Power of a spherical refracting surface

[Amith2006]

The power of a spherical refracting surface is given as, P = (n2 – n1)/R
Where n1 = refractive index of object space, n2 = refractive index of image space
R = Radius of curvature of the spherical surface
I tried to derive the expression but I got a different expression.
Let u and v be the object and image distance from the spherical surface respectively.
Suppose the spherical refracting surface produces the image of a real object placed at its first principal focus at infinity. Then,
u = +f1, v = infinity
For refraction at a spherical surface,
(n2)/v + (n1)/u = (n2 – n1)/R
(n2)/(infinity) + (n1)/f1 = (n2 – n1)/R
0 + (n1)/f1 = (n2 – n1)/R
f1 = [(n1)(R)]/( (n2 – n1)
Power = 1/f1
Therefore P = (n2 – n1)/[(n1)(R)]
Could you please tell me where I have gone wrong?

 

[Andrew Mason]

The power of a spherical refracting surface is given as, P = (n2 – n1)/R
Where n1 = refractive index of object space, n2 = refractive index of image space
R = Radius of curvature of the spherical surface
I tried to derive the expression but I got a different expression.
Let u and v be the object and image distance from the spherical surface respectively.
Suppose the spherical refracting surface produces the image of a real object placed at its first principal focus at infinity. Then,
u = +f1, v = infinity
For refraction at a spherical surface,
(n2)/v + (n1)/u = (n2 – n1)/R
(n2)/(infinity) + (n1)/f1 = (n2 – n1)/R
0 + (n1)/f1 = (n2 – n1)/R
f1 = [(n1)(R)]/( (n2 – n1)
Power = 1/f1
Therefore P = (n2 – n1)/[(n1)(R)]
Could you please tell me where I have gone wrong?

It looks like f1 is the focal length in air (n = 1). The power is the reciprocal of the focal length in the medium. That focal length is f1/n1 in a medium with index of refraction n1. So the power is P = n1/f1

AM

 

[kyle8921]

Your derivation is correct up to the point where you set u = +f1 and v = infinity. This assumption is only valid for a specific case where the object is placed at the first principal focus of the spherical surface and the image is formed at infinity. In general, the object and image distances can vary and the expression for power will be different.

To derive the general expression, we can use the thin lens equation:

1/u + 1/v = (n2 - n1)/R

Rearranging this equation, we get:

1/v = (n2 - n1)/R - 1/u

Substituting this in the power equation, we get:

P = n2/v - n1/u

Since v and u can vary, we cannot simplify this any further. This is the general expression for the power of a spherical refracting surface.