Question

For a thin lens find the formula \(\frac{1}{f} = (\mu - 1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)\), where the meaning of symbols is general.

Hint:

The derivation of the Lens Maker's Formula is simply the application of the single spherical surface refraction formula twice, back-to-back. The key step is realizing that the image from the first surface becomes the object for the second surface. Be very careful with the sign convention for radii of curvature (\(R_1\) is usually positive for a convex surface, \(R_2\) is usually negative).

Answer 1

This formula is known as the Lens Maker's Formula. It relates the focal length of a lens to the refractive index of its material and the radii of curvature of its two surfaces.
Step 1: Assumptions:
1. The lens is thin, meaning its thickness is negligible compared to the radii of curvature.
2. The aperture of the lens is small.
3. The object is a point object placed on the principal axis.
4. The incident and refracted rays make small angles with the principal axis.
Step 2: Derivation using Refraction at Spherical Surfaces:
Let's consider a thin convex lens of refractive index \(\mu_2\) placed in a medium of refractive index \(\mu_1\). Let \(R_1\) and \(R_2\) be the radii of curvature of the first and second surfaces, respectively.

The general formula for refraction at a single spherical surface is:
\[ \frac{\mu_{\text{image space}}}{v} - \frac{\mu_{\text{object space}}}{u} = \frac{\mu_2 - \mu_1}{R} \] Refraction at the First Surface (surface 1 with radius \(R_1\)):
- An object O is placed in the medium (\(\mu_1\)) at a distance \(u\) from the lens.
- Light travels from the medium (\(\mu_1\)) to the lens material (\(\mu_2\)).
- A real image \(I'\) would be formed at a distance \(v'\) if the second surface were absent.
Applying the formula:
\[ \frac{\mu_2}{v'} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R_1} \quad \cdots(1) \] Refraction at the Second Surface (surface 2 with radius \(R_2\)):
- The image \(I'\) formed by the first surface acts as a virtual object for the second surface, at a distance \(v'\).
- Light travels from the lens material (\(\mu_2\)) back to the medium (\(\mu_1\)).
- The final image \(I\) is formed at a distance \(v\).
Applying the formula (note the swapping of \(\mu_1\) and \(\mu_2\)):
\[ \frac{\mu_1}{v} - \frac{\mu_2}{v'} = \frac{\mu_1 - \mu_2}{R_2} \quad \cdots(2) \] Step 3: Combining the Equations:
Now, add equation (1) and equation (2):
\[ \left(\frac{\mu_2}{v'} - \frac{\mu_1}{u}\right) + \left(\frac{\mu_1}{v} - \frac{\mu_2}{v'}\right) = \frac{\mu_2 - \mu_1}{R_1} + \frac{\mu_1 - \mu_2}{R_2} \] The term \(\frac{\mu_2}{v'}\) cancels out.
\[ \frac{\mu_1}{v} - \frac{\mu_1}{u} = (\mu_2 - \mu_1)\left(\frac{1}{R_1}\right) - (\mu_2 - \mu_1)\left(\frac{1}{R_2}\right) \] \[ \mu_1\left(\frac{1}{v} - \frac{1}{u}\right) = (\mu_2 - \mu_1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \] Divide the entire equation by \(\mu_1\):
\[ \frac{1}{v} - \frac{1}{u} = \left(\frac{\mu_2}{\mu_1} - 1\right)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \] Step 4: Final Formula:
From the thin lens equation, we know that \(\frac{1}{v} - \frac{1}{u} = \frac{1}{f}\), where \(f\) is the focal length of the lens.
Let \(\mu = \frac{\mu_2}{\mu_1}\) be the refractive index of the lens material with respect to the surrounding medium.
Substituting these into the equation, we get the Lens Maker's Formula:
\[ \frac{1}{f} = (\mu - 1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \] Meaning of Symbols:
- \(f\): Focal length of the thin lens.
- \(\mu\): Refractive index of the material of the lens relative to the surrounding medium.
- \(R_1\): Radius of curvature of the first surface where light enters.
- \(R_2\): Radius of curvature of the second surface where light exits.
(Note: A proper sign convention must be used for \(u, v, R_1, R_2\).)