An inverted image is magnified by 2 when object is placed 22 cm in front of a concave mirror. Determine the image distance and focal length of the mirror.
To determine the image distance and focal length of the concave mirror in this scenario, we can use the mirror equation:
1/f = 1/do + 1/di
where:
f = focal length of the concave mirror
do = object distance (distance of the object from the mirror)
di = image distance (distance of the image from the mirror)
Given that the object is placed 22 cm in front of the concave mirror and the image is magnified by 2, we can use the magnification formula:
m = -di/do
where:
m = magnification
Since the image is inverted, the magnification is negative.
Let's solve this step by step:
1. Determine the magnification:
m = -di/do
2 = -di/22 cm
Since the magnification is 2, we have:
2 = -di/22 cm
Multiply both sides by 22 cm to isolate -di:
-44 cm = di
So, the image distance is -44 cm (negative sign indicates it is on the opposite side of the object).
2. Determine the focal length using the mirror equation:
1/f = 1/do + 1/di
Substituting the values we have:
1/f = 1/22 cm + 1/(-44 cm)
Simplify the equation:
1/f = 2/44 cm - 1/44 cm
1/f = 1/44 cm
Multiply both sides by 44 cm to isolate f:
f = 44 cm
Therefore, the image distance is -44 cm and the focal length of the concave mirror is 44 cm.