A concave spherical mirror has a radius of curvature 15 cm. A 1 cm tall bulb is placed in front of the mirror such that its image is formed 10 cm in front of the mirror. If you measure distance from the optic center of the mirror,
(A) What is the focal length of the mirror?
(B) What is the position of the object?
(C) What is the magnification of the image?
(D) What is the height of the image?
To answer the given questions, we can use the mirror formula and magnification formula for concave spherical mirrors. Here's how to find the answers:
(A) Focal length of the mirror:
The mirror formula is given by:
1/f = 1/v - 1/u
Where:
- f is the focal length of the mirror,
- u is the object distance from the mirror (negative for objects placed in front of the mirror),
- v is the image distance from the mirror (also negative for virtual images).
Given that the object distance u = -10 cm (since it is placed 10 cm in front of the mirror), and the radius of curvature R = -15 cm (negative for concave mirrors), we can use the relation u = R/2 to find the object distance u.
So, u = -15 cm / 2 = -7.5 cm
Now we can substitute the values into the mirror formula:
1/f = 1/v - 1/u
1/f = 1/(-10 cm) - 1/(-7.5 cm)
Simplifying the equation will give us the value of the focal length, f.
(B) Position of the object:
The position of the object is given by its distance from the mirror's optical center. Since the distance is measured from the optical center, and we have the object distance u, the position of the object is -7.5 cm.
(C) Magnification of the image:
The magnification formula for spherical mirrors is given by:
m = -v/u
Where m is the magnification, v is the image distance from the mirror, and u is the object distance from the mirror.
Substituting the given values:
m = -(-10 cm) / (-7.5 cm)
Simplifying the equation will give us the value of the magnification, m.
(D) Height of the image:
The height of the image can be found using the magnification formula:
m = -h'/h
Where h' is the height of the image and h is the height of the object. Given that the height of the bulb is 1 cm, and we have the magnification m from the previous calculation, we can find the height of the image h'.
h' = m * h
Substituting the values will give us the height of the image.