lines equipotential surfaces, on the hypo-
thesis that the density is uniform over both planes, produced to
infinity in all directions.
196.] Electrical condensers are sometimes formed of a plate
placed midway between two parallel plates extending considerably
beyond the intermediate one on all sides. If the radius of curvature
of the boundary of the intermediate plate is great compared with
the distance between the plates, we may treat the boundary as
approximately a straight line, and calculate the capacity of the
condenser by supposing the intermediate plate to have its area
extended by a strip of uniform breadth round its boundary, and
assuming the surface-density on the extended plate the same as
it is in the parts not near the boundary.
Thus, if S be the actual area of the plate, L its circumference
and B the distance between the large plates, we have
a = i-B, (13)
77
278 CONJUGATE FUNCTIONS. [196.
and the breadth of the additional strip is
so that the extended area is
(15)
The capacity of the middle plate is
Correction for the Thickness of the Plate.
Since the middle plate is generally of a thickness which cannot
be neglected in comparison with the distance between the plates,
we may obtain a better representation of the facts of the case by
supposing the section of the intermediate plate to correspond with
the curve ^ = \//.
The plate will be of nearly uniform thickness, /3 = 2b\}f', at a
distance from the boundary, but will be rounded near the edge.
The position of the actual edge of the plate is found by putting
/= 0, whence #'= j \oge cos ^ ^ 7j
The value of $ at this edge is 0, and at a point for which #'= a
it is a + t>loge2
~~b~
Hence, altogether, the quantity of electricity on the plate is the
same as if a strip of breadth
-# /i W/3\
— (log. 2 + log, cos —g) ,
i.e. ^loge(2cos||), (18)
had been added to the plate, the density being assumed to be every-
where the same as it is at a distance from the boundary.
Density near the Edge.
The surface-density at any point of the plate is
/ f±_
\/eb — :
47rB
1 9 7.] DENSITY NEAR THE EDGE. 279
The quantity within brackets rapidly approaches unity as x'
increases, so that at a distance from the boundary equal to n times
the breadth of the strip a, the actual density is greater than the
normal density by about 2M+1 of the normal density.
In like manner we may calculate the density on the infinite planes
When %'= 0, the density is 2~* of the normal density.
At n times the breadth of the strip on the positive side, the
density is less than the normal density by about - n+1 •
At n times the breadth of the strip on the negative side, the
density is about — of the normal density.
2
These results indicate the degree of accuracy to be expected in
applying this method to plates of limited extent, or in which
irregularities may exist not very far from the boundary. The same
distribution would exist in the case of an infinite series of similar
plates at equal distances, the potentials of these plates being
alternately -f Fand — V. In this case we must take the distance
between the plates equal to B.
197.] (2) The second case we shall consider is that of an infinite
series of planes parallel to xz at distances B = vb, and all cut off by
the plane of yz, so that they extend only on the negative side of this
plane. If we make <J> the potential function, we may regard these
planes as conductors at potential zero.
Let us consider the curves for which $ is constant.
When y ' — mrb> that is, in the prolongation of each of the planes,
we have %' = b log \ (e+ + *-*) (21)
when y-= (n+^)bir, that is, in the intermediate positions
af= a log i («*-«-*). (22)
Hence, when <£ is large, the curve for which (/> is constant is
an undulating line whose mean distance from the axis of y is
approximately a = b (</>— loge 2), (23)
and the amplitude of the undulations on either side of this line is
280 CONJUGATE FUNCTIONS.
When $ is large this becomes be*2*, so that the curve approaches
to the form of a straight line parallel to the axis of y at a distance
a from that axis on the positive side.
If we suppose a plane for which #'= #, kept at a constant
potential while the system of parallel planes is kept at a different
potential, then, since b<p = a + b\oge2, the surface-density of the
electricity induced on the plane is equal to that which would have
been induced on it by a plane parallel to itself at a potential equal
to that of the series of planes, but at a distance greater than that
of the edges of the planes by b loge 2.
If B is the distance between two of the planes of the series,
B = TI b, so that the additional distance is
a = 2»^. (25)
198.] Let us next consider the space included between two of
the equipotential surfaces, one of which consists of a series of parallel
waves, while the other corresponds to a large value of </>, and may
be considered as approximately plane.
If D is the depth of these undulations from the crest to the trough
of each wave, then we find for the corresponding value of $,
JD
0=ilog^. (26)
eb-l
The value of of at the crest of the wave is
& log *(** + <?-*). (27)
* Hence, if A is the distance from the crests of the waves to the
opposite plane, the capacity of the system composed of the plane
surface and the undulated surface is the same as that of two planes
at a distance A -f a', where
«'=lo?e— ^-5- (28)
* Let $ be the potential of the plane, <£ of the undulating surface. The quantity
of electricity on the plane per unit area is 1-5-4 TT&. Hence the capacity
= 1 -5- 4 TT &(*-<£),
= 1 -f- 47r (A + a'), suppose.
Then ^4 + a' = 6 (#-<£).
But 4 + 61og£(e* + e-*) = 6(*-log2);
.-. a' = -60 + 6 (log 2 + log £ (e* + «-*))
= b log (1 + e~2^
(26).
200.] A GROOVED SURFACE. 281
199.] If a single groove of this form be made in a conductor
having the rest of its surface plane, and if the other conductor is
a plane surface at a distance A, the capacity of the one conductor
with respect to the other will be diminished. The amount of this
diminution will be less than the -th part of the diminution due
n
to n such grooves side by side, for in the latter case the average
electrical force between the conductors will be less than in the
former case, so that the induction on the surface of each groove will
be diminished on account of the neighbouring grooves.
If L is the length, B the breadth, and D the depth of the groove,
the capacity of a portion of the opposite plane whose area is 8 will be
S-LB LB S LB a' , .
'
If A is large compared with B or a, the correction becomes by (28)
*.-*-,. <»>
l+e B
and for a slit of infinite depth, putting D — oo, the correction is
.*• (31)
To find the surface-density on the series of parallel plates we
must find <r = -- ~-f when d> = 0. We find
47T dx
l =. (32)
i *_
\/e »-l
The average density on the plane plate at distance A from the
edges of the series of plates is o1 = — = • Hence, at a distance from
4776
the edge of one of the plates equal to na the surface-density is
of this average density.
200.] Let us next attempt to deduce from these results the
distribution of electricity in the figure formed by rotating the
plane of the figure about the axis /= — R. In this case, Poisson's
equation will assume the form
dv , .
~
Let us assume F=<£, the function given in Art. 193, and de-
282 CONJUGATE FUNCTIONS. [2OO.
termine the value of p from this equation. We know that the first
two terms disappear, and therefore
J_ _l _ d(f)
''
If we suppose that, in addition to the surface-density already
investigated, there is a distribution of electricity in space according
to the law just stated, the distribution of potential will be repre-
sented by the curves in Fig. XI.
Now from this figure it is manifest that -p is generally very
«/
Provenance
- Shelf
- Reference library
- Author
- James Clerk Maxwell
- Rights
- Published in 1881, before 1929, and therefore in the public domain in the United States.
- Collected By
- StanBot reference library