Question

Establish the given formula for a convex surface:​

Answer 1

Answer:

Refraction at Convex Surface

PO = -u

PC = +R

PI = +v

Using Snell's Law

μ = sin i / sin r =

For small angle

μ = i / r

i = μr

(γ + α) = μ(γ - β)

angle =  arc / radian

γ = AP / PC

α = AP / PO

β = AP / PI

(AP / PC) + (AP / PO) + μ[(AP / PC) - (AP / PI)]

(1/PC) + (1/PO) = μ[(1/PC) - (1/PI)]

(1/R) + (1/-u) = μ[(1/R) - (1/v)]

1/R - 1/u = μ/R - μ/v

μ/R - 1/R = μ/v - 1/u

(μ - 1)/R = μ/v - 1/u

μ = μ₂ / μ₁

We get,

(μ₂ - μ₁)/R = μ₂/v - μ₁/u

So, the formula for refraction at convex surface is $\frac{\mu_2 - \mu_1}{R} = \frac{\mu_2}{v} - \frac{\mu_1}{u}$