Problem 15 - An Introduction to Physical Science

Question

An object is placed \(60 \mathrm{cm}\) in front of a converging lens with a focal length of \(20 \mathrm{cm} .\) Draw a ray diagram. Estimate the image distance and give the image characteristics.

Step-by-Step Solution

Answer

The image is 15 cm on the opposite side, real, and inverted.

Step 1: Understand the Lens Formula

The lens formula relates the object distance \( u \), the image distance \( v \), and the focal length \( f \) by the equation: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] We need to find the image distance \( v \) using this formula.

Step 2: Substitute Values into the Lens Formula

The object distance \( u \) is given as \(-60\,\text{cm}\) (we take it as negative for lens formula conventions), and the focal length \( f \) is \(20\,\text{cm}\). Substitute these values into the lens formula: \[ \frac{1}{20} = \frac{1}{v} + \frac{1}{-60} \]

Step 3: Solve for Image Distance

Reorganize the equation to solve for \( \frac{1}{v} \): \[ \frac{1}{v} = \frac{1}{20} + \frac{1}{60} \] Finding a common denominator (\(60\)), we have: \[ \frac{1}{v} = \frac{3}{60} + \frac{1}{60} = \frac{4}{60} = \frac{1}{15} \] Therefore, \( v = 15\,\text{cm} \).

Step 4: Image Characteristics

Since the image distance \( v \) is positive, the image is real and formed on the opposite side of the lens from the object. The image is inverted in relation to the object.

Step 5: Draw Ray Diagram

Draw the principal axis and the lens. Place the object 60 cm to the left of the lens. Draw a ray parallel to the principal axis from the top of the object, which passes through the focal point on the other side after refraction. Another ray passes through the lens center without deviation. The intersection of these rays on the opposite side of the lens shows the position of the image, confirming our calculated image distance.

Key Concepts

Ray Diagram

A ray diagram is a helpful tool in understanding how lenses form images. To begin drawing a ray diagram for a converging lens, start by sketching a vertical line to represent the lens, with a horizontal line passing through the center for the principal axis. The focal points, located on either side of the lens at the focal length, should be marked clearly.
Next, place the object 60 cm to the left of the lens on the principal axis. Draw a ray from the top of the object, parallel to the principal axis. After passing through the lens, this ray should be refracted through the focal point on the other side.
Draw a second ray from the top of the object directly through the lens's center, which will not deviate. The point where these two refracted rays intersect on the opposite side of the lens showcases the image's location. This visual method helps confirm both the image distance and characteristics.

Image Distance Calculation

Calculating the image distance for a converging lens involves using the lens formula, which establishes a relationship between the object distance (\(u\)), image distance (\(v\)), and focal length (\(f\)). The formula is: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
First, substitute the known values into the formula. Here, \(u\) is \(-60\) cm (taken as negative by convention), and \(f\) is \(20\) cm. Plugging these values in gives:\[ \frac{1}{20} = \frac{1}{v} + \frac{1}{-60} \]
Solving this equation starts by isolating \( \frac{1}{v} \), leading to:\[ \frac{1}{v} = \frac{1}{20} + \frac{1}{60} \]
Simplify using a common denominator (\(60\)):\[ \frac{1}{v} = \frac{3}{60} + \frac{1}{60} = \frac{4}{60} = \frac{1}{15} \]
Thus,\(v\) equals \(15\) cm, indicating the image is located 15 cm on the opposite side of the lens.

Lens Formula

The lens formula is a fundamental equation in optics, essential for calculating image distance and understanding lens behavior. It links three crucial parameters:

• Object Distance (\(u\))
• Image Distance (\(v\))
• Focal Length (\(f\))

The formula is mathematically expressed as: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Understanding this formula requires recognizing that:

• The object distance (\(u\)) is measured from the object to the lens.
• The image distance (\(v\)) is from the lens to the image.
• The focal length (\(f\)) is the distance from the lens to the focal point.

The sign conventions are important here, as they impact how values are substituted into the equation. For lenses, typically, the object distance is taken as negative when measured against the direction of incoming light rays. Applying and rearranging this formula helps find unknown distances, providing insight into how an image is formed.

Image Characteristics

The characteristics of an image formed by a converging lens hinge on key attributes such as its orientation, size, and nature. Using the calculated image distance, we can infer several characteristics.
With an image distance (\(v\)) of \(15\) cm, this particular image is:

• Real: Because the image is formed on the opposite side of the lens to the object.
• Inverted: The image orientation is upside down compared to the upright object.
• Reduced in Size: Typically, the image is smaller than the object, given the object was placed further from the focal point compared to the image distance.

Understanding these characteristics is crucial when analyzing the results of a ray diagram. They provide a comprehensive understanding of how lenses manipulate light to focus and project images.