Understanding Concave Mirrors: Magnification, Formula, and Examples

A concave mirror or a converging mirror is a type of mirror with a reflecting surface that is recessed inward, directing the light inward to a single focal point. This article delves into the formula and calculations related to concave mirrors, providing insights and examples to simplify the understanding of magnification and image formation.

Magnification Produced by a Concave Mirror

Magnification Formula

The magnification produced by a concave mirror can be calculated using the formula, where m is the magnification, v is the image distance, and u is the object distance:

[m -frac{v}{u}]

Example Calculation

For instance, if a concave mirror with a focal length f of 8 cm has an object placed at a distance of 12 cm, the magnification can be calculated as follows:

Using the mirror formula:

[frac{1}{f} frac{1}{v} - frac{1}{u}]

Given f 8 cm and u 12 cm:

[frac{1}{8} - frac{1}{12} frac{1}{v}]

[frac{3 - 2}{24} frac{1}{24}]

[frac{1}{v} frac{1}{24}]

[therefore v 24 text{ cm}]

Thus, the magnification m is:

[m -frac{v}{u} -frac{24}{12} -2]

This indicates a real inverted image that is twice the size of the object.

Mirror Formula and Magnification Examples

Mirror Formula

The mirror formula is given by:

[frac{1}{v} - frac{1}{u} frac{1}{f}]

Example Calculation

Consider a concave mirror with a focal length of 10 cm. If an object is placed 15 cm away from the pole (vertex) of the mirror, the magnification and image distance can be calculated as follows:

Using the mirror formula:

[frac{1}{v} - frac{1}{15} frac{1}{10}]

Rearranging to solve for v:

[frac{1}{v} frac{1}{10} frac{1}{15} frac{3 2}{30} frac{5}{30} frac{1}{6}]

[therefore v -30 text{ cm}]

Using the magnification formula:

[m -frac{v}{u} -frac{-30}{15} 2]

The image is real, inverted, and twice the size of the object.

Magnification in Concave Mirrors

Magnification in concave mirrors is the ratio of the height of the image to the height of the object, and it is also the negative of the ratio of the image distance to the object distance. According to the Cartesian sign convention, distances towards the object are considered negative, and distances from the mirror to the opposite side are positive. Heights above the principal axis are positive, and below are negative.

Hence, for a real image:

[m -v/u

And for a virtual image:

[m v/u > 0]

Magnification is mathematically defined as:

[m -frac{v}{u}]

With this, one can calculate the magnification of a concave mirror given the object and image distances.