Question

Derive Lens Maker's formula for a thin lens.

Hint:

The derivation hinges on the "thin lens approximation," where the thickness of the lens is considered negligible. This allows us to measure all distances from a single point (the optical center) and to use the image distance from the first surface directly as the object distance for the second surface. Remember to use the Cartesian sign convention consistently for \(u, v, R_1,\) and \(R_2\) when solving problems.

Answer 1

Step 1: Understanding the Concept and Setup:
The Lens Maker's formula relates the focal length (\(f\)) of a thin lens to the refractive index (\(\mu\)) of its material and the radii of curvature (\(R_1\) and \(R_2\)) of its two surfaces. It is derived by applying the formula for refraction at a single spherical surface twice.
Consider a thin convex lens of refractive index \(\mu_2\) placed in a medium of refractive index \(\mu_1\). A point object O is placed on the principal axis.
\begin{center} \begin{tikzpicture} % Principal Axis \draw[<->] (-4,0) -- (4,0); % Lens \draw (0,1.5) .. controls (0.5,0) .. (0,-1.5); \draw (0,1.5) .. controls (-0.5,0) .. (0,-1.5); \node at (0, -2) {Lens ($\mu_2$)}; \node at (2.5, -2) {Medium ($\mu_1$)}; % Object \node[label=below:O] at (-3,0) {$\bullet$}; % Image 1 \node[label=below:$I'$] at (3.5,0) {$\circ$}; % Image 2 \node[label=below:I] at (2,0) {$\bullet$}; % Rays \draw[->, thick, red] (-3,0) -- (0,0.8); \draw[->, thick, red, dashed] (0,0.8) -- (3.5,0); \draw[->, thick, red] (0,0.8) -- (2,0); % Distances \draw[<->] (-3,-0.5) -- (0,-0.5) node[midway, below] {$u$}; \draw[<->] (0,-0.5) -- (2,-0.5) node[midway, below] {$v$}; \draw[<->] (0,-1) -- (3.5,-1) node[midway, below] {$v'$}; \end{tikzpicture} \end{center}

Step 2: Refraction at the First Surface (Radius \(R_1\)):
Light travels from the medium (\(\mu_1\)) to the lens (\(\mu_2\)). For the first surface, the object is at O (distance \(u\)). An intermediate image \(I'\) is formed at a distance \(v'\). The formula for refraction at a single spherical surface is: \[ \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R} \] Applying this to our case: \[ \frac{\mu_2}{v'} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R_1} \cdots \text{(1)} \]

Step 3: Refraction at the Second Surface (Radius \(R_2\)):
The image \(I'\) formed by the first surface acts as a virtual object for the second surface. Light now travels from the lens (\(\mu_2\)) back to the medium (\(\mu_1\)). The final image is formed at I (distance \(v\)).
For this surface, the object distance is \(v'\). Applying the single surface formula (with \(\mu_1\) and \(\mu_2\) interchanged): \[ \frac{\mu_1}{v} - \frac{\mu_2}{v'} = \frac{\mu_1 - \mu_2}{R_2} \] \[ \frac{\mu_1}{v} - \frac{\mu_2}{v'} = -\frac{\mu_2 - \mu_1}{R_2} \cdots \text{(2)} \]

Step 4: Combining the Equations:
Adding equation (1) and equation (2): \[ \left( \frac{\mu_2}{v'} - \frac{\mu_1}{u} \right) + \left( \frac{\mu_1}{v} - \frac{\mu_2}{v'} \right) = \left( \frac{\mu_2 - \mu_1}{R_1} \right) - \left( \frac{\mu_2 - \mu_1}{R_2} \right) \] The term \(\frac{\mu_2}{v'}\) cancels out: \[ \frac{\mu_1}{v} - \frac{\mu_1}{u} = (\mu_2 - \mu_1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \] Dividing 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 5: Introducing Focal Length (\(f\)):
By definition, when the object is at infinity (\(u = \infty\)), the image is formed at the focal point (\(v = f\)). Substituting this into the above equation: \[ \frac{1}{f} - \frac{1}{\infty} = \left(\frac{\mu_2}{\mu_1} - 1\right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \] Since \(\frac{1}{\infty} = 0\), we get: \[ \frac{1}{f} = \left(\frac{\mu_2}{\mu_1} - 1\right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \] If the surrounding medium is air or vacuum (\(\mu_1 = 1\)) and the lens material has refractive index \(\mu_2 = \mu\), the formula simplifies to its most common form: \[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \] This is the Lens Maker's formula.