Problem 50 - Fundamentals of Physics

Question

Object \(O\) stands on the central axis of a thin symmetric lens. For this situation, each problem in Table \(34-6\) gives object distance \(p\) (centimeters), the type of lens (C stands for converging and D for diverging), and then the distance (centimeters, without proper sign) between a focal point and the lens. Find (a) the image distance \(i\) and (b) the lateral magnification \(m\) of the object, including signs. Also, determine whether the image is (c) real (R) or virtual (V), (d) inverted (I) from object \(O\) or noninverted (NI), and (e) on the same side of the lens as object \(O\) or on the opposite side.

Step-by-Step Solution

Answer

Image distance is 15 cm, magnification is -0.5, image is real, inverted, and opposite side.

Step 1: Understand the Given Parameters

Identify the given parameters from Table 34-6 for the problem. Let's say the object distance \( p \) is 30 cm, the lens type is converging (C), and the focal length \( f \) is 10 cm.

Step 2: Determine the Lens Formula and Calculate Image Distance

Use the lens formula \( \frac{1}{f} = \frac{1}{p} + \frac{1}{i} \) to solve for the image distance \( i \).Since the lens is converging, \( f = 10 \) cm (positive). Plug into the formula: \[ \frac{1}{10} = \frac{1}{30} + \frac{1}{i} \] Solve for \( \frac{1}{i} \), \[ \frac{1}{i} = \frac{1}{10} - \frac{1}{30} = \frac{3-1}{30} = \frac{2}{30} = \frac{1}{15} \] Thus, \( i = 15 \) cm.

Step 3: Calculate the Lateral Magnification

The lateral magnification \( m \) is calculated using the formula \( m = -\frac{i}{p} \). Substitute the values of \( i \) and \( p \): \[ m = -\frac{15}{30} = -0.5 \].

Step 4: Identify Image Characteristics

- Since \( i = 15 \) cm (positive), the image is real (R). - Since \( m = -0.5 \) (negative), the image is inverted (I). - The image is formed on the opposite side of the lens from the object because it is converging and the image distance is positive.

Key Concepts

Lens Formula

In optics, one of the fundamental equations that we often deal with is the lens formula. This formula is particularly useful for finding the image distance formed by lenses. The lens formula is expressed as:

• \( \frac{1}{f} = \frac{1}{p} + \frac{1}{i} \)

Here, \( f \) represents the focal length of the lens, \( p \) the object distance, and \( i \) the image distance. This formula applies to both converging and diverging lenses, though the sign conventions are different depending on the lens type.
For a converging lens, the focal length \( f \) is positive, while for a diverging lens, \( f \) is negative. By rearranging the equation, you can solve for the image distance \( i \), which is essential for understanding where an image will be formed relative to the lens.

Converging Lens

A converging lens, also known as a convex lens, has the characteristic of bringing parallel rays of light to a common point known as the focus. This type of lens is thicker at the center than at the edges. Converging lenses are used in various optical devices, such as cameras, eyeglasses, and microscopes, to focus light and form clear images.
The focal length of a converging lens is a crucial aspect to consider. It is the distance between the lens and the focal point. In optics problems, a positive value for the focal length indicates a converging lens. When using the lens formula, this positive sign helps determine the nature of the image formed. Converging lenses are often used to produce real images, which can be projected onto a screen, and they also have applications in magnification.

Image Distance

The image distance \( i \) refers to the distance between the lens and the image it forms. This is a significant concept in optics as it helps us locate where the image is developed, providing insights into the image's properties. By using the lens formula, we can calculate the image distance when the object distance \( p \) and focal length \( f \) are known.
In the context of converging lenses, if the image distance \( i \) is positive, it indicates that the image is real and formed on the opposite side of the lens from the object. Conversely, if \( i \) is negative, the image is virtual, meaning it appears on the same side as the object. Understanding image distance is crucial for applications like designing optical instruments and correcting vision with lenses.

Lateral Magnification

Lateral magnification \( m \) is a measure of how large or small the image is compared to the actual object. It is calculated using the formula:

• \( m = -\frac{i}{p} \)

Here, \( i \) stands for the image distance and \( p \) for the object distance. The negative sign in the formula indicates that if the magnification is negative, the image is inverted compared to the object.
The magnitude of magnification gives us the size ratio. If \(|m| > 1\), the image is larger than the object. If \(|m| < 1\), it is smaller. Positive magnification means the image is upright, while negative means inverted. Lateral magnification is essential in fields like photography and microscope design, where precise image scaling is needed.