Let us consider a portion of a spherical surface AB separating two media having refracting indices ?1 and ?2 (Fig. ). This is symmetrical about an axis passing through the centre C and cuts the surface at P.

*Refraction at a
spherical surface*

Let us consider a portion of a spherical surface AB separating
two media having refracting indices ?_{1} and ?_{2} (Fig. ).
This is symmetrical about an axis passing through the centre C and cuts the
surface at P.

The point P is called the pole of the surface. Let R be the
radius of curvature of the surface.

Consider a point object O on the axis in the first medium.
Consider two rays OP and OD originating from O. The ray OP falls normally on *AB* and goes into the second medium,
undeviated. The ray *OD *falls at* D *very close to* P*. After refraction, it meets at the point* I *on* *the axis, where the
image is formed. *CE* is the normal
drawn to the point *D*. Let *i* and *r* be the angle of incidence and refraction respectively.

Let angle DOP = α, angle DCP =
β, Angle DIC = γ

Since D is close to P, the angles α, β and γ are all small. From
the Fig.

tan α = DP/PO

tan β = DP/PO

tan γ = DP/PI

α = DP/PO

β = DP/PO

γ = DP/PI

From the ∆ODC, i = α + β
??..(1)

From the ∆DCI, β = r + γ or r = β ? γ ???(2)

From Snell?s Law, ?_{2}/ ?_{1} = sin i/ sin r
and for small angles of i and r, we can write, ?_{1} i = ?_{2}r ...(3)

we get ?_{1} (α +
β) = ?_{2} (β − γ) or ?_{1}
α + ?_{2} γ = (?_{2} - ?_{1} )β ????(4)

Substituting the values of α, β and γ in equation (4)

?_{1}(DP/PO) + ?_{2 }(DP/PI) = (?_{2} -
?_{1}) . DP/PC

?_{1}/PO + ?_{2}/PI = ( ?_{2} - ?_{1 })
/PC

As the incident ray comes from left to right, we choose this
direction as the positive direction of the axis. Therefore u is negative,
whereas v and R are positive substitute PO = ?u PI = +v and PC = +R in equation
(5),

( ?_{1} / -u ) + (?_{2}/v) = (?_{2}- ?_{1})/R

(?_{2}/v) - ( ?_{1} / u ) + = (?_{2}- ?_{1})/R ???.(6)

Equation (6) represents the general equation for refraction at a
spherical surface.

If the first medium is air and the second medium is of
refractive index ?, then

(? / v) ? ( 1/u) = (? -1)/R

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