AM-GM Inequality in Optical Lens Systems

Problem: AM-GM Inequality in Optical Lens Systems

In geometric optics, a fundamental constraint exists regarding the distance between a real object and its corresponding real image formed by a thin convex lens. Given a lens with a fixed focal length ##f##, we define ##u## as the object distance and ##v## as the image distance. For a real object and a real image, we seek to determine the minimum value of the total separation distance ##D = u + v##.

Using the thin lens equation:

Prove that the minimum distance ##D## between the object and the real image is equal to ##4f##.

Worked Solution & Step-by-Step Explanation

To determine the minimum value of ##D##, we treat the lens equation as a constrained optimization problem.

**Step 1: Expressing the separation distance ##D##**

The total distance ##D## is defined as the sum of the object distance and image distance:

From the lens formula, we have:

By substituting ##D = u + v## into the equation above, we obtain a relationship between the product ##uv## and the focal length ##f##:

**Step 2: Applying the AM-GM Inequality**

The Arithmetic Mean-Geometric Mean (AM-GM) inequality states that for any two non-negative real numbers ##u## and ##v##:

This inequality provides a lower bound for the sum of two variables when their product is constant.

**Step 3: Substitution and Algebraic Manipulation**

We substitute ##D = u + v## and ##uv = fD## into the AM-GM expression:

To solve for ##D##, we square both sides of the inequality. Since distance is a positive quantity (##D > 0##), the inequality sign remains unchanged:

Dividing both sides by ##D## (as ##D \neq 0##):

**Step 4: Determining the Condition for Minimum Distance**

The AM-GM inequality reaches its equality condition when the two terms are equal:

Substituting ##u = v## into the original lens formula:

Since ##u = v##, we find ##v = 2f##. Thus, the minimum separation ##D_{min}## occurs when the object is placed at a distance of ##2f## from the lens, resulting in a real image at ##2f## on the opposite side.

Conceptual Significance in Physics

This result is a cornerstone for experimental optics, particularly in the displacement method used to measure the focal length of a convex lens.

1. **Existence of Real Image:** For a real image to form, the object must be placed beyond the focal point (##u > f##).

2. **The 4f Constraint:** If the screen (where the image is captured) is placed at a distance less than ##4f## from the object, no real image can be formed by moving the lens between the object and the screen.

3. **Symmetry:** The configuration ##u = v = 2f## represents a symmetric optical state where the magnification ##m = -v/u = -1##. The image is the same size as the object but inverted.