Problem 65 - College Physics

Question

For a lens combination, show that the total magnification \(M_{\text {total }}=M_{1} M_{2}\). [Hint: Think about the definition of magnification.

Step-by-Step Solution

Answer

The total magnification is the product of the individual magnifications: \( M_{\text{total}} = M_1 M_2 \).

Step 1: Understand Magnification

The magnification of a single lens, \( M \), is defined as the ratio of the height of the image \( h_i \) to the height of the object \( h_o \), which can also be expressed in terms of the object distance \( d_o \) and the image distance \( d_i \): \( M = \frac{h_i}{h_o} = \frac{d_i}{d_o} \).

Step 2: Apply Magnification to the First Lens

For the first lens, consider the magnification \( M_1 \) which relates the image height after the first lens to the object height: \( M_1 = \frac{h_{i1}}{h_o} = \frac{d_{i1}}{d_{o1}} \). Here, \( h_{i1} \) is the height of the image formed by the first lens.

Step 3: Use the Image as an Object for the Second Lens

The image formed by the first lens becomes the object for the second lens. Hence, the object height for the second lens is \( h_{i1} \). The magnification of the second lens is \( M_2 = \frac{h_{i2}}{h_{i1}} = \frac{d_{i2}}{d_{o2}} \).

Step 4: Derive Total Magnification

The total magnification is the product of the magnifications of each lens. Calculate \( M_{\text{total}} = M_1 \times M_2 = \left(\frac{h_{i1}}{h_o}\right) \times \left(\frac{h_{i2}}{h_{i1}}\right) \). Simplifying this gives, \( M_{\text{total}} = \frac{h_{i2}}{h_o} \).

Step 5: Conclude the Total Magnification Formula

Because the intermediate image height cancels out, the total magnification of the lens system is \( M_{\text{total}} = M_1 \times M_2 = \frac{h_{i2}}{h_o} \). This shows that the total magnification is indeed the product of the individual magnifications of the lenses.

Key Concepts

Lens Combination

In optics, a lens combination refers to the use of two or more lenses in sequence to achieve a desired magnification or focus.
By examining a pair of lenses, we can understand how they collectively modify the path of light to form images. When light passes through the first lens, it forms an image. This image then acts as the object source for the second lens.

• The behavior of light through multiple lenses can change depending on the focal lengths and the arrangement.
• Each lens contributes based on its own properties, like refractive index and curvature.

Understanding how lens combinations work enables us to create complex optical systems. This understanding is crucial in fields like photography, microscopy, and even astronomy, where lenses are combined to enhance detail and clarity.

Image Distance

Image distance refers to the distance from the lens to the point where the image is formed. This is typically denoted as \( d_i \) in lenses problems.
It plays a crucial role in determining the clarity and size of the image formed by a lens or lens system.

• When using a single lens, the image distance depends on both the focal length of the lens and the object's distance from the lens.
• In a lens combination, the first lens creates an image that becomes the object for the second lens, altering the second lens's image distance.

To calculate image distance accurately, it's important to apply the lens formula:
\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]
Here, \( f \) is the focal length, \( d_o \) is the object distance, and \( d_i \) is the image distance. This formula helps in deciphering complex lens systems and their optical behavior.

Object Distance

Object distance is the measure from the object to the lens, denoted as \( d_o \). Understanding object distance is fundamental to calculating the behavior of a lens, as it collaborates with focal length to determine where the image forms.

• Closer objects (small \( d_o \)) result in a larger, magnified image by the lens.
• When an image formed by the first lens acts as the 'object' for the second lens, we refer to the distance from this intermediate image to the second lens as the new \( d_o \).

The sum of all lenses’ influences produces a final image, giving insight into how lens combinations affect overall image distance and magnification. Always using the lens formula helps connect all these distances:
\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]
This formula is the bedrock upon which lens equations are built, aiding in systematically exploring objects and their corresponding images in optical systems.