The Radius of Newton's Rings

Show that the radius r of the mth dark Newton's Ring as viewed directly from above is given by r=(mλR)^1/2, where R is the radius of curvature of the spherical surface and λ is the wavelength. Assume that the air gap is everywhere much less than R and also r< If a dark spot occurs at a radius r, then the thickness of the air-gap at this radius, t satisfies:

2t=mλ

but, also:

R-t=(R²-r²)^1/2

R-t=R(1-r²/R²)^1/2

R-t~R(1-r²/(2R²))

R-t=R-r²/2R

Then:

-t=-r²/2R

t=r²/2R=mλ

r²=mλR

r = (mλR)^1/2

QED