Question

Derive the equation for refraction at a single spherical surface.

Answer 1

1. Let us consider two transparent media having refractive indices n₁and n₂ are separated by a spherical surface.
2. Let C be the centre of curvature of the spherical surface. Let a point object O be in the medium n₁.
3. The line OC cuts the spherical surface at the pole P of the surface.
4. As the rays considered are paraxial rays, the perpendicular dropped for the point of incidence to the principal axis is very close to the pole or passes through the pole itself.
5. Light from O falls on the refracting surface at N. The normal drawn at the point of incidence passes through the centre of curvature C.
6. As n₂ > n₁, light in the denser medium deviates towards the normal and meets the principal axis at I where the image is formed.
7. By Snell’s law,
   n₁ sin i = n₂ sin r
   As the angles are small,
   n₁ i = n₂ r
   Let the angles,
   ∠ NOP = α, ∠ NCP = β ∠ NIP = γ
   tan α = `"PN"/"PO"`;
   tan β = `"PN"/"PC"`;
   tan γ = `"PN"/"PI"`
8. As these angles are small, tan of angle could be approximated to the angle itself.
   α = `"PN"/"PO"`;
   β = `"PN"/"PC"`;
   γ = `"PN"/"PI"`
   for the triangle, ΔONC,
   i = α + β
   for the triangle, ΔINC,
   β = r + γ (or) r = β – γ
9. on sub i & r, we get
   n₁ ( α + β) = n₂ (β – γ)
   Rearranging
   n₁α + n₂γ = (n₂ – n₁)β
   `"n"_1 ("PN"/"PO") + "n"_2 ("PN"/"PI") = ("n"_2 - "n"_1)("PN"/"PC")`
   Further simplifying
   `"n"_1/"PO" + "n"_2/"PI" = ("n"_2 - "n"_1)/"PC"`
   `"n"_1/(-"u") + "n"_2/"v" = ("n"_2 - "n"_1)/"R"`
   If the first medium is air then, n₁ = 1 and
   the second medium is taken just as n₂ = n,
   then the equation is reduced to.
   `"n"/"v" - 1/"u" = ("n - 1")/"R"`