Problem 7 - Fundamentals of Physics

Question

A concave shaving mirror has a radius of curvature of \(35.0 \mathrm{~cm}\). It is positioned so that the (upright) image of a man's face is \(2.50\) times the size of the face. How far is the mirror from the face?

Step-by-Step Solution

Answer

The mirror is approximately 24.5 cm away from the face.

Step 1: Understand the Problem

We are given a concave mirror with a radius of curvature of \(35.0 \text{ cm}\) and need to determine the object distance from the mirror, where the image of the man's face is \(2.50\) times the size of the face.

Step 2: Use the Mirror Equation

The mirror equation is \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\), where \(f\) is the focal length of the mirror, \(d_o\) is the object distance, and \(d_i\) is the image distance.

Step 3: Find the Focal Length

The focal length \(f\) is half of the radius of curvature, so \(f = \frac{35.0}{2} = 17.5 \text{ cm}\).

Step 4: Use the Magnification Relation

The magnification \(m\) is given by \(m = \frac{d_i}{d_o} = -\frac{h_i}{h_o}\). It's given that the magnification \(m = 2.50\) (upright image, so \(-\) sign is not used).

Step 5: Find Image Distance in Terms of Object Distance

From \(m = \frac{d_i}{d_o}\), we have \(d_i = 2.50d_o\).

Step 6: Plug into the Mirror Equation

Substitute \(d_i = 2.50d_o\) into the mirror equation: \(\frac{1}{17.5} = \frac{1}{d_o} + \frac{1}{2.50d_o}\).

Step 7: Solve the Equation for Object Distance

Combine the fractions: \(\frac{2.5}{2.5d_o} + \frac{1}{2.5d_o} = \frac{3.5}{2.5d_o}\). This equates to \(\frac{1}{17.5} = \frac{3.5}{2.5d_o}\).

Step 8: Calculate the Object Distance

Solving \(2.5d_o = 3.5 \times 17.5\) gives \(2.5d_o = 61.25\). Solving for \(d_o\), we find \(d_o \approx 24.5 \text{ cm}\).

Key Concepts

Radius of Curvature

In optics, the radius of curvature is an important concept when dealing with mirrors, especially concave and convex mirrors. It refers to the radius of the sphere from which the mirror is a small segment. For a concave mirror, the reflective surface bulges inward, like a spoon.
The radius of curvature is directly related to the focal length (f) of the mirror, which is the distance from the mirror to the focal point.

• The radius of curvature (R) is typically twice the focal length: \[ R = 2f \]
• In our example, the concave mirror has a radius of curvature of \(35.0\text{ cm}\), and this helps us find the focal length easy peasy by using the same formula.
• This radius plays a crucial role in determining how the mirror forms images of objects placed at various distances.

Understanding the radius of curvature helps us grasp how the curvature of a mirror affects its ability to focus light.

Magnification

The term 'magnification' in mirror problems describes how much larger or smaller the image is compared to the object itself. It is a vital concept in understanding how optical devices, like concave mirrors, alter our perception.
For a mirror, the magnification (m) is calculated using the formula:

• \[ m = \frac{h_i}{h_o} = \frac{d_i}{d_o} \]
• Here, \(h_i\) is the height of the image, \(h_o\) is the height of the object, \(d_i\) is the distance to the image, and \(d_o\) is the distance to the object.
• A positive magnification indicates an upright image. In our scenario, the man's face appears 2.5 times larger, so the magnification value is 2.50.

Understanding magnification allows us to predict the size and orientation of the image produced by a concave mirror, an imperative skill for tackling such problems.

Mirror Equation

The mirror equation is a fundamental formula used to relate the object distance, image distance, and focal length of a mirror. Understanding it equips us to solve a variety of problems involving mirrors, like finding where the image will appear.
The mirror equation can be expressed as:

• \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]
• Where \(f\) is the focal length, \(d_o\) is the object distance, and \(d_i\) is the image distance.

In a concave mirror scenario, the values are generally positive when considering real images.
When provided with either the image or object distance and the focal length, you can easily determine the missing variable using this equation.
In practice, like in our example, you might be required to substitute known values into the mirror equation to find either \(d_o\) or \(d_i\). This makes the mirror equation an indispensable tool in optics!

Focal Length

The focal length is a key parameter in the study of mirrors and lenses, especially when analyzing concave mirrors. It represents the distance between the mirror's surface and its focal point, where parallel rays of light either converge or appear to diverge from.

• The focal length \(f\) is determined by half of the radius of curvature:\[ f = \frac{R}{2} \]
• In our exercise with a mirror having a radius of curvature of \(35.0\text{ cm}\), the calculated focal length is \(17.5\text{ cm}\).
• A positive focal length in concave mirrors indicates that the focal point is located on the side of the reflecting surface.

Understanding the focal length is crucial because it helps predict the paths of light rays as they reflect off the mirror, determining where the image forms.
In problem-solving, once the focal length is known, we can apply it in the mirror equation to find unknown distances and fully describe the optical behavior of the setup.