Solving 2 Slit Diffraction: Find Central & 10th Maxima Intensity

[stunner5000pt]

A coherent, monochromatic HeNe laser beam 1cm in diameter at lambda = 634nm illuminates a pair of slits. The separation of the slits is 2mm and the slits are each 0.1mm across. An interference pattern is cast on a screen 1m away from the slits. Index of refraction of air = 1.0002926 and CO2 (at STP) = 1.000449.

Find the location of the central maximimum and the 1st 2nd and 10th maxima
Now normally this isn't a big deal all i use is the formula
$$sin \theta = \frac{m \lambda}{d}$$ and $$\frac{\Delta x}{L} = \frac{\lambda}{d}$$
d = 2.1mm
lambda = 634nm
L = 1m
the thing which throws me off is the diameter of the beam. Does that have any effect on the way this pattern appears??

b) Find the intensity of the 10 th maximum compared to the intensity of the central maximum

using this formula $$I(\theta) = I_{m} (cos^2 \beta) (\frac{sin \alpha}{\alpha})^2$$
where $$\alpha = \frac{\pi a}{\lambda} sin \theta$$
$$\beta = \frac{\pi a}{\lambda} sin \theta$$
and a = width of slit = 0.1mm
for the central max the angle theta is zero and for the 10th the angle is determined from above

c) Now the room is filled with CO2 uniformly. What is the index of refraction of the Co2 gas? In three sentence or less explain how the intensity of the pattern will change

well since carbon di oxide has a different index of refraction (which is 1.000449)
should i use this little relation
$$\frac{n_{air}}{n_{CO_{2}}} = \frac{v_{CO_{2}}}{v_{air}}$$
and this is also the reation of the wavelengths and the frequencies so equate them all

Please tell me if I am wrong, i quite stumped on the width of the beam... How would i deal with that??

 

[stunner5000pt]

can anyone help??

my only problem is what to do width the width of the beam?? Any help would be greatly appreciated!

 

[thepatientmental]

The width of the beam will not have a significant effect on the interference pattern. It will only slightly broaden the peaks and troughs of the pattern, but the overall location and intensity of the maxima will not change significantly.

a) Using the formula for the location of maxima, we can find the central maximum at \theta = 0. We can then plug in values for m (1, 2, and 10) to find the angle \theta for the 1st, 2nd, and 10th maxima. The corresponding distances from the central maximum can then be found using the formula \frac{\Delta x}{L} = \frac{\lambda}{d}.

b) To find the intensity of the 10th maximum compared to the central maximum, we can use the formula I(\theta) = I_{m} (cos^2 \beta) (\frac{sin \alpha}{\alpha})^2, where \beta and \alpha are determined using the values for a, \lambda, and \theta for the 10th maximum. The intensity of the 10th maximum will be significantly lower than the central maximum due to the decrease in intensity with increasing distance from the central maximum.

c) The index of refraction for CO2 at STP is 1.000449, which is slightly higher than that of air (1.0002926). As a result, the wavelength of the laser beam will decrease slightly in the presence of CO2, causing the interference pattern to shift slightly and the intensity to decrease. This change in the index of refraction will have a small effect on the interference pattern, but it will not significantly change the overall appearance.