# 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.