Finding the angle of phase difference - two slit model

[vetgirl1990]

Homework Statement

Light of a wavelength 548nm illuminates two slits separated by 0.25mm. At what angle would one find the phase difference between the waves from two slits to be 2 rads?

Homework Equations

σ / λ = ΔΦ / 2π

σ: path difference
λ: wavelength
Δφ: phase difference

The Attempt at a Solution

Based on geometry of a two slit model, σ = dsinθ.
I thought this would be a simple substitution problem, but I'm not getting the correct answer (Answer = 0.04°), so I'm worried that I'm overlooking something.

σ / λ = ΔΦ / 2π
(dsinθ) / λ = ΔΦ / 2π
sinθ = (ΔΦ / 2π)(λ / d)
θ = sin^-1(ΔΦ / 2π)(λ / d)

I converted rads to degrees: ΔΦ = 2 rad * π/180

θ = 0.000697°

I also tried making the small angle approximation (sinθ ≈ θ) , but still didn't get the right answer: θ = 0.0000122°

 

[blue_leaf77]

θ = 0.000697°

The unit of the above value should be in radians, not degree.

 

[vetgirl1990]

The unit of the above value should be in radians, not degree.

The final answer is reported in degrees (0.04°)

 

[blue_leaf77]

The final answer is reported in degrees (0.04°)

Yes, I know, but the value of 0.000697 for $\theta$ should be in radians? Then convert it to degrees.
What's the value you got when calculating θ = sin-1(ΔΦ / 2π)(λ / d)?

 

[vetgirl1990]

Yes, I know, but the value of 0.000697 for $\theta$ should be in radians? Then convert it to degrees.
What's the value you got when calculating θ = sin-1(ΔΦ / 2π)(λ / d)?

θ = sin-1(ΔΦ / 2π)(λ / d)
= sin^-1 (2rads / 2π)(5.48x10^-7m / 2.5x10^-4m)
= sin^-1 (0.31831 * 0.002192)
= 0.03998 rads

Converting rads to degrees:
0.03998 rads * 180/π = 2.29

The answer is still wrong even when I leave it as radians until the end.

 

[blue_leaf77]

= sin-1 (0.31831 * 0.002192)
= 0.03998 rads

Check again the output of your calculator if it is in radians or degrees. If it's in degrees already, then you are getting the correct answer after rounding up to two figures behind the decimal.

 

[vetgirl1990]

Check again the output of your calculator if it is in radians or degrees. If it's in degrees already, then you are getting the correct answer after rounding up to two figures behind the decimal.

Ah I see! sin^-1 is automatically calculating my answer in degrees already. Thank you for your help and patience!

 

[blue_leaf77]

Thank you for your help and patience!

You are welcome.