# In The lens equation can be used to

In 1621, Snell derived Snell’s law. Usingthis equation, and finding the 2 angles as light passes through 2 differentmediums, allows us to find the unknown refractive index of the medium. Therefractive index calculated for glass was 1.

72±0.230 andfor Perspex was 1.52±0.0698 – which are both valid answers.                  Usingthe lens equation, the focal length of a lens can be determined by alternatingthe object-lens (u) distance and the lens-image distance (v). Using the parallaxmethod, as shown later in diagram 2, the focal length of the 50mm converginglens was found to be 86.

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36±1.160 mm – which is 1.7272 times greaterthan its true focal length.  INTRODUCTION IbnSahl was the first person to be recorded who understood a basic, stripped-backversion 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 “thelaw of refraction” 13, which links how light diffracts between two differentdensities.   The lens equation can be used to make correctivelenses, which can fix eyesight problems such as myopia and hyperopia. Combiningboth the lens equation and Snell’s law can help make microscopes and telescopesin which diffract at angles to be able to view at angles.  THEORY ‘Refraction’ is the change in a wavespathway due to a change in density, which affects the waves speed 1. The amount of refraction is determined by the anglewhich 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 lightspathway through the transparent glass/perspex block, which is traced.

Anumerical value is assigned to how the density affects the wave, which isreferred to as a ‘refractive index’. The refractive index is a value whichindexes how the speed of a light changes as it enters the material compared tothe speed of light in a vacuum 3. The refractive index of a material can beexperimentally found by measuring the angle of incidence and the angle ofrefraction, and then substituting them both into Snell’s Law, (1).

We can later compare our experimentally values tothe designated refractive indexes where: nair is 1.00 4, nglassis 1.52 5,Refractiveindex.info, 2018 and nperspexis 1.50 6. Total internal reflection occurs when a wave hits aboundary at an angle which is greater than the critical angle 7. To measure the critical angle in which ‘TIR’ canoccur, the equation  (2).                   With an ‘object’ pin and an ‘image finder’ pin oneither 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 ifobject is further from the lens than the focal point 8. The independent variable, the object pin, can bechanged to adjust ‘u’ – the distance between the lens and the ‘object’ pin.Following the parallax method, when the ‘image finder’ pin is adjusted so thatthe principle rays can be applied (as shown in diagram 1). Then, fromeye-level. through the focal lens, the two pins will look as though the tipsare just touching (shown in diagram 2). Using the lens equation; (3)  9, the equation (4) can be derived. Following this equation, if agraph of uv against (u+v) is plotted, the gradient is equivalent to the focallength.

When plotted, the focal length can mathematically be found using;   (5)                                 Experimental methods Part A: finding the refractive index Before starting, drawing around the prismensures that the prism can be readjusted if knocked or moved. An additionalinsurance can be made by performing this experiment in a dark room, as then thelight is more visible and therefore less mistakes are to be made when followingthe lights pathway.                  Awhite light directed through the one of the two shorter edges of the triangularblock. The light then came out of the block’s hypotenuse. The pathway should bemarked 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 wherethe block once laid, the entry and exit points can be joined. The point wherethe block leaves the hypotenuse, a perpendicular line should be draw.

Then, twoopposite angles should be taken between the normal and the pathway of light (asdemonstrated in diagram 3)                                                                                          The two angles can then seesubstituted into Snell’s law (see equation 1). Rearranging the equation allowsthe unknown medium’s refractive index to be solved, as we know the refractiveindex of air is 1.00. This methodology can be repeated several times to gain agreater 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 aconvex lens should be set up.

Using this set up, the focal length of the lenscan be solved. Changing the object pin, the independent variable, should bechanged several times to different lengths of ‘u’ – distance between the pinand 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 beadjusted so at eye-level the two pins just meet. Due to error associated withthis point being visually decided, a second opinion should be taken to whetherthey are just touching. The distance between the ‘image-finder’ pin and thelens can be labelled as ‘v’. This can then be repeated, so there are severalpairs of u and v values. As discussed in the theory, using equation 4, a graphcan be plotted to find the focal length.

Comparing this to the general equationof a straight line, , the gradient of the line of best fit can be allocated to thefocal 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 andthe angle of refraction into Snell’s law (see equation 1), and thenrearranging, allowed singular refractive indexes associated with each pair ofangles. The errors for each measured angle were ±0.5°, as the resolution of the protractor was 1°.

The overallerror for the refractive index was calculated using the errors ofmultiplication equation:  (6).   In experimentB, 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 inhaving the same error of ±0.707. The error found for ‘u’·’v’ was found using equation 6. Plotting each refractiveindex 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 theranges of the other values. For both glass and Perspex, none of the values werediscounted as invalid. Through visual comparison of the two graphs, Perspexgenerally has greater values of error than glass and it can therefore beconcluded that glass has a more accurate experimentally-found refractive indexthan Perspex.                   Subsequent to findingrefractive 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 eachmaterial.

Using equation 2,The critical angle using our experimental value ofglass 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 adefinite positive correlation from visual reference. Using Pearson’sproduct-moment correlation coefficient for all the points found, r=0.

90986106., This implies a very strong positivecorrelation. However, the insignificantly small error barsimply that 5 points are invalid.

Calculating the r value for the trenddiscarding the ‘invalid’ points, marked in orange on graph 3, causes the ‘r’ valueis 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 thestrongest positive correlation.                   Usingequation 4, the gradient of the line of best fit can then be found to be equalto the focal length – following the equation Rearranging this to singular out the focal length (see equation 5), and theninserting the points (128, 3772) and (242, 13617), which are two points thatare located with a range around the line of best fit, the focal point is foundto be 86.36±1.

160 mm.      DISCUSSION: In experiment A, the refractive index foreach material was distinguished by taking 4 pairs of angles and inserting theminto Snell’s law (see equation 1). Then, from the 4 individually foundrefractive indexes, an average was calculated.                   Primarily,when looking at the refractive indexes for glass, the value 2.

09 was queried asan anomaly. However, after plotting the refractive indexes with their associatederrors (as seen in graph 1). Doing this allows the visual conclusion that 2.09is valid, and therefore is included within the average.

Therefore, the average refractiveindex for glass is calculated as 1.72±0.230, which hasthe set refractive index of 1.

52 5 within its range.                    Withthe defined refractive index of 1.50 6 being within its range, theexperimentally found value of 1.52±0.

0698 is valid.                  Toimprove the accuracy and reliability of both of the found refractive indexes,and therefore making the results repeatable, the average should have beenformulated 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 fairlyinsignificant (as seen in graph 3).

The small error bars account for 5 of the 9points being discounted as invalid, as they are not visually in range of theline of best fit. Dismissing these values is validated by calculating the Pearson’sproduct-moment correlation coefficient with and without these 5 errors (asdiscussed previously).                   The reliabilityof the focal point experimentally found is therefore compromised, due to havingless values – as only 4 points can be used to find the gradient. This isreflected in the calculated focal length of 86.36±1.160, whichis 1.

73 times greater than the actual focal point (50mm). In addition toexperiment A, repeating this experiment would increase the reliability of thefocal length. From the plotted and dismissed points, a 44.4% repeatability canbe assumed. Therefore, using more sets of values would increase the precisionof the found focal length.

CONCLUSION:Glass (with afound refractive index of 1.72±0.230) is a denser medium than Perspex (1.

52±0.0695), which is reflected by glass have agreater refractive index. The significance of this value is related to thespeed of light in the medium in comparison to the speed of light in a vacuum3, which is valued at c = 3.00x108m/s 10. The greater density of glass has a visible effect on the amount ofdiffraction when performing the experiment, as the diffraction is greater –which is reflected generally using Snell’s law (see equation 1).

Due to the greaterdensity, and therefore higher valued refractive index, glass has a smallercritical angle (35.5±4.747°) than the critical angleof Perspex (41.1±1.887°). Due to glass having asmaller angle, the angle at which total internal reflection occurs is smallerthan for Perspex. This means that light is reflecting back onto the firstmedium rather than passing through to the other 7. The calculated focal lengthof the lens we found (86.

36±1.160 mm) is 1.7272 times greater than the actualvalue of 50mm. However, despite our value being invalid, they both follow thesame pattern – shown by using equation 3. T he values of ‘u’ and ‘v’, as shownin 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 assmall as the real values of the focal length of 50mm.