Question

Some rear-view mirrors produce images of cars behind you that aresmaller than they would be if the mirror were flat. Are the mirrorsconcave or convex? What is a mirrors radius of curvature if cars20.8 m away appear 0.33x their normal size?

Answer 1

Concepts and reason

The concept used to solve this problem is Lens maker's formula. Initially, the image distance can be calculated by using the relation for the magnification formulae of image distance and object distance. Later, the focal length of the lens can be calculated by using the lens formula. Finally, the radius of curvature can be calculated by using the relationship between radius of curvature and the focal length.

Fundamentals

The expression for the image distance is as follows:

\(m=-\frac{d_{i}}{d_{o}}\)

Here, \(m\) is the magnification of the object, \(d_{i}\) is the image distance, and \(d_{o}\) is the object distance. The expression for the focal length of the lens is as follows:

\(\frac{1}{f}=\frac{1}{d_{0}}+\frac{1}{d_{i}}\)

Here, \(f\) is the focal length. The expression for the radius of curvature is as follows:

\(R=2 f\)

Here, \(R\) is the radius of curvature.

(a.1) The expression for the image distance is as follows:

\(m=-\frac{d_{i}}{d_{o}}\)

Substitute \(0.33\) for \(m\) and \(20.8 \mathrm{~m}\) for \(d_{o}\).

\((0.33)=-\frac{d_{i}}{(20.8 \mathrm{~m})}\)

\(d_{i}=-6.864 \mathrm{~m}\)

The image distance is negative as the image is in the same side of the mirror. Since the image distance is less than object distance, the focal length must be negative. Thus, the mirror is a convex lens. The ratio of the image distance to object distance is the magnification of the image.

(a.2) The expression for the focal length of the lens is as follows:

\(\frac{1}{f}=\frac{1}{d_{0}}+\frac{1}{d_{i}}\)

The above expression can be rewritten as follows:

\(f=\frac{d_{o} d_{i}}{\left(d_{o}+d_{i}\right)}\)

Substitute \(20.8 \mathrm{~m}\) for \(d_{o}\) and \(-6.864 \mathrm{~m}\) for \(d_{i}\)

\(f=\frac{(20.8 \mathrm{~m})(-6.864 \mathrm{~m})}{(20.8 \mathrm{~m})+(-6.864 \mathrm{~m})}\)

\(=-10.25 \mathrm{~m}\)

Hence, the focal length is negative. Thus, the mirror is a convex lens.

Part a The mirror is a convex lens.

The focal length of the lens is depends on the value of the image distance and object distance. The value of the focal length

is negative. Thus, the lens is a convex lens.

(b) The expression for the radius of curvature is as follows:

\(R=2 f\)

Substitute \(10.25 \mathrm{~m}\) for \(f\) \(R=2(10.25 \mathrm{~m})\)

\(=-20.5 \mathrm{~m}\)

Part b

The radius of curvature of the lens is \(-20.5 \mathrm{~m}\).

The radius of curvature is equal to the radius of the circular arc. The radius of curvature is depends on the focal length of the lens. The value of the radius of curvature is negative as the focal length is negative.