Question

Can you please answer and explain the rationale.

Find the Number of interference maxima in the central diffraction envelope given the information below

a = 0.04 mm,   d = 0.125 mm, l = 650 nm, l = 90 cm

y_m = ___2.35 mm_________(measured)

y_c = _____4.68 mm_______ (calculated)

Percent difference = 100% = ___49.78%___________

θ_i = tan^-1(4.68x10^-3/0.90) = 5.19x10^-3 rad

x_m = ___11.05 mm_________(measured)

x_c = _____14.63 mm_______ (calculated)

Percent difference = 100% = ___24.47%___________

θ_d =tan^-1(0.01463/0.90)

θ_d = 1.62x10^-2 rad (compare with single slit measurement)

Answer 1

From the given data
a = 0.04 mm
d = 0.125 mm
l = 650 nm
D = 90 cm

now, for youngs double slit experiment
the location of first minima is
y = lambda*D/2d
so, from the given data
y = 650*10^-9 * 0.9 / (2*0.125*10^-3)
y = 0.00234 m = 2.34 mm

so, yc should be 2.34 mm and not 4.68 mm
% difference will be (yc - ym)*100/yc = 0.01*100/2.34 = 0.42735042 %

now, thetai = arctan(yc/D)
thetai = arctan(2.34/900) = 0.0025999941 rad
thetai = 2.59999414*10^-3 rad

for diffraction,
first order minima location
xc = lambda*D/a = 650*10^-9*0.9/0.04*10^-3 = 0.014625 m = 14.625 mm
hence, % diff (xc - xm)*100/xc = (14.625 - 11.05)*100/14.625 = 24.44444444444 %
  
the angle for the first minima of the diffraction is
thetad = arctan(xc/D) = 0.0162485699 rad = 1.62485699*10^-2 rad

so we can see, thetad/thetai = 6.2494640468
number of interference maximas inside central diffraction maxima is 2*6.24946 = 12.49
hence there are 12 interference maximas inside the central diffraction maxima