Spacing between interference maxima

[t_n_p]

Homework Statement

a) Light of wavelength 651 nm illuminates a grating with 3,424 slits per cm. If the screen is 1.08 m away, what is the spacing between the interference maxima?

b) A laser cavity of length 0.2400 m, has reflecting mirrors at each end. What is the frequency, in GHz, of the 20-th allowed mode? Answer to at least 3 significant figures. Assume that the speed of light in the cavity is 2.998E+8 m/s. First find the fundamental frequency for standing waves in this cavity. This is a basic standing waves question!

The Attempt at a Solution

a) So far I have tried this:
d=1/N=1/342400=2.92e-6m

then dsinθ=mλ
I know d and λ, but now I have two unknowns in m and theta. Where to from here?

b) I used 3 formulas, namely

L=mλ/2 (1)
V=mc/2L (2)
and f=v/λ (3)

where m = mth mode = 20
L= length of laser cavity = 0.24m
c=2.998e8

I calculated λ from equation 1 to be = 2L/m = 2*0.24/20 = 0.024m and velocity using equation 2 to be = 20*2.998e8/2*0.24 = 1.24e10m/s
Then using equation 3, f = 1.24e10/0.024

I get an answer of 520GHz as opposed to the correct answer of 12.49GHz.
Am I using the wrong formula/formulae?

 

[Redbelly98]

a) So far I have tried this:
d=1/N=1/342400=2.92e-6m

then dsinθ=mλ
I know d and λ, but now I have two unknowns in m and theta. Where to from here?

Hmmm. They're asking for the spacing between different interference maxima. Maybe calculate θ for the 1st two maxima, and see how far apart they are on the screen?

b) I used 3 formulas, namely

L=mλ/2 (1)
V=mc/2L (2)
and f=v/λ (3)

where m = mth mode = 20
L= length of laser cavity = 0.24m
c=2.998e8

I calculated λ from equation 1 to be = 2L/m = 2*0.24/20 = 0.024m and velocity using equation 2 to be = 20*2.998e8/2*0.24 = 1.24e10m/s
Then using equation 3, f = 1.24e10/0.024

I get an answer of 520GHz as opposed to the correct answer of 12.49GHz.
Am I using the wrong formula/formulae?

The velocity is c.

Equation (2) is actually giving "nu", a Greek letter that resembles v, but is often used for frequency.