To understand the image of an object formed by a concave mirror, learn example problems and solutions below. In this issue, the object is assumed at a certain distance from the concave mirror, then draw the image formation by the concave mirror,
the image distance from the concave mirror and the magnification of the image formed by the concave mirror.
Consider a concave mirror has a focal length of 20 cm. Draw the image formation and then determine the image distance and the image magnification if:
a) the distance of objects is smaller than the distance of the focal length (do <f)
b) objects are located in the focal point of the mirror (do = f)
c) objects are between the focal points and the center point of the mirror curvature (f < do <R)
d) objects located at the center of the mirror curvature (do = R)
e) the distance of objects is greater than the radius of curvature of the mirror (do > R)
Known:
The focal length of the concave mirror (f) = 20 cm
The focal length of the concave mirror is positive because the beam of light passes through the focal point of the concave mirror.
Radius of curvature (R) = 2 f = 2 (20) = 40 cm
Solution:
Object distance is smaller than the focal length of the concave mirror (do <f)
Suppose the distance of objects from the mirror is 5 cm, 10 cm, and 15 cm.
The focal length (f) = 20 cm and if the object distance (do) = 5 cm
Image distance (di) :
1/di = 1/f – 1/do = 1/20 – 1/5 = 1/20 – 4/20 = -3/20
di = -20/3 = -6.7 cm
Image distance (di) has a negative sign means that the shadow is virtual, or the image is behind a concave mirror where the beam of light does not pass through the image.
The image distance (di) 6.7 cm is greater than the object distance (do) 5 cm
Magnification of image (M) :
M = -di / do = -(-6.7)/5 = 6.7 / 5 = 1.3 times
The linear magnification of the image (M) exceeds 1 means the size of the image is greater than the size of the object.
The linear magnification of the image (M) has a positive sign, means the image is upright.
Focal length (f) = 20 cm and if the object distance (do) = 10 cm
Image distance (di) :
1/di = 1/f – 1/do = 1/20 – 1/10 = 1/20 – 2/20 = -1/20
di = -20/1 = -20 cm
Image distance (di) has a negative sign means that the image is virtual, or the image is behind the concave mirror where the beam of light does not pass through the image.
The image distance (di) 20 cm is greater than the object distance (do) 10 cm
Image magnification (M) :
M = -di / do = -(-20)/10 = 20/10 = 2 times
The linear magnification of the image (M) exceeds 1 means that the size of the image is greater than the size of the object
The linear magnification of the shadow (M) has a positive sign, means the image is upright.
The focal length (f) = 20 cm and if the object distance (do) = 15 cm
Image distance (di) :
1/di= 1/f – 1/do = 1/20 – 1/15 = 3/60 – 4/60 = -1/60
di = -60/1 = -60 cm
Image distance (di) has a negative sign means that the image is virtual, or the image is behind a concave mirror where the beam of light does not pass through the image.
The image distance (di) 60 cm is greater than the object distance (do) 15 cm
Image magnification (M) :
M = -di / do = -(-60)/15 = 60/15 = 4 times
The linear magnification of the image (M) exceeds 1 meaning the size of the shadow is greater than the size of the object.
The linear magnification of the image (M) has a positive sign, means the image is upright.
Objects located at the focal point of the concave mirror (do = f)
Objects located at the focal point of the concave mirror, therefore the object distance (do) = the focal length (f) = 20 cm
The image distance (di) :
1/di = 1/f – 1/do = 1/20 – 1/20 = 0
di = 0
The magnification of image (M) :
M = di / do = 0
If the object distance (do) is equal to the focal length of the mirror (f) then there is no image formed by the mirror.
The object is between the focal point and the center of the concave mirror curvature (f < do <R)
Suppose the object distance from the mirror is 25 cm, 30 cm, and 35 cm.
1) Focal length (f) = 20 cm and if the object distance (do) = 25 cm
Image distance (di) :
1/di = 1/f – 1/do = 1/20 – 1/25 = 5/100 – 4/100 = 1/100
di = 100/1 = 100 cm
The image distance (di) is signed positive means that the image is real, or the image is in front of a concave mirror where the beam of light passes through the image.
The image distance (di) 100 cm is greater than the object distance (do) 25 cm
The image magnification (M) :
M = -di / do = -(100)/25 = -4 times
The linear magnification of the image (M) exceeds 1 means the size of the image is greater than the size of the object.
The linear magnification of the image (M) has a negative sign means that the image is inverted.
2) The focal length (f) = 20 cm and if the object distance (do) = 30 cm
Image distance (di) :
1/di = 1/f – 1/do = 1/20 – 1/30 = 3/60 – 2/60 = 1/60
di = 60/1 = 60 cm
The image distance (di) is signed positive means that the image is real, or the image is in front of a concave mirror where the beam of light passes through the image.
The image distance (di) 60 cm is greater than the object distance (do) 30 cm
The linear magnification of the image (M) :
M = -di / do = -60/30 = -2 times
The linear magnification of the image (M) exceeds 1 means that the size of the image is greater than the size of the object
The linear magnification of the image (M) has a negative sign means that the image is inverted.
3) The focal length (f) = 20 cm and if the object distance (do) = 35 cm
The image distance (di) :
1/di = 1/f – 1/do = 1/20 – 1/35 = 7/140 – 4/140 = 3/140
di = 140/3 = 46.7 cm
The image distance (di) is signed positive means that the image is real, or the image is in the front of the concave mirror where the beam of light passes through the image.
The image distance (di) 46.7 cm is greater than the object distance (do) 35 cm
The linear magnification of the image (M) :
M = -di / do = -46.7 / 35 = -1.3 times
The linear magnification of the image (M) exceeds 1 means that the size of the image is greater than the size of the object.
The linear magnification of the image (M) has a negative sign means that the image is inverted.
The object located at the center of the concave mirror curvature (do = R)
An object located at the center of the concave mirror curvature hence the object distance (s) = the radius of curvature of the mirror (R) = 40 cm
The image distance (di) :
1/di = 1/f – 1/do = 1/20 – 1/40 = 2/40 – 1/40 = 1/40
di = 40/1 = 40 cm
The image distance (di) is signed positive means that the image is real, or the image is in the front of a concave mirror where the beam of light passes through the image.
The image distance (di) 40 cm is equal to the object distance (do) 40 cm
The linear magnification of the image (M) :
M = -di / do = -40/40 = -1
The linear magnification of the image (M) equals to 1 means that the size of the image is equal to the size of the object.
The linear magnification of the image (M) has a negative sign means that the image is inverted.
The object distance is greater than the radius of curvature of the concave mirror (do > R)
Suppose the distance of objects from the mirror is 45 cm, 50 cm and 60 cm.
1) The focal length (f) = 20 cm and if the object distance (do) = 45 cm
The image distance (di) :
1/di = 1/f – 1/do = 1/20 – 1/45 = 9/180 – 4/180 = 5/180
di = 180/5 = 36 cm
The image distance (di) is signed positive means that the image is real, or the image is in the front of the concave mirror where the beam of light passes through the image.
The image distance (di) 36 cm is smaller than the object distance (do) 45 cm
The linear magnification of the image (M) :
M = -di / do = -(36)/45 = -0.8 times
The linear magnification of the image (M) is smaller than 1 means that the size of the image is smaller than the size of the object.
The linear magnification of the image (M) has a negative sign means that the image is inverted.
2) The focal length (f) = 20 cm and if the object distance (do) = 50 cm
The image distance (di) :
1/di = 1/f – 1/do = 1/20 – 1/50 = 5/100 – 2/100 = 3/100
di = 100/3 = 33.3 cm
The image distance (di) is signed positive means that the image is real, or the image is in the front of the concave mirror where the beam of light passes through the image.
The image distance (di) 33 cm is smaller than the object distance (do) 50 cm.
The linear magnification of the image (M) :
M = -di / do = -33/50 = -0.7 times
The linear magnification of the image (M) is smaller than 1 means that the size of the image is smaller than the size of the object
The linear magnification of the image (M) has a negative sign, means the image is inverted.
3) The focal length (f) = 20 cm and if the object distance (do) = 60 cm
The image distance (di) :
1/di = 1/f – 1/do = 1/20 – 1/60 = 3/60 – 1/60 = 2/60
di = 60/2 = 30 cm
The image distance (di) is signed positive means that the image is real, or the image is in the front of a concave mirror where the beam of light passes through the image.
The image distance (di) 30 cm is smaller than the object distance (do) 60 cm
The linear magnification of the image (M) :
M = -di / do = -30/60 = -0.5 times
The linear magnification of the image (M) is smaller than 1 means that the size of the image is smaller than the size of the object.
The linear magnification of the image (M) has a negative sign, means the image is inverted.
