LIGHT AND GEOMETERIC OPTICS
ATWOOD’S MACHINE
1. ELECTROMAGNETIC WAVES
White light is a mixture of all visible colors. When a beam of white light passes
through a glass prism, the resulting beam is dispersed into a spectrum of colors.
Dispersion of light in a transparent material occurs because the index of refraction of the
material varies with wavelength, with higher indices corresponding to shorter wavelengths.
Wavelengths in the narrow, visible light spectrum range from 750 nm for the color
red down to 400 nm for the color violet as shown below.
700 nm
4x10 14 Hz
Wavelength, λ
400 nm
Frequency, f
7.5x10 14 Hz
However, visible light makes up just a tiny portion of the large electromagnetic spectrum.
Types of waves with longer wavelengths than visible light include infrared radiation,
microwaves, and radio waves. Those with shorter wavelengths include ultraviolet
radiation, x-rays, and gamma-rays.
ELECTROMAGNETIC WAVES
λ (m)
f (Hz)
5
radio waves
0.34 - 570
5x10 - 9x108 Hz
microwaves
1x10-3 - 0.1
3x109 - 3x1011 Hz
infrared radiation (IR)
7x10-7 - 1x10-3
3x1011 - 4x1014 Hz
visible light (VIS)
4x10-7 - 7x10-7
4x1014 - 7.5x1014 Hz
ultraviolet (UV)
1x10-8 - 4x10-7
7.5x1014 - 3x1016 Hz
x-rays
1x10-11 - 1x10-8
4x1016 - 3x1019 Hz
gamma-rays
< 1x10-11
> 3x1019 Hz
Because light is made up of transverse waves, it can be polarized. The direction of
polarization is taken to be the direction of the electric field vector. A special filter known as
a polarizer produces polarized light from unpolarized light. The same filter can be used as
an analyzer to determine whether or not light is polarized. When the axis of the analyzer is
parallel to the plane of polarized light, the greatest amount of light is transmitted. When the
analyzer is perpendicular to the plane of polarized light, no light transmission occurs.
2. REFLECTION & REFRACTION
KEY CONCEPT – LIGHT & GEOMETERIC OPTICS
Light is assumed to travel along straight-line paths known as rays. When light
strikes a surface, some of it undergoes reflection. A light ray strikes a surface at a certain
angle of incidence, θi relative to the surface normal, and is reflected at an identical angle
of reflection, θr. This is the law of reflection as given by:
θi = θ r
θi
θr
If the surface of reflection is a flat, shiny mirror, the process is known as specular
reflection. The law also holds for diffuse reflection from a rough surface as long as each
small section of surface is considered separately.
If a light ray passes into a transparent medium at an angle θ1 that is not equal to
90°, the path of the ray bends. This bending of light is known as refraction, and the angle
θ2 at which it is bent is known as the angle of refraction.
θ1
θ2
Light travels at a speed of c = 3.00 × 108 m/s in a vacuum, but travels slower in
other transparent media such as water or glass. It is the wavelength of light, not the
frequency, that varies from medium to medium. The index of refraction n is the ratio of
the speed of light in a vacuum c to the speed of light in another medium v as given by:
c
n=
v
The index of refraction can never be less than one because v can never be greater than c.
When light passes from one medium to another medium, the angle of refraction θ2
depends on the angle of incidence θ1 and the speed of light in the two media involved.
Therefore, the index of refraction in first medium n1 is related to that in the second medium
n2 by the equation:
n1 sin θ1 = n2 sin θ2
This relationship is known as Snell’s Law.
If light rays traveling in a medium with a high index of refraction strike a second
medium with a lower index of refraction at an angle that is large enough, the rays cannot
KEY CONCEPT – LIGHT & GEOMETERIC OPTICS
refract into the second medium, and all of the rays are reflected at the interface. This
phenomenon is known as total internal reflection. The angle at which it occurs is given
by:
n2
n1
where the angle θc is known as the critical angle.
sin θC =
2. MIRRORS & LENSES
When looking at a plane mirror, the image appears to be located behind the mirror.
Because light rays do not actually pass through this image, it is known as a virtual image.
In the case of a plane mirror, the image distance si from the mirror is the same as the
object distance so from the mirror.
The most common curved mirror is a spherical mirror. A concave or converging
mirror has its reflecting surface on the inside of the sphere. Parallel light rays reflecting on
a converging mirror converge to a focal point F which is located midway between the
mirror and its center of curvature C. Therefore, the focal length f of the mirror is half the
radius of curvature R. A convex or diverging mirror has its reflecting surface on the
outside of the sphere. Parallel light rays that are reflected by a diverging lens diverge as if
their origin were a focal point behind the mirror.
C F
F C
Diverging Mirror
Converging Mirror
There are certain principle rays that are defined according to their path with respect to the
mirror’s geometry. These include:
•
A parallel ray that is incident along a path parallel to the mirror’s axis and is
reflected through the focal point.
•
A chief ray that is incident through the center of curvature and is reflected back
along its incident path.
•
A focal ray that passes through the focal point and is reflected parallel to the
mirror’s axis.
C F
Parallel ray
C F
Chief ray
C
F
Focal ray
KEY CONCEPT – LIGHT & GEOMETERIC OPTICS
The image formed by a converging mirror can be real or virtual, whereas the image formed
by a diverging mirror can only be virtual. The image characteristics depend on the position
of the object relative to the mirror. For a converging mirror these characteristics can be
summarized as follows:
•
For so > R the image is real, inverted, and reduced.
•
For so = R the image is real, inverted, and the same size.
•
For R > so > f the image is real, inverted, and enlarged.
•
For so = f the image is at infinity.
•
For so < f the image is virtual, upright, and enlarged.
f
object
image
C
real
Inverted
enlarged
F
R
si
so
The object distance, the image distance, and the focal length are related by the spherical
mirror equation:
1 1 1
+ =
si so f
and the magnification M of the mirror is given by:
M =−
si hi
=
so ho
where hi is the image height and ho is the object height.
A lens is a piece of curved, clear material that bends light by refraction. A biconvex
spherical lens is a converging lens with both surfaces convex. A biconcave spherical lens
is a diverging lens with both surfaces concave. These lenses have two focal points and
two centers of curvature, one for each lens surface.
KEY CONCEPT – LIGHT & GEOMETERIC OPTICS
F
F
F
F
Diverging lens
Converging lens
A converging lens can form either a real or a virtual image, whereas a diverging lens is
capable of forming only a virtual image. The image characteristics depend on the position
of the object relative to the lens. For a converging lens these characteristics can be
summarized as follows:
•
For so > f the image is real and inverted, and it can be enlarged, reduced, or the
same size.
•
For so = f the image converges at infinity.
•
For so < f the image is virtual, upright, and enlarged.
image
real
inverted
reduced
object
ho
F
F
so
hi
si
The object distance, the image distance, and the focal length are related by the lens
equation:
1 1 1
+ =
si so f
and the magnification M of the lens is given by:
M =−
si hi
=
so ho
4. DIFFRACTION
When waves encounter an object, they bend around its edge into the region directly
behind the region. This phenomenon is known as diffraction. Diffraction occurs at the
KEY CONCEPT – LIGHT & GEOMETERIC OPTICS
edges of openings or slits. The diffraction effect is greatest when the slit width is on the
order of the wavelength of light that is being diffracted.
In Young’s double-slit experiment, light emerging from the two slits is used as two
coherent sources. When the light is projected onto a screen, an interference pattern
appears that consists of a series of alternating bright and dark fringes around a central
maximum. Because the bright fringes indicate constructive interference, the path
difference between two waves must be an integral number of wavelengths. The positions
of the bright fringes are given by the equation:
dsinθ = mλ m = 0,1,2,3,...
where d is the distance between slits, θ is the angle of the bright fringe from the center line,
λ is the wavelength of the illuminating light and m is the order number. Dark fringes occur
when the path difference is an odd number of half wavelengths, and their positions are
given by the equation:
dsinθ = m(λ 2) m = 1,3,5,...
For a single slit, the width of the central maximum of the diffraction pattern is twice
that of the side fringes. The positions of the dark side fringes are given by the equation:
wsinθ = mλ m = 1,2,3,...
where w is the slit width.
A diffraction grating is a device that consists of thousands of slits per centimeter.
The conditions required for a diffraction grating to produce bright fringes are the same as
for the double-slit setup. Diffraction gratings produce very sharp bright lines and are used
in spectrometers to make precise measurements of light wavelengths.
When light reflects off the two surfaces of a thin film, an interference pattern is
formed. This phenomenon is known as thin film interference. A thin coating is applied to
lenses to make them nonreflective. For the coating to be nonreflective to light of
wavelength λ, the minimum thickness t of the coating is given by the equation:
t=
λ
4n
where n is the refractive index of the lens glass. When a curved lens is placed on a flat
glass plate, the wedge of air that is trapped between the two produces an interference
pattern consisting of concentric bright and dark circular fringes known as Newton’s rings.