Derive equation for convex surface having real object and virtual image in terms of refractive index and various distances
Answers
Explanation:
Let MPN be concabe spherical surface of a medium of refractive index μ. P is the pole o is the centre of curvature PC is the principle axis. Let O be a point object or it's principle axis kept in a rarer medium first incident ray travelling through C is normal to the spherical surface therefor it will not deviate and goes along PX another mident ray OA will refract at Point A and it bends towards the normal. AB and PX are proceeded behind then at I, a virtual image will be form.
Let α,β,γ are the ray angle normal with the principle axis respectively.
then by snell's law
μ=
sinr
sini
........(1)
but here i and r are the small then we put.
sini=i sinr=r in equation (1)
μ=
r
i
i=μn ...........(2)
now by using exterior angle theorem in ΔAOC
γ=1+a
i=γ−α .........(3)
now in ΔIAC by using exterior angle theorem
γ=b+r
rho=γ−β ..........(4)
putting value of i and r in equation (2)
(γ−α)=μ(γ−β) ...........(5)
now angle=
radius
arc
α=
OP
PA
β=
IP
PA
now γ=
CP
PA
using value of α,β,γ in equation (5)
PC
PA
−
PO
PA
=μ(
PC
PA
−
PI
PA
)
PA(
PC
1
−
PO
1
)=m.PA(
PC
1
−
PO
1
)
PC
1
−
PO
1
=μ(
PC
1
−
PI
1
) .........(6)
now by using sign convention
PC=−R PI=V
PO=−u
using these value in equation (6)
(
−R
1
)−(
−u
1
)=μ(
R
−1
+
V
1
)
R
−1
+
u
1
=μ(
R
−1
+
v
1
)
R
−1
+
u
1
=
R
μ
+
v
μ
R
−1
+
R
μ
=
v
μ
−
u
1
R
μ−1
=
v
μ
−
u
1
This is the required expression for the refraction formula for the concave spherical surface.