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.