(a) Find the angle between the first minima for the two sodium vapor lines, which...

Gregory Emery

Gregory Emery

Answered

2022-01-11

(a) Find the angle between the first minima for the two sodium vapor lines, which have wavelengths of 589.1 and 589.6 nm, when they fall upon a single slit of width 2.00μm.
(b) What is the distance between these minima if the diffraction pattern falls on a screen 1.00 m from the slit?
(c) Discuss the ease or difficulty of measuring such a distance.

Answer & Explanation

raefx88y

raefx88y

Expert

2022-01-12Added 26 answers

Step 1
Known wavelength of sodium vapor lines:
λ1:589.1nm
λ2:589.6nm
Slit width (D) is 2μm
Dsinθ=nλ (1)
y=nλLD (2)
where D is slit width, n is order of minimum; λ is wavelength of light, θ is angle at which minimum is present, y is distance of minimum from center maximum, and L is distance between slit and screen
Step 2
Solution (a)
use equation (1) to calculate θ1 and θ2 corresponding to wavelength λ1 and λ2
sinθ1=(1)(589.1nm)(2μm)
sinθ1=0.2945
θ1=17.127
sinθ2=(1)(589.6nm)(2μm)
sinθ1=0.2948
θ1=17.145
Difference between two angles θ2θ1 is 0.018.
Solution (b)
Use equation 2 to calculate distance of first minimum corresponding to the wavelength λ1 and λ2.
y1=(1)(589.1nm)(1m)(2μm)
y1=249.5mm
y2=(1)(589.6nm)(1m)(2μm)
y1=294.8mm
distance between these minimum is y2y1=0.3mm
Solution (c)
These measurements are difficult because length and angles are too small to measure.
braodagxj

braodagxj

Expert

2022-01-13Added 38 answers

a) Calculation:
We use equation (1) to calculate θ1 and θ2 corresponding to wavelengths values λ1 for 589.1nm, λ2 for 589.6nm and D for 2μm,
sinθ1=(1)(589.1×109)2×103
=0.2954
θ1=17.1270
Similarly,
sinθ2=(1)(589.6×109)2×106
=0.2954
θ2=17.1450
Difference between two angles θ2θ1 is 0.0180.
Conclusion:
When the two sodium vapor lines fall on a single slit of width 2.00μm the angle between the first minima is 0.0180.
b) Calculation:
We use equation (2) to calculate distance of first minima corresponding to the values of wavelengths λ1 for 589.1nm, λ2 for 589.6nm and m1.
y1=(1)(589.1×109)(1)2×106
=249.5mm
Hence, the value of y1 is 249.5mm
Similarly,
y2=(1)(589.6×109)(1)2×106
=249.8mm
Hence, the value of y2 is 249.8mm
The division among the two minima y2y1 is 0.3mm.
Conclusion:
There is a distance from the two minima of 0.3mm.
c) It is seen from sub part (a) and (b) that the angle is 0.0180 and distance between two minima is 0.3mm respectively which is very small. Such small cannot be measured precisely. Hence, these are difficult measurements.
Conclusion:
These are difficult measurements, since the length and angles of these measurements are small.

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