Practical Lab 2 The Diffraction Grating

Transcription

1 Practical Lab 2 The Diffraction Grating OBJECTIVES: 1) Observe the interference pattern prouce when laser light passes through multiple-slit grating (a iffraction grating). 2) Graphically verify the wavelength of the laser light using the ata collecte from the interference pattern prouce when the light passes through a iffraction grating. CAUTION! The laser is a evice that can prouce an intense, narrow beam of light at one wavelength. NEVER look irectly into the laser beam or its reflection from a mirror, etc. INTRODUCTION Interference occurs when two or more coherent waves overlap. (Coherent means that the waves have a fixe phase relationship.) Constructive interference takes place at certain locations where two waves are in phase (for example, both waves have maximum). Destructive interference takes place where two waves are out of phase (for example, one wave has maximum, the other has minimum). The simplest interference patterns involve plane waves (collimate or parallel light beams). In this experiment, we will use a laser as our light source. A laser prouces collimate an coherent light beams at one wavelength.

2 Multiple-slit interference (a iffraction grating) Figure 1: Intensity istribution of a iffraction grating Although a multi-slit grating is commonly referre to as a iffraction grating, a more appropriate name for it is an interference grating. The phenomenon that is observe is interference an not as its name suggests iffraction. The conition here for interference maximum is the same as for ouble-slits, but the pattern may be very ifferent because (the slit spacing) for gratings is very small. sin θ m = mλ or mλ sinθ = (1) If a graph of sinθ m vs. m is constructe, it has the form of a straight line with m playing the role of x an sinθ m playing the role of y. The line shoul have an intercept of zero λ 1 an a slope of. Note: is the ruling ensity which will be given in lines/mm. Using the slope of the line an the ruling ensity, the wavelength of the laser light can be calculate.

3 The angles involve when using the iffraction grating are large; therefore you cannot use the small angle approximation here. In Figure 1 above, sinθ (for m = 3) is expresse as the ratio of the opposite sie of the right triangle ( x 3 ) to the hypotenuse of the right triangle ( D x 3 ) +. The istances x m are measure from the location of the central maximum, x 0. Equation 1 can be expresse as: x D + x The istances x m are positive for positive values of m an are negative for negative values of m. The orer of the maxima can take on the values of m=0, ±1, ±2, PROCEDURE The wavelength (λ) of the green ioe laser use in this experiment is 532 ± 10 nm = 5.32 ± mm. The screen to iffraction grating istance D an the ( ) 4 = mλ (2) ruling ensity 1 you will use will be given to your answer sheet of the practical lab. 1. Place the grating in the laser beam at the istance D specifie on your answer sheet in front of the screen an recor this istance in your Excel spreasheet. 2. Recor the labele ruling ensity (grooves/mm) in your Excel spreasheet. 3. Tape a piece of paper across the screen. Mark carefully the positions of the principal maximum an the interference maxima. You shoul mark an recor the location of the central maximum, both first orer maxima an both secon orer maxima. Remove the paper from the screen an attach it to your lab report. 4. Measure the istance of each interference maximum from the principal maximum (x m ) an recor them in your Excel spreasheet. Have Excel calculate ( D 2 x 2 xm + ) ansinθ = for each of the maxima. m D + xm 5. Transfer your ata table into Kaleiagraph an make a plot of sinθ m vs. m. Fit your plot with a best-fit line an have Kaleiagraph isplay the equation of the line along with the uncertainties in the slope an intercept. QUESTIONS 1. Sketch the pattern you observe when the laser light passe through a iffraction grating (i.e. attach the piece of paper from your screen). Label each of the interference maxima. 2. What is the slope of your graph?

4 3. From the slope of your graph an the ruling ensity, calculate the wavelength of 1 δ δ slope the laser light an its uncertainty (show calculations). δλ = λ + slope 1 4. Discuss the consistency of your measure value of the wavelength from question 3 with the accepte value of 532 ± 10 nm. 5. If the grating ensity, 1 were halve what woul be the highest possible orer of the resulting interference maxima? Comparing Data It is often necessary to compare two ifferent pieces of ata or results of two ifferent calculations an etermine if they are compatible (or consistent). In just about every experiment in this course you will be aske if two quantities are compatible or consistent. The following escribes how to etermine if two pieces of ata are consistent (or compatible). Use this proceure to answer the question at the en an use it as a reference whenever you are aske if two pieces of ata are compatible or consistent. Let s enote the pieces of ata by 1 an 2. If 1 = 2 or 1-2 = 0, clearly they are compatible. We often use Δ (pronounce Delta ) to enote the ifference between two quantities: Δ = 1 2 (8) This comparison must take into account the uncertainties in the observation of both measurements. The ata values are 1 ± δ 1 an 2 ± δ 2. To perform the comparison, we nee to fin δδ. The aition/subtraction rule for uncertainties is: δδ = δ 1 + δ 2 (9) Our comparison becomes, is zero within the uncertainty of the ifference Δ? This is the same thing as asking if: Δ δδ (10) Equation (9) an (10) express in algebra the statement 1 an 2 are compatible if their error bars touch or overlap. The combine length of the error bars is given by (9). Δ is the separation of 1 an 2. The error bars will overlap if 1 an 2 are separate by less than the combine length of their error bars, which is what (10) says. Sometimes rather than a secon measure value you are comparing your ata to an expecte value. If this is the case, replace 2 ± δ 2 with e ± δe, where e ± δe is the expecte value incluing its uncertainty.

5 Excel Commans Operation or Mathematical escription Excel comman Function Aition = Subtraction =29 21 Multiplication =30 * 15 Division 44/12 =44/22 Example =3 + 4/(5*2) (3*7) Square root 5 or 7 (5 / 3) =sqrt(5) or =sqrt(7*5/3) Power 6 3 or =6^3 or 7^(0.5) Pi π =pi() Sum of numbers a =sum(a i i ) where a i can be a list of cells Examples A1+A2+A3+A4+A5 =sum(a1,a2,a3,a4,a5) or* =sum(a1:a5) Mean value A 1+ A2 + A3 + A4 + A5 =average(a1:a5) 5 Stanar eviation 2 ( x i x) =stev(series of cells) N 1 Sine Sin (x) or Sin(2 πx) =sin(x) or =sin(2*pi()*x) Cosine Cosine (x) =cos(x) *- This secon option can be use when the Excel comman references cells in the same column an ajacent rows, or in the same row an ajacent columns. You can also combine methos of efining cells. For example, if you wante to fin the sum of the contents of cells B3 through B28, B32 an B40 through B100 the Excel comman you woul use is: =sum(b3:b28,b32,b40:b100) Some other useful hints: If in oubt, use parentheses to make sure things get calculate in the right orer. For example, =3+5/2 results in 5.5. But, =(3+5)/2 results in 4. In the first case, it woul be better to use =3+(5/2) in Excel. Pushing the Ctrl + ~ keys will isplay the formulas for the entire spreasheet. Pressing these two keys again reverts back to the calculate numbers.

6 Spreasheet for Practical Lab: Diffraction Grating Practical Lab Measure or Given Quantities: Calculate Quantities Double Slit Interference Distance from the slit to the screen D Ruling ensity mm mm -1 orer of istance from sin (θ m ) = interference principle maximum x m x m (mm) D + x m D + xm m slope uncertainty wavelength from slope uncertainty mm wavelength from slope uncertainty nm