"‘1?
DESIGN. CONSTRUCTION. AND '
CALIBRATION or A SIXTY-INCH
INTEGRATING SPHERE PHOTOMETER
THESIS FOR THE DEGREE OF M. 8.
Walter Alfred Had-rich
I932 '
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DESIGN, CONSTRUCTION, AND CALIBRATION
OF A SIXTY-INCH
INTEGRATING SPHERE PHOTOIETER
A Thesis
Submitted to the Faculty
of
iichigan State College
of
Agriculture and Applied Science
by
welter Alfred Hedrich
Candidate for the Degree
of
master of Science
June 1932
{HESAS
ACKNOWLEDGMENT
The writer is indebted to the members of the Department
of Electrical Engineering and Engineering Sheps for their
worthy seeperation in this project. He also wishes to express
his gratitude to members of the Department of Mathematics
whose services have been invaluable in the preparation of the
theoretical treatment of the subject. For the generous
Opinions, advice, and criticisms contributed by Dr. J. E.
Powell and Dr. I. S. Kimball of this department I am partic—
ularly grateful.
ly gratitude is especially due to Professor L. S. Foltz
of the Department of Electrical Engineering for helpful
criticisms and generous attention in checking the manuscript.
'0 A. H.
ENIII II“. - , h ., IIIINAJIWIIIIA
CONTENTS
Introduction
General Considerations
Theoretical Considerations
Design and Construction, with Photographs
Photometric Uses and Methods
Calibration
Page
10
57
68
80
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This thesis comprises the design, construction, and calibration
of a sixty-inch integrating sphere photoneter. Unless otherwise indi-
cated, the expression "sphere photometer" in this work shall invariably
refer to the sphere as an auxiliary unit in the photometry of light
sources.
If the sphere photometer is to constitute part of the photometric
equipment of a laboratory, it is desirable to design and dimension the
sphere in accordance with the principles of laboratory precision and
commercial practice. The features incorporated in the design of this
photometer afford the possibility of photonetering over a wide range
of luminous intensities as associated with various types of commercial
lamps and lightinr units. The sphere must provide for rapid handling
of lamps in quantity tests. As a mechanical unit, it should be suffi-
ciently flexible to permit ready replacement of parts, ease of convey-
ance, and should combine lightness with stabiliti.
It is the purpose of this work to attain a desirable balance in
the design, construction, and calibration of a sphere intended to meet
the requirements of laboratory and commercial tests.
. I I u. gun‘s-v.33}! . 1». 1' .I to . .EhaNHNﬂI I I-
o!‘
CU7LPBIJCXHYfIDLWD'ICCT‘
historical Development
Early determinations of total luminous flux from a light source
were made without the use of1ntegratiI devices. The point by point
method, consisting of the measurement of candle power of the source at
different angles in a vertical plane,Isas then employed. From these
candle power values the total luminous output was ob ained by sunnation
of the various zonal fluztes previouslv computed. The measurerent of
candle power at various angles in a vertical plane through the source
4.1
is still in vogue today, lar gely for the purpose of deterraininr the
.0
distribution of luminous intensity. The unsynnetrical nature of the
source makes it necessary to rotate the lanp about its vertical axis,
or to conduct measurements in a number of vertical planes throuch the
source. Very accur te results can be O“t91Pn8d by the point by point
method. Equally reliable results obtained through the use of integra-
ting devices have enpha51zed the lahorious aSpect of the point by point
method, and unquestionably “ ustifi ecl tlte use of the integrating sphere
photometer.
A nunber of photometric a: M ances were devised and used for ob-
taining the mea- Sihecical candle power of a lamp. They‘rere designed
to give t} Is I. S. C. P. in a single reading. Disadvantages inherent to
these forms of integrators and their practical linitations lead early
investigators in this field to turn their attention to the hollow sphere,
r. Ulbricht in the year 1900. At that
(-4
lThiCh was fix st investigated bg'
time Dr. Ulbricht was uninformed of an earlier publication on the dif-
fusion of light, which appeared in 1893 in the Phil. lag. and was writ-
ten by Dr. Sumpner. His photometric calculations revealed the striking
conclusion that the illumination on the inner wall of the hollow sphere
is everywhere the same, due to diffusely reflected light. Obtained un-
der the hypothesis of Lambert's cosine law of emission, this result was
the theoretical foundation for the development of the sphere photometer.
Apparently Dr. Surpner did not make Specific use of this important de-
duction, however, as applied to photometric considerations and particu-
larly to the problem of measuring the K. S. C. P. hy means of the hol-
low sphere. Contemporary investigators who did much to promote valu-
able experimental work on spheres of different sizes were Bloch, Corse-
pius, Karchant, Dyhr, and honasch. There has been very little change
in, or addition to the theory of the integrating sphere beyond that em-
bodied in the beginning treatment of the subject. much has been done,
however, to make the sphere better suited to pra tical application. In
recent years the sphere has served remarkably well in connection with
other methods of photometry which are quicker and more reliable than
the long established visual connarison process.
.‘
Descr'ntion of the Sphere Photoneter
General.
The integrating sphere is, as its name implies, an integrating
device. It integrates the illuminating effect of any source of light
mounted or suspended within its enclosing spherical wall. The latter
must be as nearly perfectly diffusing as possible. A small window of
diffusion glass replaces the equivalent surface area on the inner wall
of the sphere. Lccording to the theory of the sphere, if a source of
light be placed anywhere within the enclosure, every point on the in-
ner surface will receive both direct and reflected light. If a white,
diffusely reflecting, and opaque screen be interposed between the source
of light and the window in such a manner that the latter is shielded
from direct radiation only the illumination on the window will be due
9
to diffusely reflected light, and will be directly preportional to the
’?
mean spherical candle power of the source. since the total luminous
3
(”D
output of the source is preportional to the L . P., the brightness
of the window is a measure of the total lumens emitted by the source.
Probably the most general use of the sphere photometer is confined to
the measurement of this total luminous output, although it serves a num-
ber of other practical purposes which will be mentioned later.
The mathematical deveIOpment of the theory assumes an empty
sphere with.continuous inner surface obeying Lambert's cosine law of
emission. In order to realize a practical development, certain varia-
tions are unavoidable. They are caused by the necessary use of screens,
imperfect diffuse reflection from sphere surface, selective absorption
by paint, presence of non-luminous bodies within the sphere, imperfect
diffusion of window, position of lamps and luminaires in the Sphere,
inconstancy of wall paint, and difference in window and wall absorption
factors. These are some of the most important sources of error in the
sphere. While it is possible to minimize these errors through the se-
lection of suitable methods of measurement or the application of gen-
eral corrective measures to overcome such departures, it is neverthe-
less essential to investigate the order and magnitude of some of these
sources of error inasmuch as they may become appreciable in certain
photometric measurements. This quantitative treatment of the matter
of errors is discussed in the section on theoretical considerations.
C
screens.
The screen, unich is necessary for the proper functioning of
the sphere photoneter, reduces the illun‘nation on the window in two
ways. All direct light from the source intercepted by the screen rust
first be reflected from the latter before it can reach the diffusing
wall of the sphere. Hence this flux is diminished by an amount equal
to the screen absorption before it illuminates the sphere, from whence
it is reflected to the window. The screen acts as a secondary source
of lower intensitgu The screen also obscures a portion of the surface
of the sphere from the window; Light from this hidden area must be re-
flected to some other portion of the surface which re-reflects it to
the window. Such light flux reaching the window has suffered greater
absorption and in consequence the window illumination due to this con-
,onent is lower. Since the screen and the inner surface of the sphere
have the sane diffusing quality, that side of the screen facing the
window will reflect light from the sphere to the window and thus com-
pensate for some of the loss. By increasing the size of the suhere,
the screen error nay'be sufficiently reduced to render ibs effect nec-
[—1
Ho
gible. The ratio of screen surface to sphere surface is less. The
screen absorption is less in the sane proportion. Two other important
considerations in reducing the screen error are its dimensions and re-
lative position in the sphere. This matter is given quantitative at-
I
. c o o o 0‘ o "
tention in the discuSSion under "Theoretical ConSiuerations.
hon-luminous Bodies within the Sphere.
Included in this classification are screens and their -ittings,
lamp supports, lamp fittings, shades and reflectors. All of these will
absorb ,trt of the direct light from the source. The error involved
due to the presence of these bodies is materially reduced by giving all
non-luminous bodies except the source and accessories integral to it, a
matte white finish like that used on the inner wall of the sphere. Non-
uniform diffuse and specular reflection due to lamp shades, reflectors,
etc., constitute a departure from ideal conditions. This and other er-
rors resulting from causes heretofore outlined are minimized by follow-
titution method in photometering.
6’)
ing the sub
Sphere Paint.
The material used to render the inner surface of the sphere dif-
fusely reflecting must closely approximate theoretical requirements.
erfect diffusing quality,
\—
The latter demand non-selective absorption, p
and a unifO‘m and minimum absorbing power. Departure from these speci-
fications must be expected. Furthermore, the diffusing paint should be
permanent over a long period of time and not appreciably affected by
moisture or moderate changes in temperature. From time to time the sur-
face rust be refinished, as it becomes soiled due to collection of dust
,
nin should not be done over the soiled sur-
E“
Q
U)
and foreign matter. Refini
face. A sphere coating made up of a permanent oil base should form the
foundation for the finishing paint. The latter should be readily renov-
able when soiled, by the application of a suitable and quick-acting sol-
vent. It is obvious that such a finishing paint will expedite the re-
finishing process. It is not warranted, however, unless it possesses
d'
o a high degree the qualities prescribed by theoretical requirements.
*Vindowu
The window consists of an appropriate diffusion glass, which may
be located almost anyWhere on the inner surface of the sphere. It is
desirable to have the window flush.with the inner surface in order to
minimize the error that otherwise would exist through the discontinuity.
In practice, the offset is small. In order to facilitate measurements
with a bar photometer, the window is generally located so that the line
through its center and normal to the surface lies in the equatorial
plane. This position of the window affords greater flexibility in ob-
n
taining measurements with various types oi photometric devices. The
J.
window Opening is small compared to the inner surface of the sphere.
Glass used for this purpose must be uniformly'and highly diffusing as
well as non-selective in absorbing power. It is important that the in-
terior surface be matte, preferably of the same diffusing quality as
that of the sphere paint. The window error can.be minimized by care-
ful workmanship and selection of glass.
Kethod of Obtaining Photometric Balance.
Two general methods of obtaining a photometric balance are cus-
J." "
tomarily classified as the "Direr and Substitution" procedures. The
. - o o i .
former, sometimes called the "Direct Comparison' method, involves the
direct comparison of the test lamp with a calibrated standard lamp. The
nethod is simple, but Open to objection owing to the possibility of er-
rors which cannot be eliminated from the measured results. Although the
errors inherent to the integrating sphere become less as the diameter
of the latter is increased, the substitution method should be used to
insure maximum accuracy under the most unfavorable conditions. This
method involves the use of a third light source known as a comparison
lamp. It must be a constant source but need not be a calibrated stand-
ard. A standard lamp and a test lamp are interchanged as sources of il-
lumination within the sphere. Conditions within the sphere will then
have the same effect on the window illumination in both cases. This il-
lumination in each case is compared with the illunination produced on
the photonetric screen from the comparison lamp. The readings obtained
from the settings for photometric balance in each case are evaluated by
the application of the Law of Inverse Squares, whereby'the candle power
of the test lamp is obtained in terms of that of the standard lanp. In
some cases the scale on the photometer bar is graduated according to the
Inverse Square Law or in lumens. This is done to simplify the work.
The use of either method for the purpose of determining the H. S. C. P.
or total luminous output of the source, assumes a knowledge of the con-
stant of the Sphere as determined by calibration.
Then the candle power of the test lamp is much in excess of that
of the comparison lamp, the quantity of light transmitted from the win-
dow to the photometer head is reduced by means of an iris diaphragm.
This is, in effect, a variable aperture for control of the light flux
from the window.
Frequently the light from the test source and that from the com-
parison lamp show slight difference in color. In making a photometric
balance this color difference is a disturbing factor, often preventing
satisfactory comparison. Colored glass plates, known as color filters,
of known transmission coefficients for the ranges of spectrum over
which they are to be used, are interposed either between the window
V
—nd the photonetric head or between the latter and the conparison
$3
3
lamp. Color filters and iris dianhrarms serve to obtain a well de-
(-J
F
fined and reliable photometric balance.
IO
THEORETICAL COUSIDERATIOYS
Theory of the Sphere
Illunination Due to Diffusely Reflected Lith.
Let r represent the radius of the sphere. Assume inner sur-
face of sphere to give perfectly diffused and non-selective reflection,
and let L be a source of light of mean spherical candle power I
Let ¢ represent the total lumens flux emitted by L.
Then ¢= 9.". regardless f the candle power distribution of L.
It is desired to investigate the illumination at any point A on the
surface of the sphere.
11
Let B represent any other point on the surface, and consider
the differential element of area dB at B. Since ¢ is the total
flux from L, let 4.? represent the lumens flux incident on dB. The
illumination at B is 68.1% lumens per square centimeter (phots).
By hypothesis there is no specular reflection, hence let k represent
the diffuse reflection factor, where 1: is defined as the ratio of light
flux diffusely reflected from a surface to that incident on it. Then
AL - IQ represents lumens per sq. cm. reflected from dB. Since
the surface of the sphere gives a reflected flux distribution following
Lambert's Cosine Law of Emission, this quantitq,r represents the bright-
ness of the surface at B.
#:4d—L—oﬂ lamberts (emitted lumens per sq. cm.) and is the
same in all directions from within the sphere. Then
=- -—--k is the brightness in candles per sq. cm.
ﬁ.d¢= 41¢, gives the reflected flux from dB, and the bright-
ness at B may also be formulated according to
ﬁz—L-ié’Wandles per sq. cm.
W 48
The luninous intensity of dB in direction B0 is
l . 1
50- =r‘d7§ *' ‘8: I‘d; * canciles.
Luminous intensity,r of dB in direction PA is
g4=#.d¢. i cad: candles.
The Inverse Square Lat-r gives the illuiiination at A:
E _ d¢ detect-4M;
— 8
A. ’74
lumens per sq. cm. (phots),
but datach ,
amt-hue”: _ 44H: _ dg-g
4-wh-‘Coo‘t " 4m.“ " .3 ’
wheres = 4V4} is the area of the sphere.
therefore EA =
'
EA is the illumination at A due to the flux d¢ incident on B
I
and once reflected. The expression for EA, is obviously independent
A
of 8 , and hence EA is the same for all positions of A. It follows
I
that all elenents of area on the surface of the sphere are equally il-
luminated by diffused}: reflected light emitted from. any; surface element
on the sphere and ore ip; Lantert's Cosine Law of T*l"‘10
That this is true for twice and multiply reflected light nay be
seen by considering the illumination at A due to twice reflected light.
Consider once reflected light at any point C on the surface of the
sphere. Some light flux from C will be reflected diffusely along the
path CA. From the above discussion, the illu1‘2ination at C due to
once reflected light is
54:5 —_é_d 4‘
A, 4a.”).
Brishtness of dC is 5: m3 latterts.
471%}
rig-£2
4,-2.4}.
Expressed in candles per sq. cm., 4 =
Intensity of illumination (candle power) of dC in direction CA is
_M. usOdC
41PHA.
13
Illumination at A due to twice reflected light is
g = 415' é‘ma-metc mots,
42 4,242. , [2.
Since 1 =24 (—0.3 a , therefore
E - d¢.‘a. Caste- 4-6:- dﬁf‘a’c - dﬁiad/g nhots
42— 4173/21. 44‘cos'l- (41M? " _ 52 '
Since 9 does not appear in the final expression for EAL , the il-
lumination at A due to twice reflected light is independent of the
position of .A. éih‘ is the illumination at A due to twice reflec-
ted flux 436 from so. An extension of the above to multiply reflec-
ted light leads to the following important result:
All points on the surface of a sphere are equally illuninated by
diffusely reflected light emanating frem every infinitesimal element of
that surfac e .
The total illumination at A due to all light flux from L once re-
flected is readily obtained as
¢
t I S S
o
The total illumination at A due to all flux from L twice reflected
is EA =ﬂjgg¢c=ongfd¢= f; '95
t 9 5’
tan—#55
14
The resultant illumination at A due to an infinitude of reflections
is %+%+%2+....+ﬁ+..
n1
ll
—§.£(I+é+é‘+i’+ - - - 4—3;- )
i"
II
It is known that the infinite series within the parenthesis converges
for all values of k within the interval -/< k< I and defines the
functionf(‘)= ———for all values of 1: within this interval. In
"I?
practice, I: will always lie in the interval 0< *(I ;
H. ’ “ELL
6' /-é 4M‘ I-k
therefore, E = phots.
a A
Discussion of Results.
Introducing the direct illumination from source L, the result
is I Ice:
_ ¢ . é
REA—ﬁﬁi— 4"} [-k pht.
The illumination at A due to direct rays from L is governed entirely
hy the photometric distribution of ligut from the source and its rela-
tive location within the sphere.
The significance of the expression E: ¢ * is at
M 4M at t-Ie
once apparent. The illuzitination at antipoint on the surface of tl‘e sphere
due to diff1-1se reflected light is directly proportional to the total lu-
nens enitted from the source. (Terfect Diffusion)
.f'_"_I_°_-_l‘;_ _
434‘ l-‘k A} 1-16
"Ie may write E
R A
15
from which it is seen that the illumination is proportional to the mean
spherical candle power of
tte source. If at the point A a small win-
dow of diffusion glass is inserted and shielded from the direct rays of
light coming from L, the illumination of the window will be directly
prOportional to the total luminous output of the source. Thus by mea-
suring the illumination at the window, either the mean spherical candle
power or the total luminous output of the source can.he readily calcula-
ted. It is liken-.r'se significant to note that RE". is tne sane regard-
less of the position of the source L within the sphere. In practice,
however, the source should not he made to approach the wall of the sphere
too closely; otherwise, the error introduced in consequence of later con-
siderations may become appreciable. A single measurement of‘eli‘ ,
carefully executed, is sufficient to give a very fair value of the I. S.
C. P. or total luminous output of the light source under consideration.
Certain refinements in construction and methods serve to minimize the
error within the limits conrensurate with the size of sphere employed.
The following consideration will serve to clarify'the simple re-
lations involved in the direct and reflected illumination. The average
illumination due to direct light from L is
I
The average total illunination is
_ _ I ‘ [0- I. I
TEN-5+5.- “/[T‘Trz+7?- A" ,..,., '
rnonyrnous with REA as employed. in the ahove.
H.
t!
3.3,
,.J
H.
O
:3“
D1
3
H-
(A
m
g A
16
Assuming an average k, for example, k = 0.80,
6R = I 0' 90 : 4—— A lumens per sq. on.
733mg
5‘ = + _1 ={—I'— llmens oer sq. cm.
TA! 4—41;
The average total illumination would be five times as great as the il-
fJo
lunination f the null if the latter Here black with a zero reflect on
factor. Again bv placinc; the source L at the center of the adhere,
1a a _ .- d ' .- ' . -..~° . ' 7d £ _. I.
the averabe irect 1ll\1.1nat1on wou- be .0" t
L is a source of uniform intensity in all directions, the illumination
and in the event
l
on the wall nould be the same at all points and equal to
t? :: JED lumens per sq. on.
U A‘—
Earring invisible radiation, the source supplies enerjv in the
form of visible radiation sufficient to balance the loss due to absorp-
tion. If a is the absorption factor for the inner sur ace of the
sphere, the total absorbed light fhxxzhs 95 , and the total reflected
O I.4 . I
flux 18 ’b~ . These deductions follow from elementary'considera-
ac
tions.
Light flux abs creed Light flux reflected
first absorption 4¢ First reflection (/— 4) ¢
Second " d(’ -a) ¢ Second " (I - a} ‘ ¢
Third " 4(I-aj‘¢ Third " [I 14):?
nth " 4 ’“dj "75 nth " ( I -a jn¢
cf +all-ay¢ +a[z-4j‘¢+--- 1- 1-4) ¢*"'
4% [I +(I-a)+(I—af' +- - - -+ (“‘91“)
Total absorbed flux
n H I?
ll
17
Let 3 represent the sum of the first n terms of the series in the
n
parenthesis. Then 5 = 4 _ _C_/ ‘19.)”
a ‘1 ‘L
4”” 5,, = Li/h—L— L/m M”
”*w $19004 19-“: a.
since (I—a}< | I I , Lim Milo
n-ND d ’
Therefore, [aim 5,, = —-’—
w-No 4'
Hence, total absorbed flux = 63¢ 0.1. = ¢
62
Similarly,
total reflected flux
(I-a)¢+ (I—a}‘¢+ - ~-+(I-a)“¢+--
" " " .(I-a)¢[1+(l~a)+(I—a)‘+---+(I-a)3.-]
t n u _ I “a
. ¢__a
The source supplies energy to overcmie the loss qb , which is constant
a Q -
as long
as the quality of the reflecting medium and the lurinous out-
put of the lamp remain the same.
The window in the wall of the sphere may have an area of A sq.
cm. The flux incident on the window is ¢~= ER-A .
96 := .Ziﬁi.. 3‘ - .fﬂfi...___1£__. lumens
W 4M3 /- k 3 1-)?
Prightness of window as viewed from outside is t; .
5,, = £} 3‘ lamberts, in which I is the coefficient of transmission
for diffusion glass.
6..
t...
-S%}ir candles per sq. cm.
5 7(l-k)=
= K¢ candles per sq. cm.
4:? '4‘6- 16)
7'
h is the so-called constant of the sphere. The constant of the sphere
as determined by calibration can be formulated mathematically in a num-
ber of ways, depending upon whether the brightness or the luminous in-
tensity of the diffusion glass window is measured. It is neither neces-
sary to know or to determine the values of k and 3' , as these quan-
,
h as determined by calibration.
k1.
tities are ezhodied
._)
K : = {w , in which 61,, is expressed in candles per sq.
”I L .L.
4 ,
cm., Qb rrprcsents total luminous output, and I; is the K. S. C. P.
f light source.
5.73.111, Since I“: ‘6‘”. AW , K: i7“: 8 75% , in ﬁzzITTiCh AW
‘9 0
is the area of the window in sq. n., and IL, is the normal candle
p wer of the window.
'4.
However, the constant of the sphere n:;'be calculated to conform With
the simple ratios -%1 , 4%"— , I" , I" ; whence it should be
. 75‘ I.
observed that the K's are essentially different. K may also be ex-
pressed as the reciprocal of these ratios. having obtained K by mea-
suring the brightness of the window as Viven by a source of known total
lumens ¢ , the luminous output of a test source May be readily ob-
tained as ¢t , in which g =K'é , where ‘6 is the brightness of
the window due to the test lamp ernressed in candles per 8 . cm.
A.’ ._ J.
Effects of Hon-Luminous Bodies within the Sphere
h
upon formulations Deduced under Ideal Conditions
‘
The presence of non-luminous oodies in the sphere such as the
screen, lamp fixtures, supporting devices for lighting units, etc.,
must have an effect on the uniform illumination produced on the dif-
fusing surface within, for these bodies absorb some of the direct as
well as reflected light flux. Point sources of illumination must be
excluded fran'the discussion, for all practical sources of light have
finite dimensions. There follows a discussion of the effect of non-
luminous bodies Within the sphere upon diffusely reflected light.
Hon-Luminous Eodies within the Sphere
and their Effect upon Diffusely Reflected Light.
Consider a non-luminous diffuselv reflecting body' h screened
v
I.
w
from all direct rays of light, and let the surface area of L be U.
Its reflection factor is K; . If R occupies any position in the
sphere such that it receives only reflected light, it will absorb a
small amount of light from each of the infinite reflections intercep-
ted by its outer surface. See fig. 2. Let dU represent a differen-
tial element of area on the surface U. A simple photometric calcula-
tion will disclose that any surface whose dimensions are negligible in
comparison with the distance from the source of light to the surface
I
will have an illumination E = nb , regardless of the orientation or
osition of this element of surface dU within the sphere. The quan-
"d
tity b represents the uniform brightness of the source; namely, the
20
interior surface of the sphere. It may be well to establish the rela-
tion E' = nb now, and then proceed with the discussion.
Referring to Fig. 2, consicer a sphere of inner radius r, and
whose inner surface has a uniform brightness b. The brightness of any
element of area on the inner surface of the sphere will, therefore, be
the same when viewed from every direction, and its luminous intensity
4% ‘n‘q
(N
Fig. 2
will vary as the cosine of the angle of emission. Kathematically,
Ia = b-dA, where Is is the candle power in any direction a from
the normal to the surface dSl, and dA is the projected area
d31°cos a. Let P be any point within the hollow sphere and d?
a differential elenent of area free to assume any fixed position at
P. Since del = In is the normal candle power of 31 in direc-
tion b0, delcos a = la is the candle power of dsl in direction
VP. But dSlcos a = an, therefore, bdA = I and
a,
illoq'lkﬂha ._ ‘ita; ;
ll‘ ._ -s the illumination or incident flux density at
I
P due to C181 0
Since A: ”2‘“ [175]” Od¢ = Afsin 919 d¢ , therefore
.d -¢ = ‘~ﬁfI/)o’¢0$0'l€‘dd .. ‘ .. ’
A; A,’ _ ﬁm05m0do¥¢_d£é
Hence ‘5'; = ‘ we'd“) , where —%— :1“) =51.” ed€l¢
I
is the solid angle subtended by dA. If 3' represents the total illu-
p
mination received on the right side of dP, then
'gar { . E
' I a - 8/010 2.
EP=/df = [£9110 waded? =2”; Slnom040=2ﬂﬁ T
o ' °
0 . .
cr [ =..L.al"=. F4 lnCident llunens per sq. on.
P 1
T73 incident flux density on the opposite side of d? is
1' ’f
V 0 1 F
l - 6 :1
£P= ﬁsbomadolf=—zrﬁ 3M0¢N040=an [JiLz
. 5 ’-
1 o
E :17; luriens per sq. on.
p
Thus the flux density is the same at every point within the
sphere, and hence the illumination or 'ncident flux density'will be
the sane at every point on the outer surface of a body K immersed in
the flux within the sphere. This result also holds for radiation of
any'wave length J1, . It is interesting to note the reciprocal case,
where the flux eritted per sq. cm. area of uniform brightness b is
E' = nb. This accounts for the existence of the conversion factor n
whereby brightness expressed in candles per sq. cm. can be converted
to lanberts (enitted lunens per sq. cm.).
how, if E represents the total uniform incident illumination
on the inner surface of the sphere, then
IO
[0
—if = t
candles per sq. cm. uniform brightness of sphere.
Therefore, nb = Ek = E' is the illumination or incident density in lu-
T7
nens per sq. cm. at every point on surface of n,
and 77“”: E‘U = ¢~ is the total flux incident on IT in lunens.
Of this, I? ' absorbs AMI-f“) = EU‘ (I ‘£n) lunens.
Total flux incident on the sphere to give it an illumination B would
be ¢ =477A‘Elunens. Of this, there is absorbed a flux equal to
47"1036 (I’ﬁ) lunens. The total loss of flux due to absorption
within the enclosed sphere must be equal to ¢.
Hence, 4F/11F (I—£)+ EUfﬂ-fu) =¢ . Solving for 13,
E = ¢ lul-nens per sq. cry.
470171-11?) 1- U750 ‘1‘“) ' " ‘
_é_
41M}
.. _ _ 46 __¢_
5’R e EAV 1Fh‘(I-£) +U£(I—‘,,) 4MP- ,
is the illumination. on the surface of the sphere due to dif-
rre have
The average direct illurli nat ion is AV: . Therefore,
whe r e E
.513
fuse re flected light. Removing the hody H from the sphere, the illu-
mination due to diffuse reflected light would then be
49 . 5
5?: 4me- 1—?
Thus the illumination on the wall of the sphere and therefore at the
windmr has been dininished by an amount AQdue to the presence of II.
_ ¢ . f? _ 95 2
‘7? " 4w 1—? «m? ﬂ- 0+ 114% t.) + 4W-
AgR: 417/L'L7é_+’_ 4M‘?£§)+LHW
A
23
q
5
5(l—é) + Liam—>6) .
zﬂéiz== :g? ’:€;§ +"'_'
46‘ — -—’—- ’ ¢
’8 (1—26)5 3(/—£)+ aka—:6.)
It was assumed that b represents the uniform brightness of the sphere,
'which in practice is not true because no matte surface obeys Lamhert's
Cosine Law exactly. Nevertheless, the ratio of diffuse reflected light
to direct light in the sphere is so large as to render the error due to
assumed uniform brightness, negligible. The-wall of the sphere can be
made to meet the requirements of perfect diffusion to a degree satisfac-
tory for all laboratory and practical reasurements.
The last expression above for AER can be written
_-= 1.. /
l—(S‘ M
/+ (Hos
and shows at once the effect on the window illumination with changes in
(1) 4Q
U. Accessories within the sphere such as screens, fittings, supports,
etc., which are not a part of the lamp or lighting unit, are customarily
coated with the same diffusing paint as employed for the inrer surface
4'.
of the sphere. For these parts then, ku = k, and the above expression
simplifies to
lunens per sq. on.
____L _ ’
(2) AE’? (1—793 I Hug—i
It follows thee the constant of the sphere K decreases with increa-
sing values of U, since K is proportional to En. K=-é!=§ﬂl
{\D
17>
The expression (1) for AERwould indicate that the reduction
at
in window illumination due to the presence of a non -lumir ous bod: n
$9
depends entirely upon 'he surface area and absorption factor of L,
and is independent of its position in the field of reflected light.
Verv accurate er perimente 1 work would undoubtedly disclose some varia-
TY
tion in AER with change of position of 1:. To account for this chan-
ge which, it is reasonable to believe,11:ust be well within the allow-
able limit of error, greater refinement is necessary in the analvsis.
The hj'pothesis underl"ir:g t‘e derivation of (ﬂtie assumec the
sultant illunination 2 due to direct and reflected lig ht to he uni-
form over the inner surfs es of the sphere. Likewise, it assumed a dif-
‘
fusely reflecting bony K with no mention relative to the degree of
diffusion. All foreign bodies in the sphere except polis ded fittL mgs
glass, porcelain, metal, and sinilar sur aces strong in specular re-
flection are more or less diffusely reflecting or transmitting, but to
a varying and lesser derree t1 n the inner surface of the sphere. fur-
ther considerations will disclose the importance of rer7
This gives the distance x for minimum total error, and is a function
of o . The limits of x for a typical case of a source suspended ver-
tically along W' with o. = 30 , obtain as follows. Let ilr. =/ and
n = l . Then the shadow is bounded by parallel rays, and thesexpression
(3) reduces to x = 0.37488r = 0.325(l-d) . Pong—2:, and n = 9 , we
have x = 0.57735r - 0.500(l-d) . 3
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