Angular separation between two wavelengths

[socoil23]

1. using differentiation,find the angular separation in arc minutes in second order for two wavelengths 589.0 and 589.6 nanometers



2. d sin($$\theta$$)=m ($$\lambda$$)



3. i try to use D=$$\delta$$$$\theta$$/$$\delta$$$$\lambda$$
by differentating d sin($$\theta$$)=m ($$\lambda$$) i got d/m sec(thetha).this equation can be further change to D=(sinA+sinB)/$$\lambda$$sinB.
i assumed D is the separation between the wavelengths.is it the correct way to approach the problem??

 

[vela]

No, the problem is asking for the angular separation Δθ.

 

[socoil23]

No, the problem is asking for the angular separation Δθ.

thanks.now i know how to solve it. is it Δlambda/Δθ= d/m cosθ? and by using small angle approximation,cos theta = 1.so Δθ= Δlambda*m/d? i forgot to put extra info:d=(1/600)mm

 

[vela]

I think you need to be a little more careful because the problem asks you to find the separation to second order, not just first order.

 

[rdl]

I would say that your approach is on the right track. However, there are a few things to consider when finding the angular separation between two wavelengths.

Firstly, it is important to note that angular separation is typically measured in radians, not degrees or arc minutes. So, we need to convert our values to radians before we can find the angular separation.

Secondly, the equation you have used, d sin(\theta)=m (\lambda), is correct for calculating the angular separation in terms of the grating spacing (d), the wavelength (\lambda), and the order of diffraction (m). However, this equation assumes that the diffraction pattern is being produced by a diffraction grating.

If we want to find the angular separation between two wavelengths in general, we can use the equation D=\delta\theta/\delta\lambda, as you have mentioned. However, this equation is typically used for finding the angular dispersion, which is the change in angle per unit change in wavelength. To find the actual angular separation, we need to integrate this equation over the range of wavelengths we are interested in.

So, to answer your question, yes, your approach is correct, but we need to make a few adjustments to account for the units and the type of diffraction pattern being produced. Additionally, it would be helpful to specify the range of wavelengths you are interested in, as this will affect the final result.