Light Intensity: How the Distance Affects the Light Intensity

Transcription

1 Light Intensity: How the Distance Affects the Light Intensity Fig 1: The Raw Data Table Showing How the Distances from the Light Bulb to the Light Probe Affects the Percent of Maximum Intensity Distance from the light bulb Percent of Maximum Intensity (± 0.01) to the light probe (m) ( ± 0.004m) Trial 1 Trial Trial The percent of maximum intensity values have all been subtracted by 0.0 because it was the average value the probe measured without the light source on. Also, the uncertainty for the distance was judged to be ± 0.004m. By separately measuring the distance from the edge of the container to the light probe in it and the distance from the light source to the probe, there was already an uncertainty of ± 0.001m. Another ± 0.003m was added because there would have been some additional uncertainties from where the ruler measured to be the edge of the light bulb. Moreover, the uncertainty of the percent of maximum intensity was ± 0.01, because a computer collected the values. Sample Calculations: Calculating the average of percent of maximum intensity of when the distance is 1.040m ± 0.004m P1 + P + P3 Averagemaximumintensity =, where P, 1 P and P respectively represents the percent of maximum intensity 3 3 in trial 1, and 3. (0.16) + (0.17) + (0.15) = = 3 = 0.16

2 Calculating the uncertainty for the maximum intensity when the distance is 1.040m±0.004m Average maximum intensity (A)=0.16, as calculated above Maximum residual=(maximum data value collected) (A) = =0.01 Minimum residual=(a) (minimum data value collected) = =0.01 The greater value between the maximum and minimum residual is the uncertainty. However, for the distance of 1.040m±0.004m, both the residuals are the same. Therefore, the uncertainty of the average maximum intensity is ±0.01m for the distance of 1.040m±0.004m. Calculating the average light intensity when the distance is 1.040m ± 0.004m: As calculated before, the average maximum intensity is 0.16 when the distance is 1.040m ± 0.004m. Since the average light intensity is multiplied by a constant, its uncertainty should be multiplied by the same constant as well. Average light intensity = (average maximum intensity) 0.75 = ( 0.16 ± 0.01) 0.75 = ( ) ± ( ) = ± W m ± 0.01W m Fig : Processed Data Table Showing the Relationship between the Distance from the Light Bulb to the Light Probe and the Average Percent of Maximum Intensity, its Uncertainty as well as the Average Light Intensity and its Uncertainty Distance from the light Average Percent Uncertainty for the Average Light Intensity Uncertainty for the bulb to the light probe of Maximum Average Percent of (W m ) ( ± 0.01W m ) Average Light (m) ( ± 0.004m) Intensity Maximum Intensity (±) Intensity ( ± W m ) There were various distances, 0.061m, 0.140m, 0.190m, 0.40m, 0.90m, 0.340m, 0.390m, 0.440m and 0.790m, where the uncertainty for the average percent of maximum intensity, as well as the uncertainty for the average light intensity, was 0. However, this is not very realistic because when performing various trials even the light probe is prone to have at least a small degree of uncertainty. Thus, for the measurements when the uncertainties were 0, an uncertainty of ±0.01m was

3 established for the average percent of maximum intensity. The uncertainty was the smallest value that had the same number of decimal places as the average percent of maximum intensity Fig 3: Average Light Intensity vs. Distance from the Light Bulb to the Light Probe Average Light Intensity Distance from the Light Bulb to the Light Probe (m) Most of the vertical error bars were very small since the uncertainties were mostly ±0.01W m, for all the data which is a very small value. The horizontal error bars are also small because the uncertainty for distance was only ± 0.004m. The first eight data points are very similar to one another and there is not great change in the average light intensity as the distance increases. However, according to the equation of Power = Intensity Area = I 4πd, as the distance increases, the intensity should decrease by a function of inverse squared. The first eight collected values do not support the equation. In fact, only from the eighth data value and the subsequent ones support the equation. Thus, if the first seven points are excluded from the graph, the data can be represented as shown in Fig 4.

4 Fig 4: Average Light Intensity vs. Distance from the Light Bulb to the Light Probe, Excluding the First Seven Data Points Average Light Intensity y = x.005 R² = Distance from the Light Bulb to the Light Probe (m) Again as aforementioned, both the horizontal and vertical error bars are very small because their uncertainties are respectively ± 0.004m and ±0.01W m which are small in comparison to the measured distance and the average light intensity. Sample Calculations: The slope of Fig 4 is neither linear nor significant. Therefore, for the slope to represent power, the distance has to be manipulated. Δy slope = = Power Δx Δy Power = I A =, Δx where I is the intensity, A the area of the sphere with distance d, and Δy the intensity I. 1 Therefore, the x-axis has to represent 4πd I A = I Δx 1 1 Δx = = A 4πd for the slope of the graph to signify power.

5 1 Calculating the value of when d=0.440m ± 0.004m: 4πd The uncertainty is divided by the distance value to calculate the ratio of the two. 1 1 = 4πd π (0.440± ) ± = 4π Then the ratio of the uncertainty to the distance was multiplied by because the was raised to the power of. It was also multiplied with the processed distance of inverse squared of ( 0.440) ( ) = ( 0.440) ± 4π 0.440! ± (" ) 4! { } Since the distance is divided by a constant, the uncertainty too should be divided by the same constant.! ± " ! 4!! 0.411m ± 0.007m The processed distance should have three significant figure since the raw data had three significant figures. Then, the uncertainty should be to the smallest decimal place of the calculated inverse area which is to the thousandths. Fig 5: Processed Data Table Showing the Relationship between the Distance from the Light Bulb to the Light Probe and the Inverse of the Area of the Sphere with the Indicated Distances, as well as their Uncertainties, Excluding the First Seven Data Points in Fig 1 Distance from the light bulb to the light probe (m) ( ± 0.004m) Inverse of the Area of the Sphere with the Indicated Distances as the radius ( m ) Uncertainties for the Inverse of the Area of the Sphere ( ± m ) Average Light Intensity (W m ) ( ± 0.01W m ) Uncertainty for the Average Light Intensity ( ± W m ) Even though the distances of 0.940m ± 0.00m, 0.990m ± 0.00m and 1.040m ± 0.00m respectively had three, three and four significant figures, their inverse of the sphere s area were rounded to two significant figures to match the decimal places of the rest of the data and the uncertainty.

6 Fig 6: Average Light Intensity vs. Inverse of the Area of the Sphere Average Light Intensity y = x R² = Inverse of the Area of the Sphere! 1 $ # & (m ) " A % All the points have small horizontal and vertical error bars because their uncertainties were small. Another thing that should be noted is that as the inverse area increases, the uncertainty for the horizontal error bar increases which can be attributed to the squaring of the distance and taking the inverse of the area. Sample calculations: Calculating for slope uncertainty: To calculate the slope uncertainty, one has to know the maximum and minimum slopes. Firstly, to calculate the maximum slope, it is ideal to use the point A( , ), which is equivalent to (0.36, 0.66), and point B ( , ), which is equivalent to (0.075, 0.11). The point of the smallest and second to largest inverse of the area values should be used to best represent the maximum slope. In this case, using the value with the greatest inverse area would not be recommended since using the second largest data of the inverse area would yield a greater slope. maximumslope Δy = = =.W Δx The units is W (watts) since the slope measures the power as aforementioned. Similarly, to calculate the minimum slope, it is ideal to use the point C( , ), which is equivalent to (0.418, 0.69), and ( , ), which is equivalent to (0.073, 0.13). The point of the smallest and greatest inverse of the area values should be used to represent the minimum slope. Δy minimum slope = = = 1.6W Δx Then, with the maximum and minimum slopes, the uncertainty of the slope can be calculated. The average of the slope is also needed and is dictated by what the excel program calculated as the slope of the trend line in Fig 6 which is

7 approximately 1.8W. All the slopes were rounded to two significant figures because the smallest significant figure number was two. Average slope (S)=1.8W Maximum Slope (max)=.w Minimum Slope (min)=1.6w Maximum residual=(max) (S)=. 1.8=0.4W Minimum residual=(s)-(min)= =0.W The maximum residual is greater than the minimum residual. Therefore, the uncertainty of the slope is 0.4W. The slope of Fig 6 is 1.8W±0.4W. Conclusion: As supported by the collected data, as the distance increases, the light intensity decreases by its inverse squared. First, in Fig 4, the trend line is represented by the equation y = x.005. The distance is raised approximately to the negative squared power, indicating that the average light intensity and distance have an inverse squared relationship. Fig 6 also supports the aforementioned conclusion. The inverse area of the sphere, which raises the distance to a power of negative two, has a linear relationship between the average light intensity. This illustrates that the distance and light intensity have an inverse squared relationship. However, the graph showing the trend between distance and the average light intensity, which is Fig 4, had a curved line as its trend line. Thus, to straighten it, what the x-axis represented had to be manipulated; instead of the x-axis signifying the distance, it was manipulated so that the x-axis represented the inverse of sphere s area with the measured distance as its radius. This was done not only to make the graph linear but also to make the slope of the trend line represent power. The result of the graph with the processed data is Fig 6. The trend line is a straight line but not all the uncertainty bars pass through the trend line. Also, the line does not intersect the origin. Therefore, it is not proportional. Lastly, the Fig 6 does not show that there was a very prominent anomaly that would have completely altered the data. All data points are fairly close to the trend line. The trend line in Fig 4 was manipulated, as shown in the sample calculations before, so that it became a straight line as shown in Fig 6. The trend line in Fig 6 represents power and has a slope of 1.8W with an uncertainty of ± 0.4W. The y-intercept of Fig 6, as calculated by Microsoft Excel, is approximately 0.013W. This indicates that when the inverse area of the sphere is zero, which means that the distance to the light source is infinitely large, there would still be a power of 0.013W from the light source. Fig 3 and 4 can be compared to one another. Both graphs use the values from the same set of data. However, Fig 4 plots only a portion of the data set that Fig 3 plots. Nevertheless, the inverse squared relationship is better shown in Fig 4 than in Fig 3 which has additional data that does not correspond to the expected inverse squared relationship. In Fig 4, x was raised to the power of.005, a value very close to. This indicates that inverse area graph, Fig 6, would yield a linear graph. However, when the distance from the light source is infinitely large, the power that the light probe measures should be 0W since the light probe would be too far from the source to measure any light; all the light would have been dissipated before reaching the sensor. This indicates that there was a systematic error where all the data points were about 0.013W greater than it should have been because even when error bars are considered, the y-intercept is around 0.013W. This may have occurred because the light probe may have measured light that was from the surrounding and not from the light bulb or because the initial subtraction of 0.0 to yield the data in Fig 1 was not significant enough. Moreover, the uncertainty, an indicator of the random errors, had an appropriate size even though most of them did not fall within the trend line. Anomalies, which are other indicators of random errors, were not present in the experiment. There were no glaring anomalies that would have changed the data significantly. This is supported by the R squared value of Fig 6 which is The value is very close to 1 which signifies that the data lies close to the trend line and that there were no obvious random errors. The percent error cannot be calculated because there was not an accepted value. The power the light bulb used was 40W. However, this number is not reliable in calculating the percent error because it is not the power emitted by the light bulb and because most of the power is dissipated through heat. Thus, it could not even be determined if the theoretical value falls within the experimental value. Nonetheless, the 40W that the light bulb uses indicates the efficiency of the light bulb. The efficiency, which is the power output over power input, can be calculated. The slope of Fig 6, approximately 1.8W is

8 the power output and 40W is the power output. Thus, the efficiency is approximately 0.045, which indicates that only about 4.5% of the electrical energy is converted to usable energy. This is a low value considering that the efficiency for incandescent light bulb is between 10% and 17% ("Lighting Efficiency Comparison"). Evaluation of Procedures and Improvements: The conclusion that as the distance increases, the light intensity decreases by its inverse squared is reasonable as Fig 4, as well as Fig 6, strongly supports. Moreover, the experimental value of 1.8W ± 0.4W was also reasonable according to the raw data collected. In Fig 6, the R squared value was , which implies that the data points were very close to the trend line. This degree of accuracy supports that the experimental results for power is reasonable. The aforementioned systematic error where the trend line in Fig 6 intersects the point (0, 0.013) instead of the origin might have been caused for various reasons stated below. Error How the error affected the data How the error can be improved How the improvement will affect the data When collecting the raw data, the sensor measured some percent of light intensity even though the light source was turned off. As aforementioned, 0.0 was subtracted from all the raw data in Fig 1. However, that was the approximation of what the light probe measured throughout the distances. As the light probe moved farther away from the light source which was turned off, it seem to have measured a greater percent of maximum intensity. For the greater distances, it would have caused a systematic error where the data points, in Fig 6, were greater than the actual because their percents of maximum light intensities, when the light source was turned off, were greater than the subtracted value of 0.0. It would have shifted the data points upwards. On the other hand, for the smaller distances, this error would have caused the data points to be smaller than the actual because their percents of maximum light intensities, when the light source was turned off, were smaller than the subtracted value of 0.0, shifting the data Instead of taking the average, the results would be more accurate by recording what the light probe measured at each distance with the light source turned off and subtracting that value from what the light probe measured when the light source on. It would have reduced the systematic error for all the data points in Fig 6. For the greater distances, the data would be smaller and shifted downwards. For the smaller distances, the data would be greater and shifted upwards. The slope therefore would be smaller and the y-intercept greater but it would be a more accurate representation of the performed experiment. The amount of surrounding light, like the sunlight and the light from the computer, was not constant throughout the experiment. This control variable was not controlled because even during the one and a half hours in which the experiment was performed, more and more sunlight reached the classroom where the experiment was performed. If the variable of surrounding light was controlled, then the light probe should not have measured any light when the points downwards. By the end of the experiment, during which the most sunlight was reaching the experiment, the percent of maximum light intensity would have been greater than the actual. Thus, the raw data in the third trial would have been greater than when there would have been no influence of external light sources. This error can be mitigated by performing the experiment during a time when there is no sunlight or light from the room. For instance, the experiment was done during the first block of the school, which was during the morning. There is a lot of sunlight in the morning, which the light probe might have measured. Therefore, it is idealistic to do the lab in a room without windows or during the nighttime. The data of the third trial in Fig 1 would have been smaller than what was measured. It would have decreased all the averages of the data. The systematic error would have been reduced and the subsequent trend line in Fig 6 would have shifted downwards. Then, the trend line, when extended, would be closer to intersecting the origin.

9 light source was turned off or it should not have fluctuated. The surfaces under the light probe were different. Throughout the distance that the light probe measured the percent light intensity, there were some books right below it and none in measuring some distances. If the control variables were controlled, then the surface on which the light probe was placed on should have been the same throughout the experiment. Where there were books under the light probe, it would have had less light reflections and would have measured less percent of maximum intensity. On the other hand, where there were no books under the probe, the light could have been reflected from the desk and measured by the light probe. Thus, for some distances, the light intensity could have been greater than others due to the reflective nature of the desk. This error can be adjusted by using a long surface that barely reflects any light, such as a wooden surface. Then, the books would also have been unnecessary since the reflective surface of the desk could have been avoided. For some median distances, the values would be smaller because there would be less light reflected by external light sources such as the sun light. Thus, it would decrease the slope in Fig 6 and the y- intercept would be closer to the origin. The method with which the equipments were used maximized the usage of each material. For instance, to not measure the sun light with the light probe, it was placed within a tube. The method had maximized the usage of the tube. Also, the time was managed well since the experiment was finished during the allotted time. Lastly, the equipment used had an accuracy of ± 0%, which is a significant range. Thus, to yield more accurate results, a light sensor could have been used instead of a light probe ( TI Light Probe ). If more accurate results were yielded, then the y-intercept could have been closer to the origin. Thus, this modification in the apparatus would improve the results. The range of values was a bit too large because the distances that were closer to the light bulb were not necessary in analyzing the data. Initially, there were twenty data points to work with. However, in the processed data table of Fig 5, there were only thirteen data points that were actually needed. The seven shortest distances were not necessary. Therefore, the range of the values tested could have been smaller. A wider range would not have improved the results. However, due to the great fluctuations when measuring the percent of maximum light intensity, the number of trials should have been greater, like five trials per distance instead of three. Repetition of more trials could have made the results more accurate The collected data is quite reliable. The multiple trials in Fig 1 yielded raw data of percent of maximum light intensity that were close to one another. Even though the trend line in both Fig 4 and 6 did not fall within the uncertainties, the R squared value is close to one which indicates that the data points were all close to the trend lines. Moreover, there were no obvious anomalies supporting that the data is reliable. Lastly, there was an improvement that was made during the experiment. The surface of the desk was reflecting a lot of sunlight and light from the lamp. Thus, the lamp and the light probe was elevated and placed on top of books to lessen the impact of the reflecting light.

10 Works Cited "Lighting Efficiency Comparison." MGE. Madison Gas and Electric Company. Web. 8 Mar. 01. < "TI Light Probe." Vernier.com. Vernier. Web. 1 Mar. 01. <