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.

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).

In addition to

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.