Irradiance versus distance. Exploring the inverse square law.

Transcription

1 Iadiance vesus distance.. OBJECTIVE CONTENTS To examine the distance distibution of the visual adiation emitted by the light souces. The iadiance vs. distance dependence is analyzed in tems of the invese squae law. I. Theoy. A. Basic concepts of Radiomety and Photomety C. Exploing the iadiance vs. distance. II. Appaatus fo the expeimental exploation. III. Data acquisition. IV. Data analysis (TI83) V. Data analysis (MS Excel) THEORY A. Basic concepts of Radiomety and Photomety The desciption and measuements of the popagation of the enegy of the electomagnetic waves adiated by light souces uses twofold appoaches. The so called adiometic (physical) appoach is based on the entie adiant powe poduced by the souce while the photometic (physophysical) efes only to the pat of the adiant powe peceived by the human eye as light and human aveage esponse function. The spectum of adiant enegy waves that we call light is naow, anging fom appoximately 300nm to 750nm. Wavelengths shote (UltaViolet) o longe (InfaRed) than these do not poduce the visual esponse in the eye. Consequently we have two set of quantities and units used in measuements see table 1. The symbols fo adiometic quantities ae analogous to the photometic countepat. All photometic quantities can be obtained fom the thei coesponding adiometic quantities on the basis of the spectal luminous efficacy human eye esponse The elationship between the (physical) adiometic and the (psychophysical) photometic set units is established by the chosen value of spectal luminous efficacy fo human vision: 683 lumens/watt. In both appoaches the basic concepts ae simila and descibe: Powe of the souce The light souce is chaacteized by the total light output of a light souce - total powe adiated by it [in watts]. This is called Radiant powe o Radiant flux Φ. In adiomety the unit of Φ is watt [W]. When efeed to the sensitivity of the human eye and so called optical pat of the adiated spectum it detemines the Luminous flux (o luminous powe). The photometic unit of Φ is lumen [lm]. Intensity The commonly used tem intensity of light ( bightness ) efes to the enegy adiated by the souce into the unit solid angle in a unit time. This powe pe unit solid angle is called Radiant intensity I in watts/steadian. The photometic unit of the intensity (pointance, luminous intensity) is candela [cd]. The adiant intensity enables to descibe spatial chaacteistics of the souce. In geneal, it is given by: I(Ω)= dφ/dω. 1

2 Illumination In eveyday life ou visual eception is based on the effect poduced by the light souce at the suface of the suounding objects. This means that we ae inteested in the amount of adiated enegy, which eaches the obseved suface element in a unit time. The iadiance E expesses adiation powe (flux) eceived by the unit aea of the illuminated suface. The unit of iadiance is watt/m. The espective photometic tem is called illuminance and is given in lux = lumen/m. The iadiance (as well as the illuminance) of a suface due the point light souce depends on the adiant intensity, the distance to the suface, and the oientation of the suface with espect to the souce: adiant flux Φ I Ω I A I E = = = = aea A A A (1) whee flux in a defined solid angle is: Φ = I Ω () and solid angle: A Ω = (3) Fig.1. Radiant flux Φ and adiant intensity I Solid Angle: The solid angle Ω efes to the cone cut out fom the sphee. Solid angle is elated to the aea A intecepted by the cone on the suface of a sphee of adius centeed on the cone vetex: A/ The unit of the solid angle is steadian [s]. Fig.. Solid angle Fo the sufaces not pependicula to the diection of light popagation the illumination expession E should be modified in ode to deal with the effective (pojected) aea of the illuminated suface see the fig.(1): I E = cosφ (4) whee φ is the angle between the nomal to the suface and diection of the popagation of the light enegy.

3 Fig. 3. Iadiance E at the inclined plane The invese squae law. The illumination poduced by the point light souce, which adiates equally in all diections follows the so-called Invese squae law. It expesses the fact that amount of enegy that passes though the unit aea dops with the distance fom the souce. The total powe iadiated by of the souce (adiant flux) in all diections (into the full solid angle) emains constant while the total aea of the sphee inceases with the squae of the adius. I E (5) So, the powe pe unit aea dops compae fig. 1. The geneal chaacte of the invese squae law applies to many othe phenomena based on point souces e.g. gavity pf point masses, electostatic field of point chages etc. Table 1. Radiometic and photometic quantities. Radiomety Photomety Quantity symbol units units symbol Quantity Radiant Enegy Q J lm s Q Luminous enegy Radiant Flux Φ W lm Φ Luminous Flux Iadiance E W/m lm/m E Illuminance Radiant Intensity I W/s lm/s I Luminous Intensity Radiance L W/(m s) lm/(m s) L Luminance 3

4 The adiometic measuements can be pefomed using the device that allows one to change the distance and the elative angle at which detecto views the souce (goniomete) see fig.4. The light detecto measues the iadiance. θ = 0 θ? E B. Exploing the iadiance vs distance. Fig.4. The goniomete The iadiance vesus distance dependence can be examined by simultaneously taken measuements of the distance fom the detecto to the souce and iadiance E at the constant angula position θ=0 of the souce. The device configuation assumes a point souce. Theefoe the obsevation can be analyzed in tems of the invese squae law: I E The typical dependence is pesented on the fig. 5. Fig. 5. Illustation of the invese squae law. APPARATUS FOR THE EXPERIMENTAL EXPLORATION. Detemining the iadiance dependence on the distance can be pefomed using the simple goniomete consisting of standad lamp handle placed on the optical bench with adjustable light detecto holde see fig. 6 The light senso uses a phototansisto with an active aea positioned at a fixed distance fom a light souce. The detecto measues elative iadiance E. The units of iadiance ae milliwatts pe squae centimete. The light senso s output is a voltage that is linealy popotional to the amount of iadiance it senses. 4

5 This configuation assumes a point souce, theefoe the invese squae law holds tue. Expeiment uses: Fig.6. Diagam of the expeimental setup i. goniomete ii. lamp housing E14 iii. incandescent lamp (e.g W) iv. Calculato Based Laboatoy unit CBL o CBL v. Light pobe (standad CBL o Venie s LS BTA) vi. Calculato Based Range CBR (sonic motion detecto) vii. Gaphic calculato TI83, TI83 Plus, TI 83 Plus SE. viii. unit-to unit cable (standad). ix. CBR CBL signal cable x. Pogam: PHOT, available fo download at: xi. TI-GRAPH LINK TM (optional) cable #seialwin and softwae ml xii. Pesonal compute with TI Connect TM softwae (optional) Desciption: ml Download: 5

6 Pactical notes about the appaatus setup. - The Light pobe should be connected to the channel CH1 of the CBL unit. - The CBR sonic detecto should be connected to the dedicated SONIC channel of the CBL - The additional plana sheet should bed mounted to the light souce holde and adjusted in the plane of the light souce. It eflects the ultasonic wave of CBR and helps measuing the distance. - The CBR detecto should be placed on the same stage as the light detecto and adjusted with the active end of it. So, the measued distance epesents distance fom the souce. - Duing the measuements the detecto/cbr stage should not be placed close than 0.4 m fom the light souce. - One should adjust the height of the detecto s mount with the cente of the light souce. - The ambient light should be educed and kept stable duing the expeimental session. DATA ACQUISITION (TI 83) In the expeiment the light senso measues elative iadiance E in milliwatts pe squae centimete as a function of the distance fom the souce. The distance is being changed by slowly sliding the detecto stand with espect to the souce holde. Data will be pesented in the ectangula coodinates as the E() plot. The expeiment is contolled by means of the peloaded calculato pogam PHOT. Expeimental pocedue is divided into the pepaatoy pat and the data acquisition. Pepaation: Make all connections as shown on the Fig.6. Tun on the calculato, the CBL and pepae CBR. Measuing the ambient light should pecede the main measuements. 1. Launch the pogam PHOT by choosing its name fom the NPEK menu.. Choose ZERO PROBE fom the pogam menu Fig Keep the light souce off and expose light senso to the backgound. Stat collecting data Fig.8. Fig.7. Fig.8. The collected iadiance value will be subtacted fom any othe collected iadiance data. The descibed calibation is to be done once duing the expeimental session povided that the backgound illumination does not change. 6

7 Data collection 1. Adjust the detecto/cbr stage at a distance 0.5m fom the souce. Check position of the light detecto. Its axis should exactly match the souce.. Choose option 1: COLLECT DATA fom the main menu. 3. Tun on the light souce. 4. Pes [ENTER] key on the calculato and stat sliding the detecto stage away fom the souce. Adjust speed of that motion so, that the whole displacement (appox. 1m) takes 0 sec. Fig Data collection ends and the iadiance vs distance plot is displayed fig If you ae not satisfied with he obtained data, epeated using the same o new settings (new calibation is not necessay). Choose 1:REPEAT fom the CONTROL menu Fig Stop measuements choosing : RETURN TO MAIN and then : QUIT fom main menu Fig Collected data ae stoed in a calculato s memoy and you can poceed using the standad calculato s functions. Now you can disconnect the CBL fom the calculato. Fig. 9 Fig.10 Fig.11 DATA ANALYSIS (TI83) Futhe analysis can be pefomed using tools implemented in calculatos (o othe analytical softwae tools such as MS Excel speadsheet). Collected data ae stoed in the calculato s lists: - Time in sec - List L 1 - Distance in m List L - iadiance E in watts/cm - List L 3 Exemplay data ae available fo a download in the following calculato type files: - distance as the file - Data sample/ti83/ L - iadiance - Data sample/ti83/ L 3 Expeimental data plot is defined as Plot1 (L 3 vs. L ) and can be ecalled by STAT PLOT menu. Analysis of plots Analysis of the obtained distance dependence of iadiance can be done by gaphical exploation of the plots. The analysis can be pefomed in tems of the invese squae law (eq. 5): I E The equation states that the iadiance is linealy popotional to the invese of the distance squae. One may examine this statement in case of the collected data by making the espective plots. This can be done in two ways. 7

8 Lineaization Accoding to the eq.5. the E(1/ ) plot should have a fom of a staight line. In ode to examine this the oiginal distance data gatheed in the list L -, should be tansfomed to 1/. The tansfomed data ae stoed in a new list L 4 Fig.1. Now, one the new plot should be defined as L 4 vs. L 3 and displayed Fig. 13. The data points well match the staight line, so assumption of the linea dependence is justified and one can apply the linea egession model. The linea egession is ecalled fom the STAT CALC menu (Fig.14) with the espective aguments Fig. 15. As the esult the paametes of the linea function ae calculated and the function is stoed as the Y 1. The esult of the egession is displayed Fig. 16. The coelation coefficient expesses the quality of the appoximation. In a given example the value is close to 1, which confims good linea coelation between iadiance E and 1/. This can be seen as well in the combined plot of expeimental data and egession line Fig. 17. Fig.1 Fig.13 Fig.14 Powe egession Fig.15 Fig.16 Fig.17 The E(1/ ) expession can be validated by diect application of the powe egession model to the oiginal iadiance (distance) data. One can select the PwReg model fom the STAT CALC menu and povide the list of espective aguments Fig. 18. Fig.18 Fig.19 8

9 As the esult the paametes a and b of the powe function y=ax b ae calculated and the function is stoed as the Y. The esult of the egession is displayed Fig. 19. The coelation coefficient expesses the quality of the appoximation. In a given example the value is close to 1, which confims good powe coelation between iadiance E and. In the given example the obtained exponent b is close to. This esults states that the obseved iadiance vs distance dependence follows the invese squae law. This can be seen as well in the combined plot of expeimental data (Plot1 Fig.0) and egession cuve Fig. 1. Fig.0 Fig.1 DATA ANALYSIS (MS EXCEL) Analysis can be pefomed using tools implemented in calculatos and pesonal compute functions offeed by the MS Excel softwae. Tansfe of expeimental data to the pesonal compute. Afte completing the expeiment data could be tansfeed fom the gaphic calculato to the pesonal compute. TI GRAPH LINK TM cable suppoted by the TI Connect TM softwae offe tools enabling exploation of the contents of calculato (TI DEVICE EXPLORER) and data edition (TI DATA EDITOR). Data collected in the expeiment ae stoed in the calculato s lists: - distance in metes - List L - iadiance mw/cm - List L 3 Within the TI Connect TM pogam - the TI DEVICE EXPLORER one can save calculato s lists on the compute s had disk and then open them within TI DATA EDITOR. Option Special Lists Expot povides oppotunity to save the chosen list as the Excel Comma Sepaated type file (*CSV file). Such a file could be open and exploe within the MS Excel TM speadsheet softwae. Exemplay data ae available fo a download in the following files: - angle: as file - Data sample/msexcel /dist - iadiance: as file - Data sample/msexcel/intensity Analysis of plots Ceate a speadsheet and impot the data fom the files angle and intensity. Make the plot E() fom the aw expeimental data. Choose scatteed type of the plot Fig.. The analysis can be pefomed in tems of the invese squae law (eq. 5): I E The equation states that the iadiance is linealy popotional to the invese of the distance squae. One may examine this statement in case of the collected data by making the espective plots. This can be done in two ways. 9

10 Lineaization Accoding to the eq.5. the E(1/ ) plot should have a fom of a staight line. In ode to examine this the oiginal distance data, should be tansfomed to 1/. The tansfomed data ae stoed in a new column of the speadsheet. Now, one the new plot should be defined and displayed. The data points well match the staight line, so assumption of the linea dependence is justified and one can apply the linea egession model by adding tend line (linea type) to the plot. The tend line is dawn and esulting linea expession is displayed in the plot Fig. 3.The detemination coefficient R expesses the quality of the appoximation. In a given example the value is close to 1, which confims good linea coelation between iadiance E and 1/. E [mw/cm ] 0,55 0,45 0,35 0,5 0,15 0,05 0,4 0,8 1, distance [m] Fig.. Iadiance vs distance. Expeimental data E [mw/cm ] 0,5 y = 0,0898x + 0,08 R = 0,9966 0,4 0,3 0, 0,1 0,0 0, 1,, 3, 4, 5, distance - [m - ] Fig.3. Iadiance E vs 1/ dependence. Linea egession. Powe egession The E(1/ ) expession can be validated by diect application of the powe egession model to the oiginal iadiance (distance) data. This can be done by applying the powe type (y=ax b ) of the tend line added to the oiginal E ( ) plot. The tend line is dawn and esulting powe expession is displayed in the plot Fig. 4. The detemination coefficient R expesses the quality of the appoximation. In a given example the R value is close to 1, which confims good powe type coelation between iadiance E and. 10

11 In the given example the obtained exponent b is close to. This esults states that the obseved iadiance vs distance dependence follows the invese squae law. E [mw/cm ] 0,55 y = 0,1187x -1,6738 R = 0,9941 0,45 0,35 0,5 0,15 0,05 0,4 0,8 1, distance [m] Fig.4. Iadiance vs distance. Expeimental data and powe type appoximation Note: The complete numeical analysis is pesented in the MSExcel file: Data sample/msexcel /invsqanalysis 11