Problem 14 - University Physics with Modern Physics

Question

A spherical, concave shaving mirror has a radius of curvature of 32.0 \(\mathrm{cm}\) . (a) What is the magnification of a person's face when it is 12.0 \(\mathrm{cm}\) to the left of the vertex of the mirror? (b) Where is the image? Is the image real or virtual? (c) Draw a principal-ray diagram showing the formation of the image.

Step-by-Step Solution

Answer

(a) Magnification is approximately 0.57, (b) image is virtual, located approximately 6.86 cm behind the mirror, (c) principal rays illustrate the virtual image.

Step 1: Understanding the Mirror Formula and Magnification Formula

The mirror formula is \( \frac{1}{f} = \frac{1}{do} + \frac{1}{di} \), where \( f \) is the focal length, \( do \) is the object distance, and \( di \) is the image distance. The magnification \( m \) is given by \( m = -\frac{di}{do} \). We need to find the magnification and image distance, \( di \).

Step 2: Find the Focal Length

The radius of curvature \( R \) is 32.0 cm. The focal length \( f \) is half of the radius of curvature, since \( f = \frac{R}{2} \). Therefore, \( f = \frac{32.0}{2} = 16.0 \) cm. Since it's a concave mirror, the focal length is negative. Thus, \( f = -16.0 \) cm.

Step 3: Use the Mirror Formula to Find Image Distance

Determine the object distance \( do \), which is given as 12.0 cm. Substitute \( f = -16.0 \) cm and \( do = 12.0 \) cm into the mirror formula:\[\frac{1}{-16} = \frac{1}{12} + \frac{1}{di}.\]Solve for \( \frac{1}{di} \):\[\frac{1}{di} = \frac{1}{-16} - \frac{1}{12}.\]Calculate \( \frac{1}{di} \):\[\frac{1}{di} = -\frac{1}{16} - \frac{1}{12} = \frac{-3 - 4}{48} = \frac{-7}{48}.\]So, \( di = -\frac{48}{7} \approx -6.86 \) cm.

Step 4: Calculate the Magnification

Using the magnification formula \( m = -\frac{di}{do} \), substitute the values found:\[m = -\left(\frac{-6.86}{12.0}\right) = \frac{6.86}{12.0} \approx 0.57.\]The positive magnification value indicates the image is upright.

Step 5: Determine the Nature of the Image

Since the image distance \( di \) is negative, the image formed is virtual (formed behind the mirror). The positive magnification indicates it is upright and smaller than the object.

Step 6: Draw a Principal-Ray Diagram (Conceptual)

Create a diagram of the mirror, object, and image. The concave mirror has principal rays that: 1. A ray parallel to the principal axis reflects through the focal point. 2. A ray through the center of curvature reflects back on itself. 3. A ray aimed at the focal point reflects parallel to the principal axis. These rays show the formation of a virtual image behind the mirror.

Key Concepts

Concave Mirror

A concave mirror, also known as a converging mirror, is a type of spherical mirror that curves inward, resembling a portion of the inner surface of a sphere. These mirrors have a reflective surface on the inside, which means they can converge light. They are commonly utilized in applications where focusing light is necessary, such as in telescopes and shaving mirrors like the one described in the exercise.

The main characteristics of a concave mirror include:

• The center of curvature (C), which is the center of the sphere of which the mirror is a part.
• The focal point (F), located halfway between the mirror's surface and the center of curvature.
• The principal axis, which is an imaginary line that passes through C and F.

The behavior of concave mirrors allows for the creation of various types of images, depending on the object's position relative to the focal point and center of curvature. These mirrors can form real or virtual images, making them versatile in different optical setups.

Magnification

Magnification in mirrors refers to the measure of how much larger or smaller an image appears compared to the actual object. In the context of spherical mirrors, it is determined using the formula: \[ m = -\frac{di}{do} \]where:

• \( m \) is the magnification
• \( di \) is the image distance from the mirror
• \( do \) is the object distance from the mirror

In our exercise, the calculated magnification value was approximately 0.57, indicating the image also appeared smaller than the actual object. Since this number is positive, it denotes that the image is upright. Concave mirrors can show either enlarged or reduced images depending on how far the object is placed relative to the focal point.

Mirror Formula

The mirror formula is a crucial equation in examining the properties of images formed by spherical mirrors. It's mathematically expressed as:\[\frac{1}{f} = \frac{1}{do} + \frac{1}{di}\]where:

• \( f \) is the focal length of the mirror.
• \( do \) is the object distance.
• \( di \) is the image distance.

This equation is essential because it connects the focal length of the mirror to the object and image distances, giving insights into the position and nature of the image.In the provided problem, using the given object distance and the calculated focal length, we solved for the image distance. Understanding this relationship helps in predicting where images will appear in different scenarios with concave mirrors.

Virtual Image

A virtual image is a type of image formed by mirrors (and lenses) that diverges from the surface where it appears to be, but it cannot be projected on a screen. In the case of a concave mirror, a virtual image occurs when the object is placed within the focal length of the mirror.

Characteristics of virtual images include:

• They are always upright as opposed to inverted, as real images can be.
• They appear to be located behind the mirror surface.
• They cannot be captured on a screen as they do not converge in real space.

In the solution to our exercise, the negative image distance indicated that the image formed was virtual, which is typical for objects placed closer to a concave mirror than its focal point. Virtual images play a significant role in everyday devices like makeup and shaving mirrors, where an upright and magnified view of the object is practical.