Problem 15 - Applied Physics

Question

What is the focal length of a convex lens that produces an inverted image twice as large as the object at a distance of \(13.3 \mathrm{~cm}\) from the lens?

Step-by-Step Solution

Answer

The focal length of the lens is \(-13.3\text{ cm}\).

Step 1: Identify and Write Down the Given Values

We are given that the distance from the lens to the inverted image is \(v = 13.3 \text{ cm}\), and the magnification \(m = -2\) because the image is inverted and twice as large as the object (signifying it is a real image). The object distance \(u\) and focal length \(f\) are to be determined.

Step 2: Use Magnification Formula

The magnification \(m\) is related to the object and image distances by the formula \(m = \frac{v}{u}\). Given \(m = -2\), we have \(-2 = \frac{13.3}{u}\). Solve for \(u\): \[ u = \frac{13.3}{-2} = -6.65 \text{ cm} \]The negative sign indicates the object is on the opposite side of the light source, as expected in real situations for lenses.

Step 3: Apply Lens Formula

The lens formula is \(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\). Substitute \(v = 13.3\text{ cm}\) and \(u = -6.65\text{ cm}\):\[ \frac{1}{f} = \frac{1}{13.3} + \frac{1}{-6.65} \]

Step 4: Calculate the Focal Length

First, find the least common denominator and solve:\[ \frac{1}{f} = \frac{1}{13.3} - \frac{1}{6.65} \]\[ \frac{1}{f} = \frac{1 - 2}{13.3} = -\frac{1}{13.3} \]Thus, \(f = -13.3 \text{ cm}\). The negative sign signifies a negative focal length for real convex lenses.

Key Concepts

Convex Lens

A convex lens, known as a converging lens, is a vital tool in optics. It's bulged outwards on both sides, causing parallel rays of light that pass through it to converge or meet at a single point. This point is known as the focal point.
Convex lenses are commonly used in various optical devices, such as cameras, magnifying glasses, and binoculars, because they help focus light and form images.
The behavior of a convex lens greatly depends on its focal length and the position of an object placed in front of it. When an object is positioned farther than the focal point, the lens can produce an inverted and real image as the light rays actually meet on the other side of the lens.

• This property is utilized in device optics for capturing real images on film or digital sensors.
• It's crucial for understanding how glasses correct vision and how projectors display images.

Understanding these basics provides a foundation for diving deeper into the details of image formation in lenses.

Inverted Image

An inverted image is a mirror image flipped vertically and horizontally in comparison to the original object. In the context of lenses, such as our convex lens example, this occurs when the light rays emerging from an object pass through the lens and converge on the other side.
Inverted images are characteristic of real images formed with convex lenses. Real images can be projected onto a screen because they form at the actual intersection of light rays, unlike virtual images which cannot be captured directly.
You can observe inverted images in many scenarios like:

• The image formed by a projector that flips the content so it appears correctly on the screen.
• The images captured by cameras, which are processed to be viewed correctly.

Such practical applications of inverted images help us understand how lenses are used to interpret and capture imagery in everyday technology.

Lens Formula

The lens formula is an essential part of understanding how lenses operate. It establishes the mathematical relationship between the object distance (u), the image distance (v), and the focal length (f) of a lens. The formula is expressed as:
\[\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\]
This formula is pivotal when determining one of these three variables, given the other two. In problems involving real convex lenses, like the exercise, object distances are often considered negative when using traditional sign conventions, as the object is on the opposite side of incoming light.
Through proper substitution and algebraic manipulation, the lens formula enables precise calculation of the missing parameter, ensuring accurate image prediction and lens application in various settings such as:

• Designing optical instruments.
• Correcting vision with glasses.

Understanding this formula allows us to calculate how lenses will behave when receiving light and how they form images.

Magnification Formula

The magnification formula is another key concept when working with lenses, linking the size and orientation of an image compared to the original object. It is represented as:
\[m = \frac{v}{u}\]
In this context, magnification (m) is the ratio of the image distance (v) to the object distance (u). The sign of m reveals the image's nature: a negative magnification indicates an inverted image. For a convex lens, if the image's absolute size is greater than the object's, it's a real and magnified inverted image.
This principle of magnification is essential for understanding:

• How a microscope or telescope enlarges tiny or distant objects.
• The workings of cameras and projectors.

By mastering the magnification formula, you can predict how lenses change your viewing experience, focusing or enlarging what you see.