Derive the following expression for the refraction at concave spherical surface: μv−1u=μ−1R.
Let MPN be concabe spherical surface of a medium of refractive index μ. P is the pole o is the centre of curvature PC is the principle axis. Let O be a point object or it's principle axis kept in a rarer medium first incident ray travelling through C is normal to the spherical surface therefor it will not deviate and goes along PX another mident ray OA will refract at Point A and it bends towards the normal. AB and PX are proceeded behind then at I, a virtual image will be form.
Let α,β,γ are the ray angle normal with the principle axis respectively.
then by snell's law
μ=sinisinr ........(1)
but here i and r are the small then we put.
sini=i sinr=r in equation (1)
μ=ir
i=μn ...........(2)
now by using exterior angle theorem in ΔAOC
γ=1+a
i=γ−α .........(3)
now in ΔIAC by using exterior angle theorem
γ=b+r
rho=γ−β ..........(4)
putting value of i and r in equation (2)
(γ−α)=μ(γ−β) ...........(5)
now angle=arcradius
α=PAOP
β=PAIP
now γ=PACP
using value of α,β,γ in equation (5)
PAPC−PAPO=μ(PAPC−PAPI)
PA(1PC−1PO)=m.PA(1PC−1PO)
1PC−1PO=μ(1PC−1PI) .........(6)
now by using sign convention
PC=−R PI=V
PO=−u
using these value in equation (6)
(1−R)−(1−u)=μ(−1R+1V)
−1R+1u=μ(−1R+1v)
−1R+1u=μR+μv
−1R+μR=μv−1u
This is the required expression for the refraction formula for the concave spherical surface.