Answer to Question #83782 in Optics for Victor marcel

Question

An object is placed 1)10cm 2)4cm from a concave mirror of radius curvature 12cm. Calculate the image position in each case and the respective magnification

Answer 1

Answer on Question 83782, Physics, Optics

Question:

An object is placed 1)10 cm 2) 4 cm from a concave mirror of radius curvature 12 cm. Calculate the image position in each case and the respective magnification.

Solution:

We can find the image position of the object from the mirror equation:

$$1/d_o + 1/d_i = 1/f,$$

here,

$$d_o$$

is the distance from the object to the mirror,

$$d_i$$

is the distance from the image to the mirror and

$$f$$

is the focal length.

By the definition, the focal length of the curved mirror is half a radius of curvature:

$$f=R/2.$$

Substituting

$$f$$

into the mirror equation, we get:

$$1/d_o + 1/d_i = 2/R,$$ $$1/d_i = 2/R - 1/d_o,$$ $$d_i = 1/(2/R - 1/d_o).$$

Let's substitute the numbers for both two cases. For the first case, we get:

$$d_i = 1/(2/12 cm - 1/10 cm) = 15 cm.$$

The sign plus indicates that the image is located in front of the mirror. The image is real.

For the second case, we get:

$$d_i = 1/(2/12 cm - 1/4 cm) = -12 cm.$$

The sign minus indicates that the image is located behind the mirror. The image is virtual.

We can find the magnification of the concave mirror for both two cases from the magnification equation:

$$M = -d_i/d_o.$$

Let's substitute the numbers. For the first case, we get:

$$M = -(15 cm)/10 cm = -1.5.$$

The sign minus indicates that the image is an inverted image.

For the second case, we get:

$$M = -(-12 cm)/4 cm = 3.$$

The sign plus indicates that the image is an upright image.

Answer:

1)

$$d_i = 15 cm$$

,

$$M = -1.5.$$

2)

$$d_i = -12 cm$$

,

$$M = 3.$$