Question

Derive the magnification formula.​

Answer 1

let C be the centre of curvature.The line CA is,normal at A.Thus by the laws of reflection,angleOAC=angle CAI.Let alpha,beta , gama,denotes the angles AOP,AIP,and OAC respectively.As the exterior angle in s triangle equal to the sum of the two opposite interior angles ,we have

beta=alpha+Ø........i)

gama=alpha+2Ø......ii)

eliminating Ø from (i) and ii)

2Beta=alpha + gama

If the point A is close to P,the angles alpha,beta and gama,are small and we can write

alpha=AP/PO,beta=AP/PC, and gama=AP/PI

As,C is the centre of curvature,the equation for beta is exact whereas the remaining two r approximate...

so

from ii)

2AP/PC=AP/PO +A/PI

or

1/PO +1/PI=2/PC........III)

using sign convention

PO=-u,PI=-v,and PC=-R.

putting this in iii)

we get

1/-u+1/-v=-2/R

or

1/u +1/v=2/R

this is the spherical formula.....which we use to calculate approx value of v,u and R.....

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