Question

A convex lens focuses light rays from an object to create an image, as shown in the diagram. The image distance, \(I\), is related to the object distance, \(b,\) by the function \(I=\frac{f b}{b-f},\) where the focal length, \(f,\) is a constant for the particular lens used based on its specific curvature. When the object is placed closer to the lens than the focal length of the lens, an image is perceived to be behind the lens and is called a virtual image. a) Graph \(I\) as a function of \(b\) for a lens with a focal length of \(4 \mathrm{cm} .\) b) How does the location of the image change as the values of \(b\) change? c) What type of behaviour does the graph exhibit for its non-permissible value? How is this connected to the situation?

Short answer

For a lens with \( f = 4 \) cm, as \( b \) changes, the image goes from virtual to real. The graph has a vertical asymptote at \( b = 4 \) cm.

Step-by-step solution

- Define the function

The relationship between the image distance and the object distance is given by the function: \( I = \frac{f b}{b - f} \)where \( f \) is the focal length.

- Substitute the focal length

For this specific problem, the focal length \( f = 4 \) cm. Substitute this into the function to get:\( I = \frac{4b}{b - 4} \)

- Graph the function

To graph \( I \) as a function of \( b \), plot \( I \) versus \( b \) using the equation \( I = \frac{4b}{b - 4} \). The behavior of the graph can be studied in intervals, especially around the non-permissible value.

- Analyze the image distance changes

Observe how the value of \( I \) (image distance) changes as \( b \) (object distance) varies. For values of \( b < 4 \), \( I \) will be negative, meaning the image is virtual and on the same side of the lens as the object. For \( b > 4 \), \( I \) is positive, indicating a real image on the opposite side of the lens from the object.

- Identify the non-permissible value

Identify the non-permissible value of \( b \) in the function. This occurs where the denominator is zero, i.e., \( b - 4 = 0 \). Therefore, \( b = 4 \) cm is the non-permissible value.

- Explain the graph's behavior at the non-permissible value

At the non-permissible value, \( b = 4 \) cm, the function is undefined, which corresponds to a vertical asymptote in the graph. This is due to the fact that when the object is placed exactly at the focal length, the lens cannot form a real image.

Key concepts

Focal Length

The focal length, denoted as \( f \), is a key property of a convex lens that describes the distance from the center of the lens to the focal point. In simple terms, it is where the lens converges parallel rays of light. The shorter the focal length, the more powerful the lens. For example, in the given exercise, the focal length is specified as 4 cm. The focal length essentially determines how the lens will behave in image formation and affects both real and virtual images. An understanding of focal length is crucial for predicting where an image will appear relative to the object and the lens.

Object Distance

The object distance, denoted as \( b \), refers to the distance between the object and the lens. This measurement is crucial because it influences how the lens focuses light to form an image. The given function, \( I = \frac{f b}{b - f} \), shows the relationship between object distance and image distance for a convex lens. As per the solution steps, varying the object distance affects whether the image is real or virtual. Understanding object distance helps in predicting the placement and properties of the resulting image.

Virtual Image

A virtual image is formed when light rays diverge, and the image appears to be located on the same side of the lens as the object. This type of image cannot be projected onto a screen as it does not actually converge. In the given example, when the object distance \( b \) is less than the focal length \( f = 4 \) cm, the image distance \( I \) is negative. This negative value signifies a virtual image perceived to be behind the lens.

Real Image

A real image is formed when light rays actually converge at a point. It can be projected on a screen, appearing on the opposite side of the lens from the object. From the exercise, if the object distance \( b \) is greater than the focal length \( f = 4 \) cm, the image distance \( I \) becomes positive, indicating that the image is real. Understanding when and how real images are formed is fundamental to applications such as photography and various optical instruments.

Mathematical Function Graphing

To visualize the relationship between object distance \( b \) and image distance \( I \), graphing the mathematical function \( I = \frac{4b}{b - 4} \) is essential. This graphical representation helps to intuitively understand how changes in \( b \) affect \( I \). For instance, plotting this equation will reveal that as \( b \) approaches 4 cm, the image distance \( I \) tends to infinity. Analyzing the graph gives insights into the nature of the images formed and the non-permissible value.

Asymptote

An asymptote is a line that a graph approaches but never touches. For the function \( I = \frac{4b}{b - 4} \), the non-permissible value of \( b = 4 \) cm represents a vertical asymptote. This indicates that at \( b = 4 \) cm, the function is undefined and \( I \) tends to infinity. Asymptotic behavior at \( b = 4 \) cm in this context means the lens cannot form a real image at this particular object distance, emphasizing the unique optical property when the object is exactly at the focal length.