Download - Experiment 2 Mirrors
Analysis
The experiment encompasses topics on optics particularly on plane and
spherical mirror. Generally, a light, like sound, behaves like a wave.
Thus, it also has properties like reflection, refraction, interference, and
diffraction. In this particular experiment, light is studied and treated like
wave with the use of a plane and spherical mirror.
Mirrors are used widely in optical instruments for gathering light and
forming images since they work over a wider wavelength range and do
not have the problems of dispersion which are associated with lenses and
other refracting elements.
For this experiment, materials that
are necessary are plane mirrors, a
pin, concave mirrors, candle, ray
table, three-surfaced mirror, meter
stick and optics table with light
source. A labeled presentation of
the experimental materials is
shown to the left (Fig. 2-1).
In the first part of the experiment, law of reflection is being proven. This
law states that the reflected ray lies in the plane defined by the incident
ray and the surface normal, and that in the plane the angle θ2 the
reflected ray makes with the normal is equal to that θ1 made by the
incident ray (See Figure 2-2 below). A beam of light is allowed to pass
through a plane mirror. As what is seen, the light reflects back at the
same angle from where the light strikes the mirror. Difficulty on
arranging the plane mirror is observed to have a perfect equal angle. The
angle from where the light strikes is called the angle of incidence while
the angle from where the light reflects back is called the angle of
reflection. If wrong set-up is made, it is noticed that the difference
between the angle of reflection and the angle of incidence increases as
the angle is increased upon rotating the mirror.
The data below (See Table 2-1)
shows the resulting angle of
incidence and its corresponding
angle of reflection. As seen, the
angles are exactly equal, thus, the
law of reflection is completely
proven.
Table 2-1. Laws of ReflectionTrial Angle of Incidence Angle of Reflection
1 10 102 30 303 50 504 70 70
Figure 2-1. Experimental Materials
Figure 2-2. Experimental Materials
The image to the left (See Fig. 2-3)
shows the actual reflection of the
light in the experiment. As seen,
the angle of incidence and the
angle of reflection form are
tantamount to each other which is
at 70o measured from the normal
line to the plane mirror.
For the next part, the number of images formed when two plane mirrors
are arranged in such a way that they are facing each other, having a
common edge and also a common angle between them are determined.
The set-up of this part is well presented below.
When two plane mirrors face each other, various reflections of images
are being seen on both sides of the mirror and is given by the equation
I=360θ
−1 equation 2−1
where I is the number of reflected images and θ is the angle between two
plane mirrors. With respect to our data, our result is somehow different
from what is theoretically observed. In the data below (See Table 2 also
associated with a graph), it is seen that when the angle is still big, the
number of reflected images can still be counted accurately but as the
angle is decreased, the possibility of miscounting also increase. It is due
there is maximum figures in the mirror which a human eye can see is
already attained.
Table 2-2. Number of ImagesTrials Angle between
plane mirrorsNumber of Images Formed
Observed Calculated1 10 24 352 15 16 233 30 10 114 45 8 75 60 6 56 75 4 47 90 2 38 120 2 2
The angle determines the number of image that can form on the two
plane mirrors. The image below (Figure 2-4) shows that a degree of 60oC
angle between two mirrors would make 5 images which follow the
equation 1. In the figure above shows the experimental and theoretical
image of the number of figure can be seen for a 60o between two mirrors.
As seen, the image is reflected as many as it can be depending on the
Figure 2-3. Reflection of angle beam at 70o
θ
Angle between them
Mirror 2Mirror 1
angle between the facing mirrors. It is observed that it is multiplied
according to a complete rotation of 360o, with respect to the angle
between the mirrors. However, the number should be decreased by one
because that “one” represents the real object.
In the graph, the violet curve line shows the theoretical number of images
formed while for the observed, it is presented by a blue line. As seen,
greater miscounting increase as the angle is decreased. It is observed that
the images formed are getting smaller and smaller that comes a point that
the image formed is cannot be seen by naked eye. Also, the brightness of
the image affects the number of images that can be seen by naked eye.
Since the experiment has dark surroundings, then it is possible to get a far
result than what is expected.
0 20 40 60 80 100 120 1400
5
10
15
20
25
30
35
40
Angle
Num
ber o
f Im
ages
For
med
Graph 2-1. Number of images formed vs. Angle obtained by calculations and experimentally.
On the third part of the experiment, the focal length of both concave and
convex mirror is obtained based on the reflected light observed on it.
These are done through mirror ray tracing. By this method, an imaginary
line which is the optic axis is drawn at the center of the mirror
perpendicular to its axis. When beam of lights are reflected by the mirror,
it all intersects at a common point (only when extended for convex
mirror). This point to the surface of the mirror through the optic axis is
the focal length of the mirror. By convention, for convex mirror the said
intersection point is located behind the surface of the mirror while for the
concave mirror it is located at front.
Figure 2-4. Experiment Part 2, Diagram on left side and actual experimentation on right side.
For convex mirror, a
ray parallel to the optic
axis and one incident
ray passing through and
reflected back from the
center of the mirror are
extended. The resulting
light rays are back-
drawn by projecting it from the rays. The distance from the common
point behind the surface of the mirror located at the principal axis to the
mirror itself is the actual measure of the focal length. So as with for the
concave mirror, the distance between the common point formed at the
front of the mirror at the principal or optical axis determines the focal
length of the mirror.
The focal length of
curved mirrors are said
to be twice the radius of
the circle fitted to the
surface of the mirror. It
is given by the equation
below. The image is also presented to allow us to have a better
understanding of this theorem.
f = R2
equation 2−2
Figure 2-6. The focal length based on the circle from where the arc of is segmented
Based on the image above, it can be said that the surface of a mirror can
be described as an arc segmented from a circle having a radius R.
Based on the focal length obtained from the actual and experimentally,
the values are very consistent with each other. It means that
experimentation is done correctly.
Table 3. Determination of Focal Length and RadiusTypes of Mirror Convex ConcaveFocal Length, f 6 7Experimental radius of curvature, R expt
12 14
Actual radius of curvature, R act
12 13.5
Percentage Error 0.0% 3.64%
Figure 2-5. Convex mirror focal length determination
Figure 2-6. Concave mirror focal length determination
R
As additional information, for concave mirrors, when the object is within
the range of the focal length, the formed image is enlarged, erected as is,
and formed behind the mirror as virtual image. While on the other hand,
when the object is outside the focal range, it is formed as a real image, in
front of the mirror, minimized and inverted. This scenario occurs in the
next part of the experiment.
The image above shows the actual experimentation in determining the
focal length of the spherical mirror. The convex mirror is presented to
left while the concave mirror is at the right.
In the fourth and last part of this experiment, a spherical mirror is used to
reflect back an object’s image through a detector which is a plane white
surface board.
As we all know, lights allows us to see object. When the light strikes the
object, its color, shape and depth can be seen. When a spherical mirror is
brought in front of the object, the light travels from the object to the
mirror, and is then reflected. If a detector is present, the reflected image
is seen at a certain degree of clarity and size. The image is also seen to be
inverted.
Figure 2-8. Theoretical observation on reflection image of an object (Part 4)
The clarity of the image and its size is related to the focal length of the
spherical mirror. The image would only be clear until when the distance
required between the mirror and the detector (the reflected image) is
attained at a known and distinct measure of distance between the object
itself and the mirror. It is given that the sum of the reciprocal of the
object distance and the image distance to the mirror is equal to the
reciprocal of mirror’s focal length. It is described by the equation below.
Figure 2-7. Part 3. Focal length determination for spherical mirrors (concave on left and convex on right)
1f= 1
p+ 1
qequation2−3
It is also given that when the sign of q is negative, virtual image is seen
which means that the reflected image is behind the spherical mirror. On
the other hand, when it is positive, the image is called a real image and is
located at front of the mirror. Also, the f and R are positive when the
spherical mirror is concave and will be negative if the mirror is convex.
In the experiment, we are dealing with real images and concave mirror.
This part is divided into four parts, namely the (a) object distance is
greater than the image distance, (b) image distance is greater than the
object distance, (c) object distance is equal to image distance, and (d)
object distance is very far that it is assumed to approach infinity.
On part A, (See Table 4), the data shows object distance greater than the
image distance. The percentage difference of the data with the actual is
1.96% which means that the data is somewhat consistent to what is
theoretically occurring. At those distances, the reflection of the image is
clear. Differences in the computed focal length are not major since they
only differ by a small amount. A sample computation of getting the data
is shown below using equation 2-3.
1f= 1
p+ 1
q
1f= 1
65.5 cm+ 1
25 cm
1f=0.05527/cm
f =18.09 cm
Table 4. Object Distance Greater than Image Distance p>qTrial Object Distance, p Image Distance, q Computed focal
length1 65.5 cm 25 cm 18.09 cm2 72.5 cm 24.5 cm 18.31 cm3 83.3 cm 23.2 cm 18.15 cm4 87 cm 22.8 cm 18.06 cm
Average focal length 18.1525 cmActual Focal Length 17.8 cm
Percentage Difference 1.960921 %
To determine the precision of the data, it is important to use a statistical
tool such as standard deviation to determine how each result is near or far
from each other. It is an effective way to know the consistency result. As
seen, the data fall under a normally skewed graph (a graph with normal
distribution). It means that the data are precise.
Sample computations for the mean, x and standard deviation, σ are
shown below.
x= 1N∑i=1
N
x i
x=14
(18.09+18.31+18.15+18.06 ) cm
x=18.1525 cm
Table 5. Data for Statistical Data Analysisx d i=¿x−x∨¿ d i
2
18.09 0.0625 0.0039062518.31 0.1575 0.0248062518.15 0.0025 6.25x10-6
18.06 0.0925 0.00855625x=¿18.1525 σ=0.111467484
σ=√ 1(N−1)∑i=1
N
d i2
σ=√ 14−1
(0.004+0.025+6.25 x 10−6+0.009)
σ=± 0.1147cm
E v=18.1525 ±0.1147
Table 6. Focal Length Standard Deviation, Part A
18.1525
0.111467
E v( x± σ ) 18.15 ± 0.11
x± σ 18.15 ± 0.11
x± 2σ 18.15 ± 0.22
x± 3σ 18.15 ± 0.33
On part B, the image distance is farther than the object distance. The data
seems to be consistent except for the fourth trial. But overall, the average
focal length is not that for from the actual focal length. Differences with
the computed results are brought about by certain errors that may be
rooted from the following sources:
Inaccuracy of the measuring materials
Parallax error on reading the measurement
Misinterpretation of a clear image seen
Table 7. Image Distance Greater than Object Distance q>pTrial Object Distance Image Distance Computed focal length1 28 cm 39 cm 16.3 cm2 25.5 cm 48 cm 16.65 cm3 24 cm 57 cm 16.89 cm4 25.5 cm 66.5 cm 18.43 cmAverage focal length 17.0675 cmActual Focal Length 17.8 cm
Percentage Difference 4.20162 %
To quantify those errors, we can use the propagation of error method to
determine how much error will make the data go wrong. The
uncertainties in measuring help us to determine how much difference it
will bring to the computed data. Based on observation, the meter stick
used is inaccurate by about ± 0.300 cm. Thus, it will be the standard
uncertainty for both distance records. The uncertainty is big since it is
patterned with the nature of the experiment where it is dark and the direct
distance measuring of “candle” to mirror is not accurate.
Table 8. Uncertainty on Measuring Distances Using Error Propagation MethodComputed focal length Uncertainty based on computed
16.3 ±0.238 cm16.65 ±0.242 cm16.89 ±0.246 cm18.43 ±0.247 cm
17.07±0.243 cm*Note: The data is computed using the rules of propagation of error
method. Formulas used are based on standard error propagation
determination method (See table below).
Equation 2-3 is reconstructed in such a way that propagation of error
standard formulas can be used. Rearrangement of equation is shown
below.
[ 1f= 1
p+ 1
q ] f
1=f ( q+ ppq )
f = pqq+ p
Step in determining the propagation is first done by determining the
propagation obtained on the numerator and the denominator,
respectively. So, division equation can be used afterwards.
Therefore, the error due to
inaccuracy in measuring using a
meter stick brought us a change of
±0.243 cm. If 0.243cm is added to
the average computed focal length,
the resulting percentage difference
will be reduced to 2.77% from
4.20% original.
On the third part, the image distance is equal to the object distance. It is
considered to be the hardest part of the experiment because two things
are need to be adjusted which are any of the white surface, the candle and
the spherical mirror. In the data shown below, it is observed that almost
in all trial, the distances measured are the same which is 35 cm. It
denotes that we are consistent with our data. It also implies that we are
correct since the focal length of mirror is constant. The percentage
difference that has been computed is not that high so, we can conclude
that we are making a careful and an accurate experimentation.
Table 9. Object Distance Equal to the Images DistanceTrial Object Distance Image Distance Computed focal length
1 35 cm 35 cm 17.5 cm2 35 cm 35 cm 17.5 cm3 33 cm 33 cm 16.5 cm4 35 cm 35 cm 17.5 cmAverage focal length 17.25 cmActual Focal Length 17.8 cm
Percentage of Difference 3.138374 %
In the figure below (See Figure 2-10) shows the relative image when the
image distance is farther and when it is nearer (arranged from left to
right). The image is somewhat darker when the object is farther. Their
distances are to be measured to determine the experimental value for f,
the focal length of the mirror.Figure 2-9. Error in measuring
Finally, in the last part the actual value for the focal length is determined
by having a very far object distance. By that, it is assumed that the object
distance q is very large that it is equal to infinity. Using this method, we
could directly acquire the value of the focal length by using the value of
the image distance. Since p=∞, then 1/p=0. Thus, q=f. From the values
we obtained, we find equivalent values of 17.8 cm.
That value would represent the theoretical value for the focal length of
the mirror. It is advised for the performers to minimize on committing
mistakes for this part because of the dependency of other data with the
data obtained in here.
Figure 2-10. Image reflected is (a) farther and (b) nearer the mirror.
Conclusion
Light also behaves like wave, so it also has certain characteristics similar
in describing waves. A mirror is an optical tool which formed images by
gathering the light.
For a plane mirror, the angle between the normal plane which is the angle
of reflection and incidence are the equivalent to each other. It shares a
common side which is a line normal to the mirror itself.
When two mirrors are arranged in such a way that they are facing each
other facing a common side, with a certain angle between them, the
image that is present in front of the mirrors are formed a definite times
depending on the inclination between them. It is found out that at larger
angles, less number of images are formed in the mirrors while at small
angle, large number of images is seen. It follows a circular path, with
respect to the angle of inclination. At smaller angle, less accuracy of
counting are done due to some of the images formed are too small or too
dim to be seen by a naked eye.
The focal length of the mirror determines how image will be projected or
reflected by the mirror depending on the distance from where the object
is located, etc. One way of determining the focal length of a spherical
mirror is through ray tracing method. For convex mirror, the ray of light
that is reflected is projected back through the mirror and is located behind
it at the optic axis. On the other hand, the ray of lights reflected back by
the concave mirror are intersected through one point that is located in
front of the mirror lying on the optic axis. The distance between the
intersected points to the center of the curvature is the focal length of the
mirror. It is also equal to the radius of the circle from where the curved
mirror is segmented.
The object when reflected its image by a mirror would become clear at a
certain distance from the image to the mirror and the object to the mirror
and is determined using the mirror’s focal length. The reciprocal of the
focal length of the mirror is equal to the sum of the reciprocal of the
image distance and the object distance. When the object distance is
greater than the focal length, the image would form in front small and
inverted, known to be a real image. But when the object distance is less,
it is reflected at the back of the mirror, enlarged and erected as is.
Experimental errors are not that serious because data are consistent with
each other. Less discrepancy from the actual or theoretical values are
obtained, which means that the performers had done a good
experimentation and yield good results. As for the experiment, it can be
said that its objectives are well attained.