Problem 23 - College Physics

Question

When a man's face is in front of a concave mirror of radius \(100 \mathrm{~cm}\), the lateral magnification of the image is \(+1.5 .\) What is the image distance?

Step-by-Step Solution

Answer

The image distance is \(-125 \text{ cm}\).

Step 1: Identify Known Values

First, let's note the known values from the problem. The radius of curvature \( R = 100 \text{ cm} \). The magnification \( m = +1.5 \). We can use the mirror formula and magnification formula to find the image distance.

Step 2: Understand Concepts and Formulae

The magnification formula for mirrors is given by \( m = -\frac{v}{u} \), where \( v \) is the image distance and \( u \) is the object distance. The mirror formula is \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \), where \( f \) is the focal length. For a mirror, the focal length \( f = \frac{R}{2} \).

Step 3: Calculate Focal Length

Given \( R = 100 \text{ cm} \), the focal length is \( f = \frac{R}{2} = \frac{100}{2} = 50 \text{ cm} \). Since the mirror is concave, \( f = -50 \text{ cm} \).

Step 4: Express Object Distance Using Magnification

Using the magnification formula, \( m = +1.5 = -\frac{v}{u} \). Rearranging gives \( u = -\frac{v}{1.5} \).

Step 5: Substitute into Mirror Formula

Substitute \( u = -\frac{v}{1.5} \) into the mirror formula: \[ \frac{1}{f} = \frac{1}{v} + \frac{3}{2v} \]\[ -\frac{1}{50} = \frac{1}{v} + \frac{3}{2v} \]

Step 6: Solve for Image Distance v

Combine terms on the right to get a common denominator:\[ -\frac{1}{50} = \frac{2 + 3}{2v} \]\[ -\frac{1}{50} = \frac{5}{2v} \]Cross multiply to solve for \( v \):\[ -50 \times 5 = 2v \]\[ -250 = 2v \]\[ v = -125 \text{ cm} \]

Step 7: Interpretation of Image Distance

The negative sign indicates that the image is formed in front of the mirror, which is expected for a concave mirror when the image is real and inverted. Hence, the image distance \( v \) is \(-125 \text{ cm} \).

Key Concepts

Lateral Magnification

Lateral magnification tells us how much larger or smaller the image of an object will appear when viewed in a mirror compared to its actual size. It is calculated using the formula:

• \( m = -\frac{v}{u} \)

Here, \( m \) is the magnification, \( v \) is the image distance, and \( u \) is the object distance.
For a concave mirror, if the magnification is positive, it means that the image is upright compared to the object. A magnification of \(+1.5\) indicates that the image is 1.5 times larger than the object's actual size.
This concept is crucial when analyzing how mirrors are used to manipulate the size of images in applications like telescopes or in optical instruments.

Image Distance

The image distance \( v \) is the distance from the mirror to the image formed. In mirror optics, this is crucial because it tells us where the image will appear relative to the mirror. For a concave mirror, when the image distance is negative, it indicates that the image forms on the same side as the object.
This convention is useful for determining if an image is real or virtual.

• Real images have a negative image distance and are inverted.
• Virtual images have a positive image distance and are upright.

In the problem, the image distance was calculated to be \(-125 \text{ cm}\), confirming a real image formed in front of the concave mirror.

Mirror Formula

The mirror formula is a fundamental equation used to relate the object distance, image distance, and the focal length of a mirror. It is expressed as:

• \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \)

Where \( f \) is the focal length, \( v \) is the image distance, and \( u \) is the object distance. By substituting the magnification expression \( u = -\frac{v}{1.5} \) into the mirror formula, we solve for \( v \) to understand the spatial relationship between these elements.
Understanding and applying the mirror formula helps in determining the nature and position of the image formed by mirrors and is a cornerstone concept in optics.

Focal Length

The focal length \( f \) of a mirror is the distance from the mirror to its focal point, where parallel rays of light either converge or appear to diverge. For a concave mirror, the focal length is negative, indicating that the focal point is located in front of the mirror.
It is calculated using the radius of curvature \( R \) by the formula:

• \( f = \frac{R}{2} \)

In this scenario, with a radius of \( 100 \text{ cm} \), the focal length is \(-50 \text{ cm}\), telling us how the mirror affects light paths and the formation of images.
Understanding focal length is important for designing optical devices, such as cameras and telescopes, which rely on precise control of light.