Problem 101 - Fundamentals of Physics

Question

The formula \(1 / p+1 / i=1 / f\) is called the Gaussian form of the thin-lens formula. Another form of this formula, the Newtonian form, is obtained by considering the distance \(x\) from the object to the first focal point and the distance \(x^{\prime}\) from the second focal point to the image. Show that \(x x^{\prime}=f^{2}\) is the Newtonian form of the thin-lens formula.

Step-by-Step Solution

Answer

The Newtonian thin-lens formula is \(xx' = f^2\).

Step 1: Restate the Problem

We need to prove the relationship \(xx' = f^2\) as the Newtonian form of the thin-lens formula, starting from the Gaussian form \(1/p + 1/i = 1/f\). The variables \(p\) and \(i\) represent the object and image distances from the lens, respectively.

Step 2: Define Image and Object Distances

In terms of the distances from the focal points, define \(p = f + x\) and \(i = f + x'\), where \(x\) is the distance from the object to the first focal point and \(x'\) is the distance from the image to the second focal point.

Step 3: Substitute Distances into Gaussian Formula

Substitute \(p = f + x\) and \(i = f + x'\) into the Gaussian formula \(\frac{1}{p} + \frac{1}{i} = \frac{1}{f}\). This gives:\[\frac{1}{f+x} + \frac{1}{f+x'} = \frac{1}{f}\]

Step 4: Simplify the Equation

Combine the two fractions on the left-hand side:\[\frac{f+x' + f+x}{(f+x)(f+x')} = \frac{1}{f}\]Thus, we have:\[\frac{2f + x + x'}{(f+x)(f+x')} = \frac{1}{f}\]

Step 5: Cross-Multiply and Rearrange

Cross-multiply to get:\[2f(f+x)(f+x') = f(f + x + x')\]Cancel \(f\) from both sides:\[2(f+x)(f+x') = f + x + x'\]Rearrange:\[(f + x)(f + x') = f^2 + fx + fx' + xx'\]

Step 6: Subtract and Simplify

Since \((f+x)(f+x')\) equaled itself, simplify using algebra:\[xx' = f^2\]This is the Newtonian form of the thin-lens formula.

Key Concepts

Gaussian form

The Gaussian form of the thin-lens formula is a fundamental concept in optics that helps describe how lenses form images. It is expressed as \( \frac{1}{p} + \frac{1}{i} = \frac{1}{f} \), where:

• \( p \) is the object distance, the distance from the object to the lens.
• \( i \) is the image distance, the distance from the lens to the image.
• \( f \) is the focal length, the fixed position where parallel rays converge after passing through the lens.

This equation balances the reciprocals of these distances, allowing us to calculate any one of them if the other two are known. Its strength lies in its capability to predict the behavior of light through lenses, a crucial element in designing optical devices like cameras and glasses. The simplicity and utility of the Gaussian form make it a cornerstone of geometric optics, enabling practitioners to manipulate light in a wide range of applications.

Newtonian form

The Newtonian form of the thin-lens formula emphasizes the distances from the focal points rather than from the lens itself. It's articulated as \( xx' = f^2 \), which links:

• \( x \): the distance from the object to the first focal point.
• \( x' \): the distance from the image to the second focal point.
• \( f \): the focal length.

In this form, the product of the distances \( x \) and \( x' \) equals the square of the focal length \( f^2 \). This version of the formula is particularly useful when dealing with problems involving paraxial rays and system setups where these focal distances are easier to measure or calculate.
The Newtonian form emerges from the Gaussian form through algebraic manipulation, illustrating an alternative perspective on light refraction that can simplify some calculations in optical systems.

Focal point

The focal point is a critical concept in optics. It is the point where parallel rays of light either converge or appear to converge after passing through a lens or mirror. Here are some key points related to the focal point:

• For converging lenses (like convex lenses), the focal point is where light rays meet after passing through the lens.
• For diverging lenses (like concave lenses), the focal point is where the rays appear to diverge from, acting as if they originate from a point behind the lens.
• The focal length is the distance from the lens to the focal point and is a key parameter in both the Gaussian and Newtonian forms of the thin-lens equation.

Understanding the focal point helps in determining how lenses should be positioned to focus light correctly. It's crucial for applications like focusing a camera or eyeglasses, where precise light manipulation impacts performance and clarity.

Object distance

In lens optics, the object distance is the distance from the object to the lens (denoted as \( p \) in the Gaussian formula). This distance is crucial because it influences the position and size of the image formed by the lens.

• If the object is within the focal length (\( p < f \)), the lens forms a virtual, enlarged image on the same side as the object.
• If the object is beyond the focal length (\( p > f \)), the lens forms a real, inverted image on the opposite side.
• The object distance determines whether an image is real or virtual, magnified or reduced, based on its position relative to the focal point.

Variations in object distance lead to significant changes in optical systems, affecting everything from basic magnifying glasses to advanced imaging equipment. Whether you're capturing a scene with a camera or focusing a telescope, understanding object distance is vital for achieving the desired image quality and focus.

Image distance

Image distance is the distance from the lens to the image formed, represented by \( i \) in the Gaussian thin-lens equation. It plays a pivotal role in determining the characteristics of the image, such as:

• Whether the image is real or virtual: Real images are formed on the opposite side of the lens from the object and can be projected onto a screen, while virtual images appear on the same side as the object and cannot be projected.
• Whether the image is upright or inverted: Real images are typically inverted, while virtual images tend to be upright.
• The image's size compared to the object: It can be magnified or reduced depending on object and image distances.

Image distance is a vital factor in designing optical systems such as microscopes, binoculars, and corrective lenses. By understanding and manipulating image distance, it's possible to control how and where an image will appear, ensuring that lenses perform effectively for their intended applications.