Understanding Image Formation in Concave Mirrors: Solving Practical Problems

When tackling problems involving concave mirrors in optics, it's essential to understand the underlying principles and equations. This article will guide you through the process of solving a specific problem, emphasizing the key formulas and steps involved.

Key Concepts and Formulas

In dealing with concave mirrors, you'll encounter several fundamental concepts and equations. Here are some important points to keep in mind:

Focal Length and Radii of Curvature

Focal Length (f) is a crucial parameter that characterizes the mirror. The focal length is defined as the distance between the mirror's principal focus and the mirror itself. A concave mirror has a focal length that is half of its radius of curvature, i.e., R 2f. However, for the problem at hand, you don't need to use this relationship.

Mirror Equation

The mirror equation is given by:

1/p 1/q 1/f

Where p is the object distance and q is the image distance from the mirror. This equation is the foundation for solving problems involving concave mirrors.

Magnification

The magnification M of the image is defined by the relationship:

M -q/p hi/ho

Here, hi is the height of the image, and ho is the height of the object.

Solving a Practical Problem

Let's delve into a specific problem that involves a concave mirror. A 2.5 cm tall object is placed at a distance of 47.5 cm from a concave mirror with a focal length of 5 cm. Your task is to determine the image distance and the height of the image.

Step-by-Step Solution

Given Information

Object distance p 47.5 cm Focal length f 5 cm The only unknown is the image distance q.

Applying the Mirror Equation

Using the mirror equation:1/p 1/q 1/f

Substitute the given values:

1/47.5 1/q 1/5

Rearrange the equation to solve for q:

1/q 1/5 - 1/47.5

1/q (47.5 - 5) / (5 * 47.5)

1/q 42.5 / 237.5

1/q 0.17963

q 1 / 0.17963

q ≈ 5.57 cm

Determining Image Height

To find the height of the image, use the magnification formula:

M -q/p hi/ho

Given ho 2.5 cm and q 5.57 cm, and p 47.5 cm, calculate M and then hi:

M -5.57 / 47.5 ≈ -0.11746

hi M * ho -0.11746 * 2.5 ≈ -0.29365 cm

The negative sign for hi indicates that the image is inverted (real and upside down), with a height of approximately 0.29365 cm.

Conclusion

Solving problems involving concave mirrors requires a clear understanding of the mirror equation and the magnification formula. In the example provided, we determined the image distance and height, taking into account whether the image is real, inverted, and magnified or diminished.

Additional Tips for Your Teacher

If you're teaching these concepts, it's important to note that focal lengths are typically measured in millimeters rather than centimeters in modern practice. This change in unit can help students better grasp the scale and practical application of these measurements.

Key Takeaways

Understanding the mirror equation 1/p 1/q 1/f Using magnification to find image height Note that the focal length is usually measured in millimeters