Question

derive the formula of refraction from rarer to denser medium at concave sperical refracting surface​

Answer 1

We have from Snell’s law −

μ =

sin i

sin r

From our assumptions made earlier we get −

sin i = i and sin r = r

Therefore,

i = μ r

From figure we can obtain −

i = α - γ and r = β - γ

∴ α - γ = μ(β - γ)

∴ α - γ = μβ - μγ

∴ α - μβ = γ(1 - μ)

Again using assumptions, we get −

∴ tan α - μ tan β = tan γ(1 - μ)

Suppose in figure point M is very close to P.

From figure −

tan α =

QM

OM

and tan β =

QM

IM

and tan γ =

QM

CM

Substituting values, we get −

QM

OM

- μ

QM

IM

=

QM

CM

(1 - μ)

Divide the equation by QM

1

OM

- μ

1

IM

=

1

CM

(1 - μ)

Substituting values of OM, IM, CM in optical terms from figure –

1

-u

- μ

1

-v

=

1

-R

(1 - μ)

∴

μ

v

-

1

u

= (μ - 1)

1

R

Where,

μ = Refractive index.

v = Image distance.

u = Object distance.

R = Radius of Curvature.