A large concave mirror has radius of 6.32m. To magnify an object 1.25m tall by a factor of +2.5
To determine the characteristics of the image formed by a large concave mirror, we need to use the mirror equation:
1/f = 1/do + 1/di
where:
- f is the focal length of the mirror,
- do is the object distance (distance between the object and the mirror), and
- di is the image distance (distance between the image and the mirror).
Since we want to magnify the object, the magnification equation can also be used:
magnification (m) = -di/do
Given:
- Radius of the mirror (R) = 6.32 m
- Height of the object (ho) = 1.25 m
- Magnification (m) = +2.5
Step 1: Calculate the focal length (f)
The focal length of a concave mirror is half the radius of curvature:
f = R/2
Substituting R = 6.32 m, we find:
f = 6.32/2 = 3.16 m
Step 2: Calculate the object distance (do)
Using the magnification equation, rearrange it to solve for do:
m = -di/do
2.5 = -di/do [Given magnification (m) = +2.5]
Rearranging the equation:
do = -di/m
Substituting di = f (since the image distance di is equal to the focal length in a mirror), we find:
do = -f/m = -3.16/2.5 = -1.264 m
Note: The negative sign indicates that the object is placed in front of the mirror, which is the convention for concave mirrors.
Step 3: Calculate the image distance (di)
Using the mirror equation:
1/f = 1/do + 1/di
Substituting the known values:
1/3.16 = 1/(-1.264) + 1/di
Rearranging the equation:
1/di = 1/3.16 - 1/1.264
1/di = 0.3165 - 0.7911
1/di = -0.4746
Taking the reciprocal:
di = -1/(0.4746)
di = -2.106 m
Step 4: Verify the image characteristics
Using the magnification equation:
m = -di/do
Substituting the known values:
2.5 = -(-2.106)/(-1.264)
2.5 = 2.106/1.264
2.5 = 1.66
The magnification should be +2.5, but we obtained a value of 1.66. This implies an error in our calculations or assumptions. Please double-check the given data and calculation steps to ensure accuracy.
Note: The positive magnification indicates that the formed image is upright. The absolute value of the magnification (1.66) suggests that the image is smaller than the object, which contradicts the initial requirement to magnify the object.