For spherical refracting surface establish the refraction formula μv−1u=μ−1R where symbols have their usual meanings.
Let APB is a cross-section of any spherical refracting surface. P is its pole and C is its centre of curvature. Left of the surface has air as a medium and in right medium has refractive index m.
Any point object is placed at point O on the principal axis of which a virtual image is formed at I due to refracting surface APB. According to the figure angle of incidence,
∠OMC=i
Angle of refraction,
∠LMN=∠IMC=r
Let ∠MOC=α,∠MIC=β and ∠MCP=θ
∴ From Snell's law,
μ=sinisinr
or
μ=ir(as i and r are very small)
or i=μr ...........(i)
From Δ OMC,
(exterior angle =sum of opposite interior angles)
or i=θ−α .......(ii)
and in ΔIMC, θ=r+β
or r=θ−β ...........(iii)
From eqns. (ii) and (iii) putting the values of i and r in eqn. (i)
θ−α=μ(θ−β)
or μβ−α=(μ−1)θ ..........(iv)
Again, α=PMPO,β=PMIP and θ=PMPC
Substituting above relations in eqn. (iv),
μPMPI−PMPO=(μ−1)PMPC
or μPI−1PO=(μ−1)PC ..........(v)
But, PI=−v, PO=−u and PC=−R
∴μ−v−1−μ=(μ−1)−R
or μv−1u=(μ−1)R
or (μ−1)R=μv−1u.