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 12.5 cm from the face.

Step 1: Identify Given Information

We are given the radius of curvature \( R = 35.0 \text{ cm} \) and the magnification \( M = 2.50 \). The image is upright, indicating a positive magnification for the concave mirror.

Step 2: Calculate the Focal Length

The focal length \( f \) of a concave mirror is related to the radius of curvature by the formula \( f = \frac{R}{2} \). Thus, \( f = \frac{35.0 \text{ cm}}{2} = 17.5 \text{ cm} \).

Step 3: Use the Magnification Formula

Magnification \( M \) is defined as \( M = -\frac{q}{p} \), where \( q \) is the image distance and \( p \) is the object distance. Since magnification is given as 2.50 (positive, because the image is upright), we have \( 2.50 = \frac{q}{p} \). Thus, \( q = 2.50p \).

Step 4: Substitute Into Mirror Equation

The mirror equation is \( \frac{1}{f} = \frac{1}{p} + \frac{1}{q} \). Substitute \( q = 2.50p \) and \( f = 17.5 \text{ cm} \) into the equation: \[ \frac{1}{17.5} = \frac{1}{p} + \frac{1}{2.50p} \].

Step 5: Simplify and Solve for Object Distance

Combine the terms: \[ \frac{1}{17.5} = \frac{1 + \frac{1}{2.50}}{p} = \frac{3.50}{2.50p} \].Simplifying gives: \[ p = \frac{2.50 \times 17.5}{3.50} \approx 12.5 \text{ cm} \].

Step 6: Conclusion

Thus, the face is \( 12.5 \text{ cm} \) from the mirror.

Key Concepts

Radius of Curvature

In the world of optics, particularly when dealing with mirrors, the concept of the radius of curvature is fundamental. The radius of curvature ( R ) of a concave mirror is essentially the radius of the sphere from which the mirror segment is derived. This helps us understand how the mirror is shaped.

• A larger radius of curvature implies a "flatter" mirror, while a smaller one indicates a more "curved" mirror.
• The radius of curvature is twice the focal length of the mirror, which is a direct and useful relationship in calculations.

For example, if a concave mirror's radius of curvature is given as 35.0 ext{ cm} , it implies that it is part of a sphere with a radius of 35.0 ext{ cm} .

Focal Length

The focal length (\( f \)) is a crucial concept when working with mirrors and lenses. For a concave mirror, the focal length is half of the radius of curvature:\[ f = \frac{R}{2} \].

• This means that if the radius of curvature \( R \) is 35.0 ext{ cm}, then the focal length is computed as 17.5 ext{ cm}.
• The focal length is the distance from the mirror's surface to its focal point, where parallel light rays converge after reflecting off the mirror.

Understanding focal length helps in predicting where the image will form, which is essential for solving problems like the one in the exercise.

Mirror Equation

The mirror equation combines all aspects of the mirror's properties to help us find object and image distances. It's expressed as:\[\frac{1}{f} = \frac{1}{p} + \frac{1}{q}\],
where \( f \) is the focal length, \( p \) is the object distance, and \( q \) is the image distance.

• This equation is vital for solving problems related to concave mirrors as it allows the calculation of unknown distances when others are known.
• In the provided exercise, using the mirror equation along with the given magnification enabled the calculation of how far the face is from the mirror.

Magnification

Magnification (\( M \)) describes how much larger or smaller the image is compared to the object. It's calculated by:\[ M = \frac{q}{p} \],
where \( q \) is the image distance and \( p \) is the object distance.

• For an upright image, as mentioned in the exercise, the magnification is positive. Conversely, negative magnification indicates an inverted image.
• The given magnification of 2.50 implies the image is 2.50 times larger than the actual object, thus indicating the concave mirror's ability to magnify.

Understanding magnification is crucial as it directly relates to how we perceive images formed by concave mirrors.