Mirrors

Solution:

Spherical mirrors:

• concave mirror

• convex mirror

Image is formed based on reflection of light from the mirror.

f = focal length of mirror
r = 2·f = radius of curvature of mirror
a = object distance
b = image distance
y = height of object
y' = height of image

Image magnification:

$$Z = \frac{y'}{y} = \frac{z}{y'} = \frac{b}{a} = \frac{b - f}{a - f} = \frac{a - f}{f}$$

Sign convention:

Concave mirror: Values in front of the mirror $a, b, r, f$ are positive, behind the mirror are negative.

Convex mirror: Values $r, f$ are negative. Values $y, y'$ above the optical axis are positive, below are negative.

Mirror equation of spherical mirrors:

$$\frac{1}{a} + \frac{1}{b} = \frac{1}{f}$$

If:

• $Z > 0$ → image is upright and virtual

• $Z < 0$ → image is inverted and real

• $|Z| = 1$ → image is same size as object

• $|Z| > 1$ → image is magnified

• $|Z| < 1$ → image is reduced

Solution:

Analysis:

The image is reduced, inverted, real, and 3.5 cm high. It lies 3 cm from the mirror’s vertex.

Solution:

Analysis:

The image is magnified, upright, virtual, and 13 cm high. It is formed 2 cm behind the mirror.

Solution:

Analysis:

The object is about 0.43 m from the mirror. The image is magnified 7 times.

Solution:

Analysis:

The focal length of the mirror is f = 36 cm. The image is magnified, inverted, and real.

Solution:

Analysis:

The object should be placed 50 cm in front of the mirror.

Solution:

Analysis:

The focal length of the mirror is f = 0.42 m, the radius of curvature r = 0.84 m, and the magnification Z = –0.43.

Solution:

Analysis:

The focal length of the mirror is f₁ = 40 cm, or f₂ = 85.7 cm.

Solution:

Analysis:

The light source should be placed 180 cm from the mirror.

Solution:

Analysis:

Their image is located at a distance x = 34.375 cm from the observer.