Question

Derive an expression for refraction at a single (convex) spherical surface, i.e., a relation between u, v, R, n₁ (rarer medium) and n₂ (denser medium), where the terms have their usual meaning.

Answer 1

In the figure, XPY is a spherical surface with centre C and radius R, O is the object, and I is the image,

OA and AI are incident and refracted rays; i and r are angles of incidence and refraction.

From the figure,

i = α + γ and r = γ − β

Now, tan α = `(MA)/(MO)` or α = `(MA)/(MO)`    ...(When A is very near to P)

Similarly, β = `(MA)/(MI)` and

γ = `(MA)/(MC)`

From snells law:

`(sin i)/(sin r) = n_2/n_1`

If i and r are small,

`i/r = n_2/n_1`

Also, MO ∼ PO, MC ∼ PC, MI ∼ PI,

PO = −u, PI = v and PC = R,

∴ α = `(PA)/(PO) = (PA)/-u`

β = `(PA)/(PI)`

γ = `(PA)/(PC) = (PA)/R`

i = α + γ

= `PA (-1/u + 1/R)` and

r = γ − β

= `PA (1/R - 1/v)`

Now, `i/r = n_2/n_1`

⇒ n₁i = n₂r

⇒ `PA (-1/u + 1/R)n_1 = PA (1/R - 1/v)n_2`

⇒ `-n_1/u + n_1/R = n_2/R - n_2/v`

⇒ `n_2/v - n_1/u = (n_2 - n_1)/R`