# In The lens equation can be used to

In 1621, Snell derived Snell’s law. Using
this equation, and finding the 2 angles as light passes through 2 different
mediums, allows us to find the unknown refractive index of the medium. The
refractive index calculated for glass was 1.72±0.230 and
for Perspex was 1.52±0.0698 – which are both valid answers.

Using
the lens equation, the focal length of a lens can be determined by alternating
the object-lens (u) distance and the lens-image distance (v). Using the parallax
method, as shown later in diagram 2, the focal length of the 50mm converging
lens was found to be 86.36±1.160 mm – which is 1.7272 times greater
than its true focal length.

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INTRODUCTION

Ibn
Sahl was the first person to be recorded who understood a basic, stripped-back
version of what is now interpreted as Snell’s law, back in 984 11. Therefore,
Sahl was the first person to build the basics of the study of refraction, or ‘dioptrics’
12. Then, in 1621 11, Snell derived Snell’s law, which is also known as “the
law of refraction” 13, which links how light diffracts between two different
densities.   The lens equation can be used to make corrective
lenses, which can fix eyesight problems such as myopia and hyperopia. Combining
both the lens equation and Snell’s law can help make microscopes and telescopes
in which diffract at angles to be able to view at angles.

THEORY

‘Refraction’ is the change in a waves
pathway due to a change in density, which affects the waves speed 1. The amount of refraction is determined by the angle
which the wave hits the boundary of the medium and the change in density 2.

Experiment A allows observation of the phenomenon – as the change in the lights
pathway through the transparent glass/perspex block, which is traced. A
numerical value is assigned to how the density affects the wave, which is
referred to as a ‘refractive index’. The refractive index is a value which
indexes how the speed of a light changes as it enters the material compared to
the speed of light in a vacuum 3. The refractive index of a material can be
experimentally found by measuring the angle of incidence and the angle of
refraction, and then substituting them both into Snell’s Law, (1). We can later compare our experimentally values to
the designated refractive indexes where: nair is 1.00 4, nglass
is 1.52 5,
Refractiveindex.info, 2018 and nperspex
is 1.50 6. Total internal reflection occurs when a wave hits a
boundary at an angle which is greater than the critical angle 7. To measure the critical angle in which ‘TIR’ can
occur, the equation  (2).

With an ‘object’ pin and an ‘image finder’ pin on
either side of a convex lens, the focal length of the lens can be determined.

By nature, a convex lens creates an image which is real and upside down if
object is further from the lens than the focal point 8. The independent variable, the object pin, can be
changed to adjust ‘u’ – the distance between the lens and the ‘object’ pin.

Following the parallax method, when the ‘image finder’ pin is adjusted so that
the principle rays can be applied (as shown in diagram 1). Then, from
eye-level. through the focal lens, the two pins will look as though the tips
are just touching (shown in diagram 2). Using the lens equation; (3)  9, the equation (4) can be derived. Following this equation, if a
graph of uv against (u+v) is plotted, the gradient is equivalent to the focal
length. When plotted, the focal length can mathematically be found using;   (5)

Experimental methods

Part A: finding the refractive index

Before starting, drawing around the prism
ensures that the prism can be readjusted if knocked or moved. An additional
insurance can be made by performing this experiment in a dark room, as then the
light is more visible and therefore less mistakes are to be made when following
the lights pathway.

A
white light directed through the one of the two shorter edges of the triangular
block. The light then came out of the block’s hypotenuse. The pathway should be
marked just before the light entered and just after the light exited the block.

After this is done, the block can then be removed and in the white space where
the block once laid, the entry and exit points can be joined. The point where
the block leaves the hypotenuse, a perpendicular line should be draw. Then, two
opposite angles should be taken between the normal and the pathway of light (as
demonstrated in diagram 3)

The two angles can then see
substituted into Snell’s law (see equation 1). Rearranging the equation allows
the unknown medium’s refractive index to be solved, as we know the refractive
index of air is 1.00. This methodology can be repeated several times to gain a
greater amount of precision, and can then be repeated for the other medium.

Diagram 3: how
to label the angles in respect to the normal

Part B: parallax method:

A bench with two pins on either side of a
convex lens should be set up. Using this set up, the focal length of the lens
can be solved. Changing the object pin, the independent variable, should be
changed several times to different lengths of ‘u’ – distance between the pin
and the lens. By changing this, we can then change the dependent variable, the
‘image-finder’ pin. As shown in diagram 2, the ‘image finder’ should be
adjusted so at eye-level the two pins just meet. Due to error associated with
this point being visually decided, a second opinion should be taken to whether
they are just touching. The distance between the ‘image-finder’ pin and the
lens can be labelled as ‘v’. This can then be repeated, so there are several
pairs of u and v values. As discussed in the theory, using equation 4, a graph
can be plotted to find the focal length. Comparing this to the general equation
of a straight line, , the gradient of the line of best fit can be allocated to the
focal length.

RESULTS

GLASS

Incident
angle/ °

Diffracted angle/ °

Refractive index value

1

44.0

59.0

1.23±0.0174

2

27.0

71.5

2.09±0.0414

3

31.0

59.5

1.67±0.0304

4

21.0

42.0

1.87±0.0498

PERSPEX

Incident
angle/ °

Diffracted angle/ °

Refractive index value

1

29.0

53.0

1.65±0.0324

2

48.5

70.0

1.25±0.0157

3

20.0

30.0

1.46±0.0439

4

24.0

44.0

1.71±0.0406

In experiment B; inserting the angle of incidence and
the angle of refraction into Snell’s law (see equation 1), and then
rearranging, allowed singular refractive indexes associated with each pair of
angles. The errors for each measured angle were ±0.5°, as the resolution of the protractor was 1°. The overall
error for the refractive index was calculated using the errors of
multiplication equation:  (6).

In experiment
B, errors associated with ‘u’ and ‘v’ were both ±0.5, as the ruler we used had a resolution of 1mm.

Therefore using the equation for the addition of errors;  (7), all of the values for ‘u’+’v’ result in
having the same error of ±0.707. The error found for ‘u’·’v’ was found using equation 6.

Plotting each refractive
index onto a graph with their associated error bars, as shown in graph 1 and 2,
meant any anomalies could be dismissed if they weren’t coherent and within the
ranges of the other values. For both glass and Perspex, none of the values were
discounted as invalid. Through visual comparison of the two graphs, Perspex
generally has greater values of error than glass and it can therefore be
concluded that glass has a more accurate experimentally-found refractive index
than Perspex.

Subsequent to finding
refractive indexes; where glass’ refractive index was 1.72±0.230 and Perspex had a refractive index of 1.52±0.0698, the critical angle could be found for each
material. Using equation 2,The critical angle using our experimental value of
glass can therefore be found as 35.5±4.747°
and the critical angle of our value for Perspex is found to be 41.1±1.887°.

‘u’ / mm

‘v’ / mm

‘u’ + ‘v’ / mm

‘u’·’v’ / mm

126

222

348±0.707

27972±127.632

89

153

242±0.707

13617±88.501

46

82

128±0.707

3772 ± 47.011

112

195

307±0.707

21840 ± 112.438

67

117

184±0.707

7839 ± 67.413

120

86

206±0.707

10320 ± 73.817

75

150

225±0.707

11250 ± 83.853

96

106

202±0.707

10176 ± 71.505

59

324

383±0.707

19116 ± 164.664

The plot used for experiment B, graph 3, has a
definite positive correlation from visual reference. Using Pearson’s
product-moment correlation coefficient for all the points found, r=0.90986106., This implies a very strong positive
correlation. However, the insignificantly small error bars
imply that 5 points are invalid. Calculating the r value for the trend
discarding the ‘invalid’ points, marked in orange on graph 3, causes the ‘r’ value
is worked out as 0.9951447, rounded as 1.00 to three significant figures.

Therefore, excluding these values cause a full coherent set of results with the
strongest positive correlation.

Using
equation 4, the gradient of the line of best fit can then be found to be equal
to the focal length – following the equation
Rearranging this to singular out the focal length (see equation 5), and then
inserting the points (128, 3772) and (242, 13617), which are two points that
are located with a range around the line of best fit, the focal point is found
to be 86.36±1.160 mm.

DISCUSSION:

In experiment A, the refractive index for
each material was distinguished by taking 4 pairs of angles and inserting them
into Snell’s law (see equation 1). Then, from the 4 individually found
refractive indexes, an average was calculated.

Primarily,
when looking at the refractive indexes for glass, the value 2.09 was queried as
an anomaly. However, after plotting the refractive indexes with their associated
errors (as seen in graph 1). Doing this allows the visual conclusion that 2.09
is valid, and therefore is included within the average. Therefore, the average refractive
index for glass is calculated as 1.72±0.230, which has
the set refractive index of 1.52 5 within its range.

With
the defined refractive index of 1.50 6 being within its range, the
experimentally found value of 1.52±0.0698 is valid.

To
improve the accuracy and reliability of both of the found refractive indexes,
and therefore making the results repeatable, the average should have been
formulated out of more values.  Consequently,
this would have led for the calculated value to be refined closer to the ‘true’
values for both mediums.

In experiment B,
each plotted point has error bars so visibly tiny that they are fairly
insignificant (as seen in graph 3). The small error bars account for 5 of the 9
points being discounted as invalid, as they are not visually in range of the
line of best fit. Dismissing these values is validated by calculating the Pearson’s
product-moment correlation coefficient with and without these 5 errors (as
discussed previously).

The reliability
of the focal point experimentally found is therefore compromised, due to having
less values – as only 4 points can be used to find the gradient. This is
reflected in the calculated focal length of 86.36±1.160, which
is 1.73 times greater than the actual focal point (50mm).

experiment A, repeating this experiment would increase the reliability of the
focal length. From the plotted and dismissed points, a 44.4% repeatability can
be assumed. Therefore, using more sets of values would increase the precision
of the found focal length.

CONCLUSION:

Glass (with a
found refractive index of 1.72±0.230) is a denser medium than Perspex (1.52±0.0695), which is reflected by glass have a
greater refractive index. The significance of this value is related to the
speed of light in the medium in comparison to the speed of light in a vacuum
3, which is valued at c = 3.00×108
m/s 10. The greater density of glass has a visible effect on the amount of
diffraction when performing the experiment, as the diffraction is greater –
which is reflected generally using Snell’s law (see equation 1).

Due to the greater
density, and therefore higher valued refractive index, glass has a smaller
critical angle (35.5±4.747°) than the critical angle
of Perspex (41.1±1.887°). Due to glass having a
smaller angle, the angle at which total internal reflection occurs is smaller
than for Perspex. This means that light is reflecting back onto the first
medium rather than passing through to the other 7.

The calculated focal length
of the lens we found (86.36±1.160 mm) is 1.7272 times greater than the actual
value of 50mm. However, despite our value being invalid, they both follow the
same pattern – shown by using equation 3. T he values of ‘u’ and ‘v’, as shown
in diagram 2, are both positive values that are greater than the focal length.

However, the values calculated for our focal length won’t be able to be as
small as the real values of the focal length of 50mm.