Question

Derive equation for convex surface having real object and virtual image in terms of refractive index and various distances ​

Answer 1

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.