Calculating the Focal Length of a Convex Mirror

The focal length of a mirror is a crucial parameter in optics, determining how it forms images. In this article, we will walk through the process of calculating the focal length of a convex mirror given certain conditions. We will use the mirror formula and the magnification formula to solve for the unknown focal length.

Given Information:

The image formed by the convex mirror is one third of the object's length. The object is placed at a distance of 40 centimeters from the mirror.

Concepts Involved:

Magnification Formula

(m -frac{v}{u})

where m is the magnification, v is the image distance, and u is the object distance.

Mirror Formula

(frac{1}{f} frac{1}{u} frac{1}{v})

where f is the focal length, u is the object distance, and v is the image distance.

Step 1: Finding the Image Distance

The magnification m is given as one third, indicating that the image is one third the size of the object. Since the image is virtual and upright, m is positive.

(m frac{1}{3} -frac{v}{u})

Substituting the given object distance u -40 cm (the negative sign indicates the object is in front of the mirror):

(frac{1}{3} -frac{v}{-40})

This simplifies to:

(frac{1}{3} frac{v}{40})

Solving for v:

(v frac{40}{3} approx 13.33) cm

Since the image is virtual and upright, the image distance v is negative:

(v -frac{40}{3})

Step 2: Using the Mirror Formula

The mirror formula is:

(frac{1}{f} frac{1}{u} frac{1}{v})

Substituting the known values:

(frac{1}{f} frac{1}{-40} frac{1}{-frac{40}{3}})

Simplifying (frac{1}{-frac{40}{3}}):

(frac{1}{-frac{40}{3}} frac{3}{40})

Now substituting back into the mirror formula:

(frac{1}{f} -frac{1}{40} frac{3}{40})

Combining the fractions:

(frac{1}{f} frac{3 - 1}{40} frac{2}{40} frac{1}{20})

Taking the reciprocal gives the focal length:

(f 20) cm

Conclusion

The focal length of the convex mirror is 20 cm.

Old Sign Convention

Using the traditional sign convention for mirrors, where:

(p 40) (object distance)

(m 1/3 -q/40)

Solving for q:

(frac{1}{40} - frac{1}{-frac{40}{3}} frac{1}{f}) (frac{1}{40} - frac{3}{40} frac{1}{f}) (-frac{2}{40} frac{1}{f})

Therefore:

(f -20) cm

Additional Insights

The focal length of a mirror can also be understood in the context of cameras and lenses. In photography, the focal length refers to the distance from the lens to the image sensor when the subject is in focus. For example, a 50 mm lens has a focal length of 50 mm. The principle is the same, with the focal length determining the point at which light rays converge or diverge.

Understanding these concepts is crucial for anyone working with optics or photography, as it helps in designing and optimizing imaging systems. Whether calculating the focal length of a mirror or a lens, the principles remain consistent across different applications, from microscopes and telescopes to digital cameras and smartphones.