Benjamin Silliman.

Principles of physics, or, Natural philosophy: designed for the use of colleges and schools online

. (page 33 of 78)
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facets of the particles composing their rough surfaces. Only part of
the light is thus irregularly reflected or dispersed, while much of it is
probably absorbed or destroyed.

(c) Rejection. When light falls upon polished surfaces, or on bodies
having naturally smooth and uniform surfaces, it is thrown off in a
regular manner, as a ball rebounds from a hard floor.

If a ray of light, S A, fig. 304, falls upon a polished surface, B C, it will be
reflected in the direction A R. If N A is drawn perpendicular to B C, S A N
will be the angle of incidence, and NAR will be
the angle of reflection, and the two angles will be
equal. The lines S A, N A, and A R, will lie in
the same plane; we have therefore the following
rules :

1st. The incident ray, the perpendicular at the
point of incidence, and the reflected ray, are all situated in the same plane.

2d. The angle of incidence and the angle of reflection are equal.

(d} Refraction. If a straight rod is placed obliquely, partly iir-
mersed in water, it appears broken or bent just where it enters the
water. If a coin, a, fig. 305, is placed in a
cup, in such a position that it is just hidden
from view, and water is then gently poured
into the cup, the coin will appear to be lifted
up and will become visible.

Let c d be the surface of the water, the ray, a b,
is so bent or refracted, at the surface of the water,
that the coin appears as if placed at a'.

This bending of the rays at the surface of any transparent medium is called

Let C B, fig. 306, be the surface of water in a vessel,
S A a ray of light incident at A, and NAN' the per-
pendicular, A R the reflected ray, and A T the direc-
tion of the ray which enters the water and is re-
fracted ; then :

The angle S A N is called the angle of incidence of
the ray S A. The angle N A R is called the angle of
reflection, which is in all cases equal to the angle of
incidence. The line N A N', is called the normal.
The angle T A N' is called the angle of refraction.

If we take A or, fig. 307, equal to A b, and draw
t m and b n, each perpendicular to N A N', then a m is the sine of the angle of
incidence, and 6 n ig the sine of the angle of refraction, and am divided by bn



of incidence is

is invariably the same for any given medium, whether the
increased or diminished. The quotient obtained by
dividing a HI by b n, is called the index of refraction,
and it is represented by n. The index of refraction
varies for different media ; thus for light passing from
air into water, it is about ^, for light passing from air
into glass, about f, and about f when light passes from
air into diamond. These fractions inverted give the
index of refraction for light passing out of water, glass,
and diamond, into air.

When light passes from a rare to a denser
medium, it is refracted towards the perpendicular
or normal, and when it passes from a dense to a rarer medium, it is
refracted from the perpendicular or normal.

The general law of the refraction of light is thus stated. The inci-
dent ray, the refracted ray, and the perpendicular to the refracting surface
at the point of incidence, lie in the same plane ; and the sine of the angle
of incidence bears a constant ratio, in the same medium, to the sine of the
angle of refraction ;


or, = n.

When a ray of ordinary daylight or sunlight is refracted by a dense trans-
parent medium, the refracted light is not confined to a single line, but it is


spread out into a fan-like form, as shown in fig. 308,
between A r and A v, and the different parts of the
refracted pencil show different colors, the most
strongly refracted part being violet, and the least
refracted part being red. The index of refraction, for
a single color, is uniform for any given medium j but
the index of refraction in the same medium varies for
differently colored light.

407. Amount of light reflected at differ-
ent angles of incidence. When light falls
upon a transparent medium perpendicular to its surface, nearly all the
light enters the medium, and only a small portion is reflected. As the
light falls more and more obliquely upon the medium, the amount, of
light refracted diminishes, and the amount reflected increases.

If we look at the image of the sun in water at midday, and again near sunset,
we shall see a remarkable difference. Near sunset the image is so brilliant, the
eyes can scarcely bear to look at it, while at midday we observe it without diffi-
culty. The image of objects at a little distance are seen in water more distinctly
than the images of near objects, because the light from distant objects falls more
obliquely upon the water and a greater amount is reflected.
If we look very obliquely at a sheet of white paper, placed before a candle, an
imagje of the flame may be seen reflected from the surface of the paper, but the
Image disappears when the rays fall upon the paper nearer to the perpendicular-



When light falls upon any polished metallic surface, the greatest
amount of reflection takes place when the incident rays are perpen-
dicular to the surface, and the amount of light reflected diminishes as
the angle of incidence increases.

Different substances, polished with equal care, differ in their power of reflect
ing light. The amount of light reflected depends also upon the nature of the
medium in which the reflecting body is placed. Bodies immersed in water
reflect less light than in air.

Table showing the number of rays of light reflected out of 100 rays incident, by
different kinds of glass and metals used for optical purposes.*

Angle of

Crown glass,
Sp. gravity,
n = 1-524.
Specific heat,

Plate glass,

n = 1-517.
Specific heat,

Flint glass,

n= 1-570.
Specific heat,

GHass of

Sp. gravity,
Specific heat,

Polished steel,
Sp. gravity,
Specific heat,











70 85



























































408. Internal reflection. When light passes through a transparent
medium, a portion of the light is reflected at each surface.

In fig. 309, S A is a ray of light incident upon the first surface of a trans-
parent medium. A portion is reflected in A R. A T is the refracted ray, and
T V the emergent ray, but a portion of the light is re-
fleeted at the second surface in the direction T A', of
which a part emerges in the direction A' R', a part
suffers a second reflection downward from A', a part
emerges from the second surface, and another portion
suffers successive internal reflections before it is either
lost by absorption or finally emerges on one or the other
side of the medium. In general only the rays A R,
T V, and A' R', have sufficient intensity to be visible to the naked eye.

409. Total reflection. When light passes from a dense to a rarer
medium, the angle of refraction is greater than the angle of incidence,
and when the angle of refraction is 90, the angle of incidence is much
less. For water it is 48 35', for ordinary glass it is 41 49 X , conse-

From Potter's Physical Optics.


quently a ray of light traversing water or glass at greater angles cannot
escape into the air, but is totally reflected, obeying the ordinary law of
reflection. The proportion of light suffering internal reflection from a
surface of glass or water, constantly increases from the perpendicular
to the point where total reflection takes place.

Since the angle of incidence for a dense medium is always greater than the
angle of refraction, when the angle of incidence is 90 the angle of refraction
must be considerably less than 90. If the angle of incidence is 90, its sine
will be unity. The sine of the angle of refraction will be unity divided by the

index of refraction, = , hence the angle of total internal reflection for any


1 310

medium is the angle whose sine = .


Fig. 310 shows light radiating from a point below
the surface of water and escaping into the air, the
angle of emergence increasing much faster than the
angle of incidence, until the light emerges parallel to
the surface of the water, after which total reflection
takes place.

To an eye placed below the surface of the water, all
objects above the horizon would be seen within an angle of 97 10', or double
the angle of total reflection for water.

410. Irregular reflection. Diffused light. The reflection from
polished surfaces, which follows the two laws already announced,
is called regular reflection; but only a part of the light is reflected
regularly from any surface, when the reflecting body is more dense
than the surrounding medium. A part of the light is scattered in
all directions, and is said to be irregularly reflected or diffused. This
is the portion of light which renders objects visible. Light regu-
larly reflected gives an image of the object which emits the light,
while light irregularly reflected gives only an image of the body
which reflects it. When a mirror becomes dim by the accumulation of
light dust, or anything which tarnishes its surface, the amount of
regular reflection diminishes, and the irregular reflection increasing,
all parts of the mirror become distinctly visible.

411. Umbra and penumbra. When an opaque object is held in a
pencil of light proceeding from a luminous 311

point, as s, fig. 311, a dark and well-defined

shadow is produced, which increases in size s "

as it becomes more distant. The dark

shadow is called an umbra. If the light proceeds from a luminous

body having a sensible magnitude, as A, fig. 312, besides the dark

shadow, or umbra, where no part of the luminous body is visible,

there will be a much broader partial shadow, called the penum*



bra, where a part only of the luminous body is visible. The breadth
of the penumbra increases with the diameter of the light, and with the
312 313

distance which the shadow extends behind the opaque object. The
darkness of the penumbra gradually increases from the extreme border,
which is too faint to be easily seen, to the umbra or full shadow, as is
shown in a section of the shadow, at fig. 313.

412. Images produced by light transmitted through small
apertures. If a white screen is placed near a small opening in a dark
chamber, the rays of light which pass 314

through the opening will form on the screen
inverted images of external objects.

It will be seen in fig. 314, that the rays
of light from the top and the bottom of the '
object cross each other in the small opening, and thus invert the image.
If the aperture is small, the image will be formed in the same manner,
whatever be the form of the aperture. But if the opening is large, the
image is indistinct, or entirely disappears.

413. Intensity of light at different distances. The intensity
of light at any distance from a luminous body, is in

an inverse proportion to the square of the distance.

Let 0, fig. 315, be a luminous point; at 1 1, place a
board one foot square ; it will cast a shadow that will cover
a space two feet square at double the distance, three feet
square at three times the distance, and four feet square at
four times the distance. The areas will therefore be, 1, 4,
9, 16, and the intensity of the light at the distances 1, 2,
3, 4, will therefore be in the proportions of 1, , ^, y 1 ^.

If /-and /' represent the intensity of a light at the
^distances D and I)', we shall have

JT)2 J/2' J> fit

Hence the intensity of a light at different distances will be inversely as the
squares of those distances.

414. Photometers are instruments employed to measure the com-
parative intensity of different lights. The principle on which they are
constructed is, to so place the lights that they will illuminate a single
surface, or two adjacent surfaces, with equal intensity. The relative



intensities of the two lights are then as the square of their distances
from the illuminated surfaces.

Huuseii's Photometer is the simplest and most convenient photometer
yet invented. A disk of paper four or live inches in diameter, is rendered trans-
lucent by washing it with paraffine or steavine, dissolved in oil of turpentine or
naphtha, except a spot about an inch in diameter at the centre. When this disk
is held between two lights, at a point where their intensity is unequal, the trans-
lucent part of the paper is easily distinguished from the ntral part, but when
moved to a point where the two lights have equal intensity, all parts of the
paper have a uniform appearance. No light appears to shine through, because
the illumination is equal on both sides. By means of a graduated bar, on which
the lights and disk are mounted, the distance of each light from the paper is
determined, and their respective intensities are calculated on the principles
above mentioned.

This principle may be applied in many ways to determine the intensities of
lights ; as, for instance, the portion which is transmitted or v reflected from dif-
ferent substances.

Rumford's Photometer. Rumford's photometer is composed of
two plates of ground glass, before which are fixed two opaque rods, A
and B, separated by a screen, fig. 316. The lights to be compared, as
a lamp and a candle, m n, are so placed opposite the rods that each


plate is illuminated by only one of the lights, and a shadow of the cor-
responding rod falls upon each plate, as shown in the figure. If the
two shadows, a and b, are of unequal intensity, by moving one of the %
lights backward or forward a position is obtained where the shadows
appear equally dark, and the glass plates are thus known to be equally
illuminated. The relative intensities of the lights are determined as in
the preceding case.

Silliman's Photometer. Silliman's photometer is the reverse of
Rumford's, comparing two discs of light thrown up by two equal trian-
gular glass prisms, upon a disc of roughened glass in the body of a dark
Chamber moving on a graduated bar. (Am. Jour. Sci. [2] XXII. 315.)


g 2. Catoptrics, or Reflection by Regular Surfaces.


415. Mirrors are solid bodies bounded by regular surfaces, highly
polished, and capable of reflecting a considerable portion of the light
which falls upon them.

The term mirror is generally applied to reflectors made of glass and
coated with an amalgam of tin and quicksilver.

416. Specula are metallic reflectors, having a highly polished
surface. The best speculum metal consists of 32 parts of copper, and 15
parts of the purest tin. Specula are also made both of silver and of steel.

In the use of glass mirrors, a portion of light reflected from the first surface,
interferes with the perfection of the image ; hence, where the most perfect instru-
ments are required, metallic reflectors are employed. In treating of reflectors,
we shall notice only the action of the principal reflecting surface, and use the
term mirror to comprehend all regular reflectors.

417. Forms of mirrors. Mirrors are either plane or curved. Curved
mirrors may be spherical, elliptical, or paraboloid. The properties of
elliptical and paraboloid reflectors have been mentioned in sections 324
and 325. A concave spherical mirror is a portion of the surface of a
sphere, reflecting from the internal side. A convex spherical mirror is
a portion of the surface of a sphere, reflecting from the outside. Curved
mirrors, whether concave or convex, may be regarded as made up of an
infinite number of plane mirrors, each per- 317
pendicular to a radius drawn through it from

the centre of the mirror.

Fig. 317 shows a plane mirror, M A N, a concave
mirror, m A n, and a convex mirror, m' A n', having
a common point, A, and the line, P A C, perpen-
dicular to each at the point A. If a ray of light,
I A, is incident upon either mirror at the point A,
the reflected ray, A R, will make the same angle
with the perpendicular as is made by the incident

ray. At any other points, as t or t', the curved mirrors will act like little piano
mirrors, perpendicular to the radii P t and Ct'.


418. Reflection by plane mirrors. Parallel rays of light, falling
upon a plane mirror, will be parallel after reflection.

If parallel rays of light, A D, A' D', fig. 318, fall upon the plane mirror, M N,
they will each make equal angles with the perpendicu-
lars, B D, E' D', and as the angles of incidence and
reflection will be equal, the reflected rays, D B, D' E',
will make equal angles with the perpendiculars, and
will consequently be parallel after reflection.

If A D represent the upper side of the beam of light before reflection, it will


become, after reflection in D B, the lower side of the beam. Hence a beam of
parallel light. is inverted in one direction by reflection from a plane mirror.

Diverging rays of light, falling upon a plane mirror, will continue to
diverge after reflection, and will appear to emanate from a point as
much behind the mirror as the luminous point is 319

before it. ^ I^B & IK

Let A be a radiant point in front of the plane mirror
M N, fig. 319. If the perpendiculars, E D, E' D', E" D",
be drawn, the reflected rays will make the same angles
with the perpendiculars as the incident rays, and hence
the reflected rays will make the same angles with each
other as they did before reflection, but they will appear to diverge from the
point A', behind the mirror.

Converging rays continue to converge after reflection from a plane
mirror. After reflection they will converge towards a point as much in
front of the mirror as the distance of the point behind the mirror,
towards which they converged before reflection.

This is easily seen by tracing the rays of light backward in the preceding

Keflection from a plane mirror changes the direction of the rays of
light, and removes the point of apparent convergence or divergence to
the opposite side of the mirror.

419. Images formed by plane mirrors. Let M N be an object
placed in front of the plane mirror, A B, fig. 320, and E the place of
the eye. From the great number of rays emitted in 320

every direction from M N, and reflected from the
mirror, a few only can enter the eye at E. These M ^
will be reflected from those portions, D F, G H, of
the mirror, so situated with respect to the eye and A
the points, M N, that the angles of incidence and
reflection will be equal. If the rays, D E, F E, are
continued backward, they will meet at m, and they
will appear to the eye to radiate from that point. In
the same manner the rays G E, H E, will appear to radiate from n ; a
virtual image of the object will therefore be formed between m and n.

This is called a virtual image, because it is not formed of rays of light
actually coming from the position of the image, but by rays so changed
in their direction, that they appear to the eye as though originating
from an object situated at m n, behind the mirror.

If the eye is moved about, the image remains stationary, hence it is
seen by means of rays reflected from other parts of the mirror. Two
or more persons may see the image at the same time and in the same
position, but by different rays of light.



The position of the image behind the mirror may be found by draw-
ing lines from prominent points in the object, perpendicular to the
mirror, extending them as far behind the mirror as the points from
which they are drawn are situated before it, then uniting the extremi-
ties of the lines, the outlines of the image will be delineated. The
images of all objects seen in a plane mirror have the same form and
distance from the mirror as the objects themselves.

420. Images multiplied by two surfaces of a glass mirror.
Glass mirrors produce several images. This may be readily demon-
strated by looking very obliquely at the image of a candle in a glass
mirror. The first image, caused by partial reflection from the first sur-
face of the glass, is comparatively faint. The second

image is formed by reflection from the quicksilver,
which covers the second surface, and is very clear and

When rays of light from any object fall upon the first
surface of a plate of glass, M N, fig. 321, a portion of the
light being reflected, forms the first image, a. The principal
part of the light penetrates the glass, and is reflected at c, by
the silvering which covers the back of the mirror, and coming to the eye in the
direction d H, produces the image, a', at a distance from the first image equal
to about once and a third the thickness of the glass. This image is much
brighter than the first, because the metallic coating of the mirror reflects a
greater amount of light than the first surface of the glass.

Other images, more and more obscure, are formed by rays which
emerge from the glass after successive interior reflections from the two
surfaces of the glass. As this multiplicity of images diminishes the
distinctness of vision, metallic reflectors are often employed in optical
instruments. 322

421. Images formed by light reflected
by two plane mirrors. Let AB, fig. 322,
be an object, and C D, E F, two plane mir-
rors, making an angle with each other less
than 180. The light falling upon the
mirror C D will form an image at a 6, the
position of which may be determined by
the method explained at section 419. A
portion of this light, after reflection, will
fall upon the mirror E F, and be reflected

as if coming from an image a' b', which will be seen by the eye at e.

To trace the course of the rays which enter the eye from any point, Q, in the
object A B ; let q be the corresponding point in a b, and q' a similar point in
o' &' ; the light will enter the eye as if it came from q' } therefore draw the lines




q' e, and they will show the final course of the pencil by which the point Q ia
seen. From the points where these lines meet the mirror, E F, draw lines to q,
and they will represent the course after reflection by C D ; from the points where
these lines meet the mirror C D, draw lines to the point Q, and they will show
the course of the rays which, after reflection by each of the mirrors C D, E F,
form the pencil by which the eye at e sees the point q' in the secondary image
a' b'.

The inversion of parts by the two mirrors are now seen to correct each other,
and all the parts of the image, a' b', have the same relation to each other as in
the object A B. The peculiar excellence of Wollaston's Camera hicida (518)
depends upon the fact that by means of two reflections all parts of the image
preserve their natural relations.

422. Multiplicity of images seen by means of inclined mir-
rors. When an object is placed between two mirrors, which make
with each other an angle of 90 or less, several 323
images are produced, varying in numbers accord-
ing to the inclination of the mirrors. If they are ,

placed perpendicular to each other, three images
will be seen, situated as in fig. 323.

The rays C and D, from the point 0, form, after a
single reflection, one the image 0', the other, the image
0" ; and the ray A, which undergoes two reflections at / ; /

A and B, gives a third image, 0'". When the inclination J,/ \? ,

of the mirrors is 60, five images are formed; and when

they are placed at an angle of 45, seven images are produced. The number
of images continues to increase as the inclination of the mirrors diminishes, and
when the mirrors become parallel, the number of images is theoretically infinite,
but as some of the light is lost at every reflection, and the successive images
appear more and more distant, only a moderate number of images are visible.

423. Deviation of light reflected by two mirrors. When a ray
of light reflected by a mirror is again reflected by a second mirror, in

Online LibraryBenjamin SillimanPrinciples of physics, or, Natural philosophy: designed for the use of colleges and schools → online text (page 33 of 78)