physics---wave and light1

(i) Use the formula: d.sin(a) = m入
where d is the slit separation
a is the angular position of a bright fringe
入 is the wavelength of light
m is a +ve integer

hence, sin(a) = m入/d

Let a1 and a2 be the angular positions of two consecutive bright fringes
sin(a1) = m入/d
and sin(a2) = (m+1)入/d
Thus, sin(a2) - sin(a1) = (入/d).(m+1 - m)
i.e. sin(a2) - sin(a1) = 入/d

If a1 and a2 are small,
sin(a2) - sin(a1) = a2 - a1
hence, a2 - a1 = 入/d

For light of wavelength 500 nm
fringe angular separation = a2 - a1 = 500x10^-9/7.5x10^-3 radians = 6.67x10^-5 radians

For wavelength 580 nm
fringe angular separation = 580x10^-9/7.5x10^-3 radians = 7.73x10^-5 radians

(ii) For samll angles,
sin(a2) - sin(a1) = tan(a2) - tan(a1)
thus, tan(a2) - tan(a1) = 入/d ---------------- (1)

But tan(a1) = y1/D
where D is the screen slit distance, and y1 is the distance of the bright fringe from the central axis on the screen.
Similarly, tan(a2) = y2/D

Hence, tan(a2) - tan(a1) = (y2 - y1)/D
Thus, from (1): (y2-y1)/D = 入/d
y2 - y1 = 入D/d

For wavelength 500 nm
fringe separation = y2 - y1 = 500x10^-9 x 0.8/7.5x10^-3 m = 5.33x10^-5 m
For wavelength 580 nm
fringe separation = 580x10^-9 x 0.8/7.5x10^-3 m = 6.19x10^-5 m