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Convex Mirrors: Understanding Image Formation and Calculations

March 31, 2025Technology1619
Convex Mirrors: Understanding Image Formation and Calculations Convex

Convex Mirrors: Understanding Image Formation and Calculations

Convex mirrors are commonly used in various applications due to their unique properties. These mirrors produce images that are always smaller than the object, making them ideal for safety and surveillance purposes. In this article, we will explore the mathematical calculations involved in determining the image distance for a convex mirror given the focal length and the magnification.

Understanding Convex Mirrors

Convex mirrors are diverging mirrors and are characterized by their outward curvature. This curvature causes parallel rays of light to diverge after reflection, resulting in virtual, upright, and diminished images. The focal length of a convex mirror is always positive.

Problem Statement

Given a convex mirror with a focal length of 20 cm, we want to find the image distance when the image is 1/5th the size of the object.

Step-by-Step Solution

Given:

Focal length, f 20 cm (positive for convex mirrors) Magnification, M 1/5

Step 1: Using the Magnification Formula

The magnification of a mirror is given by:

M - frac{v}{u}

Given that M 1/5, we can write:

frac{1}{5} - frac{v}{u}

Rearranging this equation, we get:

v - frac{u}{5}

Step 2: Using the Mirror Formula

The mirror formula for a convex mirror is given by:

frac{1}{f} frac{1}{v} frac{1}{u}

Substituting the given focal length and the expression for v, we get:

frac{1}{20} frac{1}{- frac{u}{5}} frac{1}{u}

Step 3: Simplifying the Equation

Simplifying the term frac{1}{- frac{u}{5}}:

frac{1}{- frac{u}{5}} - frac{5}{u}

Substituting this back into the mirror formula, we get:

frac{1}{20} - frac{5}{u} frac{1}{u}

Combining the terms on the right-hand side:

frac{1}{20} frac{-5 1}{u} frac{-4}{u}

Rearranging to solve for u:

frac{1}{20} - frac{4}{u}

Cross-multiplying:

u -80 , text{cm}

Step 4: Finding the Image Distance v

Now, substitute u -80 , text{cm} back into the equation for v:

v - frac{u}{5} - frac{-80}{5} 16 , text{cm}

Conclusion

The distance of the image from the mirror is 16 cm. Since the value of v is positive, it indicates that the image is formed on the same side as the object, which is consistent with the behavior of convex mirrors.

Alternative Method

Alternatively, if the object is at a coordinate u, the image is formed at v - frac{u}{4} since the magnification is:

M - frac{v}{u} 1/4

Focal coordinate f -20 , text{cm}. Using the mirror formula:

frac{1}{f} frac{1}{v} frac{1}{u}

frac{1}{-20} frac{1}{- frac{u}{4}} frac{1}{u}

Simplifying:

frac{1}{-20} - frac{4}{u} frac{1}{u} - frac{4 - 1}{u} - frac{3}{u}

Solving for u gives:

u 60 , text{cm}

Now, using the magnification formula:

v - frac{u}{4} - frac{60}{4} -15 , text{cm}

Thus, the image distance is 15 , text{cm}.

Key Takeaways:

Convex mirrors produce virtual, upright, and diminished images. The focal length of a convex mirror is always positive. The magnification formula is useful in determining the image distance when the magnification is known. The mirror formula can be used to solve for unknown distances when other parameters are known.

Understanding these principles can help in various applications of convex mirrors, such as surveillance systems and automotive safety features.