Problem 105 - University Physics with Modern Physics

Question

(a) For a lens with focal length f, find the smallest distance possible between the object and its real image. (b) Graph the distance between the object and the real image as a function of the distance of the object from the lens. Does your graph agree with the result you found in part (a)

Step-by-Step Solution

Answer

The smallest distance is 4f, confirmed by the graph.

Step 1: Understand the Lens Formula

The lens formula is given by the equation \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length of the lens, \( d_o \) is the distance of the object from the lens, and \( d_i \) is the distance of the image from the lens. For real images formed by converging lenses, both \( d_o \) and \( d_i \) are positive.

Step 2: Express Total Distance

The total distance between the object and the image is given by \( D = d_o + d_i \). Our goal is to find the minimum value of \( D \).

Step 3: Combine Equations

Substituting \( d_i = \frac{d_o f}{d_o - f} \) from the lens formula into \( D = d_o + d_i \), we get \( D = d_o + \frac{d_o f}{d_o - f} \).

Step 4: Differentiate and Solve for Minimum

To find the minimum value of \( D \), take the derivative of \( D \) with respect to \( d_o \), and set it to zero. This gives us \( D' = 1 - \frac{f^2}{(d_o - f)^2} = 0 \). Solving this, we find \( d_o = 2f \) and \( d_i = 2f \), so the smallest distance \( D = d_o + d_i = 4f \).

Step 5: Graph the Function

Graph the function \( D = d_o + \frac{d_o f}{d_o - f} \) with respect to \( d_o \) and observe its behavior. The graph will show a minimum distance, confirming the analytical result found earlier.

Step 6: Interpret the Graph

Upon graphing, the function \( D(d_o) \) has a vertex at \( d_o = 2f \), showing that the smallest distance occurs here, which agrees with the calculation from Step 4.

Key Concepts

Lens Formula

The lens formula is crucial for understanding how lenses form images. In optics, it is expressed as: \[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]where:

• \( f \) is the focal length of the lens.
• \( d_o \) is the distance from the object to the lens.
• \( d_i \) is the distance from the image to the lens.

This equation connects these three quantities, allowing you to calculate the position of the image if the object position and focal length are known. For converging lenses, which we use here, both \( d_o \) and \( d_i \) are positive values.
The lens formula is derived from the principles of geometric optics and is applicable for both real and virtual images. It is essential when determining image characteristics, such as position and nature. By substituting known values into this formula, we can calculate the unknowns for specific optical scenarios.

Focal Length

Focal length is a vital attribute of a lens that tells how strongly the lens converges or diverges light. It is the distance from the lens to the point where parallel rays of light converge to a point (focus).
An important concept to remember about focal length is:- Shorter focal lengths result in stronger convergence of light waves, producing a larger image.- Longer focal lengths mean a weaker convergence, producing a smaller image.
In our exercise, knowing the focal length \( f \) helps us determine the minimum distance between the object and its real image using the lens formula. When the object is placed at a distance of \( 2f \) from the lens, the image also forms at \( 2f \) on the opposite side.
Understanding focal length is essential for applications in photography, vision correction, and even in eyeglasses. It significantly influences the magnification and field of view of optical instruments.

Real Image

A real image is formed whenever light rays actually converge at a point. Unlike virtual images, which seem to be located at a point from which the rays appear to diverge, real images can be projected onto a screen.
Real images are typically formed by converging lenses or mirrors. In the context of the exercise, we were concerned with a converging lens creating a real image at \( d_i = 2f \). This means the light rays from the object get bent by the lens and meet at this point, creating an actual image.Key features of real images include:

• They can be captured on a film or a sensor, like in cameras.
• They are usually inverted, meaning they appear upside-down relative to the object.
• They are formed when the object is placed beyond the focal point of the lens.

When working through our problem, identifying the conditions under which a lens can form a real image helps predict where this image will appear and how it can be used in practical applications.