Problem 116 - Fundamentals of Physics

Question

Show that the distance between an object and its real image formed by a thin converging lens is always greater than or equal to four times the focal length of the lens.

Step-by-Step Solution

Answer

Distance between object and image is always \( \geq 4f \) for real images.

Step 1: Understanding Lens Formula

The lens formula is given by \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \(f\) is the focal length of the lens, \(d_o\) is the object distance, and \(d_i\) is the image distance. A real image is formed when both \(d_o\) and \(d_i\) are positive.

Step 2: Expressing Object and Image Distances

The distance between the object and its image is given by \(d = d_o + d_i\). This equation will be used to show that \(d \geq 4f\).

Step 3: Deriving Inequality

Substitute the lens formula into the distance equation to express in terms of \(f\). Rearranging the lens formula gives \( d_o = \frac{fd_i}{d_i - f} \) and \( d_i = \frac{fd_o}{d_o - f} \). Substitute these into \( d = d_o + d_i \).

Step 4: Substitute and Simplify

Substitute \(d_o\) and \(d_i\) into \(d = d_o + d_i\) to get:\[ d = \frac{fd_i}{d_i - f} + \frac{fd_o}{d_o - f} \].Simplify the expression and apply the constraint that both denominators must be positive, ensuring a real image is formed.

Step 5: Applying Arithmetic Mean - Harmonic Mean Inequality

Apply the AM-HM inequality for the positive terms \( \frac{d_o}{f} \) and \( \frac{d_i}{f} \), implying \( \frac{d_o}{f} + \frac{d_i}{f} \geq 2 \sqrt{\frac{d_od_i}{f^2}} \). Given the lens formula, \(d_od_i = f(d_o + d_i) \), leading to at least \( 4f \), so \(d \geq 4f\).

Step 6: Conclusion of Proof

We've shown that due to the AM-HM inequality and the lens formula, the distance \(d\) between the object and its image satisfies \(d \geq 4f\). This completes the proof.

Key Concepts

Lens Formula

The lens formula is a fundamental concept in optics that helps determine the relationship between the object distance, the image distance, and the focal length of a lens. The formula is represented as \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where:

• \( f \) is the focal length of the lens, indicating how strongly the lens converges light.
• \( d_o \) is the distance of the object from the lens.
• \( d_i \) is the distance of the real image from the lens.

This formula assumes a thin lens in air and helps predict where an image will form, based on the object's position. When using this formula, make sure both the object and image distances (\( d_o \) and \( d_i \)) are positive for real images. This is key because real images are those that can be projected onto a screen.

Real Image

In the context of lenses, a real image is an image that is formed when light rays converge and actually pass through the image point. When dealing with converging lenses:

• A real image occurs when both the object and the image are on opposite sides of the lens.
• The image is inverted compared to the object.
• Real images can be exactly captured on a screen because the light physically converges at the location of the image.

Understanding real images is important in applications like projectors, cameras, and the human eye. The position of the real image relative to the lens is calculated using the lens formula, providing practical insights into how lenses operate in everyday optical devices.

Focal Length

The focal length \( f \) of a lens is a crucial parameter that defines its optical power. It tells us how strongly the lens can converge or diverge light rays. Here are some features of focal length:

• A shorter focal length means the lens is stronger, converging light more quickly.
• For converging lenses, the focal length is positive, indicating that the lens brings light to a focus.
• The focal length determines the size and position of the image formed by the lens.

In optical systems, adjusting the focal length has practical applications such as changing the zoom level in cameras or adjusting focus in photographic settings. When designing lenses, manufacturers carefully calculate and consider the focal length to ensure the desired image outcomes.

Arithmetic Mean-Harmonic Mean Inequality

The arithmetic mean-harmonic mean (AM-HM) inequality is a mathematical concept used to relate different types of averages. In the proof mentioned, this inequality plays a key role:The AM-HM inequality states that for non-negative numbers \( a \) and \( b \), the following holds:\[\frac{a + b}{2} \geq \frac{2ab}{a + b}\]This inequality is applicable to the lens problem, with:

• \( a = \frac{d_o}{f} \) and \( b = \frac{d_i}{f} \).
• Substituting these values, it shows how the combined object and image distances relate to the focal length.

The inequality helps justify that the distance between an object and its real image, when calculated, can never be less than four times the focal length. It emphasizes the limitations imposed by the optical properties of lenses in forming real images.