Problem 4 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A certain lens focuses an object \(1.85 \mathrm{~m}\) away as an image \(48.3 \mathrm{~cm}\) on the other side of the lens. What type of lens is it and what is its focal length? Is the image real or virtual?

Step-by-Step Solution

Answer

The lens is a convex lens with a focal length of 38.3 cm, and the image is real.

Step 1: Identify Known Values

We are given: the object distance \( d_o = 1.85 \text{ m} = 185 \text{ cm} \) (converted to centimeters for uniformity) and the image distance \( d_i = 48.3 \text{ cm} \).

Step 2: Use the Lens Formula

The lens formula is given by \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length, \( d_o \) is the object distance, and \( d_i \) is the image distance.

Step 3: Substitute Values and Calculate

Substitute the known values into the lens formula: \( \frac{1}{f} = \frac{1}{185} + \frac{1}{48.3} \). First, calculate each component separately, then add them: \( \frac{1}{185} \approx 0.00541 \), \( \frac{1}{48.3} \approx 0.0207 \). Adding them gives \( \frac{1}{f} = 0.00541 + 0.0207 = 0.02611 \).

Step 4: Calculate the Focal Length

Find the focal length by taking the reciprocal of the sum: \( f = \frac{1}{0.02611} \approx 38.3 \text{ cm} \).

Step 5: Determine the Type of Lens

Since the image distance is positive, the lens is converging (convex).

Step 6: Determine if the Image is Real or Virtual

Because the image distance is positive, the image is real and located on the opposite side of the lens from the object.

Key Concepts

Focal Length

The focal length of a lens is a crucial aspect of geometrical optics. It represents the distance from the center of the lens to the point where parallel rays of light converge or appear to diverge after passing through the lens. This point is known as the focal point.

In the exercise, the focal length is calculated using the lens formula, which is:

• \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \)

• Where \( f \) is the focal length
• \( d_o \) is the object distance
• \( d_i \) is the image distance

The focal length tells us how strongly the lens converges or diverges light. In this particular exercise, we found that the focal length is approximately 38.3 cm, indicating the power of the lens to bring light to a focal point. A shorter focal length means stronger convergence or divergence of light beams.

Converging Lens

A converging lens, commonly known as a convex lens, is one that brings parallel rays of light to focus at a point known as the focal point. This type of lens is thicker in the middle and thinner at the edges.

Converging lenses are used in a variety of applications, such as eyeglasses, cameras, and microscopes, because they can form real images. In the exercise, since the image distance (\( d_i = 48.3 \text{ cm} \)) was positive, it indicated that the lens used was indeed a converging lens.

Such lenses produce real and inverted images if the object is placed beyond the focal length. If the object is closer to the lens than its focal length, the image will be virtual and erect. This exercise showcases a situation where the object is beyond the focal point, therefore forming a real image on the opposite side of the lens.

Real Image

A real image is formed when light rays actually converge and pass through the image location. Unlike a virtual image, which is created by light rays that appear to diverge, real images can be projected onto a screen.

The properties of real images are:

• They are inverted (upside down).
• They can be projected onto a surface.
• Formed on the opposite side of the lens from the object, when using a converging lens.

In the given exercise, the image formed by the converging lens was real since the image distance was positive. This signifies that the image is located on the opposite side of the lens from the object. This is a typical behavior of converging lenses when the object is placed beyond the lens's focal length.