0
You visited us 0 times! Enjoying our articles? Unlock Full Access!
Question

Derive the following expression for the refraction at concave spherical surface: μv−1u=μ−1R.

Solution
Verified by Toppr

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 lawμ=sinisinr ........(1)but here i and r are the small then we put.sini=i sinr=r in equation (1)μ=iri=μn ...........(2)now by using exterior angle theorem in ΔAOCγ=1+ai=γ−α .........(3)now in ΔIAC by using exterior angle theoremγ=b+rrho=γ−β ..........(4)putting value of i and r in equation (2)(γ−α)=μ(γ−β) ...........(5)now angle=arcradiusα=PAOPβ=PAIPnow γ=PACPusing value of α,β,γ in equation (5)PAPC−PAPO=μ(PAPC−PAPI)PA(1PC−1PO)=m.PA(1PC−1PO)1PC−1PO=μ(1PC−1PI) .........(6)now by using sign conventionPC=−R PI=VPO=−uusing these value in equation (6)(1−R)−(1−u)=μ(−1R+1V)−1R+1u=μ(−1R+1v)−1R+1u=μR+μv−1R+μR=μv−1uμ−1R=μv−1uThis is the required expression for the refraction formula for the concave spherical surface.

15
Similar Questions
Q1
Derive the following expression for the refraction at concave spherical surface: μv1u=μ1R.
View Solution
Q2
Derive an expression:
μv1u=μ1R
for refraction of light at spherical surface.
View Solution
Q3
For spherical refracting surface establish the refraction formula μv1u=μ1R where symbols have their usual meanings.
View Solution
Q4
Define refraction of light waves.
Draw a ray diagram for refraction at a spherical separating two media. For refraction at a spherical surface, derive the relation n2vn1u=n2n1R in object distance (u), image distance (v), refractive index of media (n1,n2) and radius of curvature (R).
View Solution
Q5
The equation of refraction at a spherical surface is

$\frac{{\mu }_{2}}{\nu }-\frac{{\mu }_{1}}{\mu }=\frac{{\mu }_{2}-{\mu }_{1}}{R}$.

Taking $R=\infty$, show that this equation leads to the equation

$\frac{\mathrm{Real}\mathrm{depth}}{\mathrm{Apparent}\mathrm{depth}}=\frac{{\mu }_{2}}{{\mu }_{1}}$

for refraction at a plane surface.
View Solution