Easy interference problem regarding Newton's rings

[LCSphysicist]

Homework Statement

All below

Relevant Equations

All below

Unfortunately, i found r² = (R1)(R2)(λ)*(n-1/2)/(R1-R2)

I imagined a difference of phase λ/2 on the blue ray.
The grey is the air maybe polluted, as currently

 

[TSny]

Please show us your work. Did you take into account any phase shifts due to reflections?

 

[LCSphysicist]

Please show us your work. Did you take into account any phase shifts due to reflections?

OOps
I thought that can the error is in the reflections yes.

t is the thickness, that is, the difference of path, but:

2t = (n-1/2)λ

where the half is the difference of phase.

 

[LCSphysicist]

To be more specific, the red ray will reflect first without change of phase, and a part of it will be transmitted.
In this transmission, it will suffer another reflection, but this case the difference of refraction indice will make a difference of phase λ/2 (here i am a little confused if is λ/2 or λ

 

[TSny]

To be more specific, the red ray will reflect first without change of phase, and a part of it will be transmitted.
In this transmission, it will suffer another reflection, but this case the difference of refraction indice will make a difference of phase λ/2 (here i am a little confused if is λ/2 or λ

If a reflection causes a phase shift, it will be a phase shift of $\lambda/2$. Yes, there is a phase shift at the second reflection but not at the first reflection.

Keep in mind that you are dealing with the dark rings.