A concave mirror has a focal length of 27.0 cm. The distance between an object and its image is 46.5 cm. Find the object and image distances, assuming the following.
(a) that the object lies beyond the center of curvature
do =
di =
(b) that the object lies within the focal point
do =
di =
(a) If the object lies beyond the center of curvature, the object distance (do) is greater than the focal length (f).
Given:
focal length (f) = 27.0 cm
distance between object and image (di) = 46.5 cm
Using the mirror formula:
1/f = 1/do + 1/di
Since the object distance (do) is greater than the focal length (f), we can assume that the image distance (di) is positive.
1/27.0 = 1/do + 1/46.5
Simplifying the equation:
1/do = 1/27.0 - 1/46.5
1/do = (46.5 - 27.0)/(27.0 x 46.5)
1/do = 19.5/(27.0 x 46.5)
1/do = 19.5/1255.5
1/do ≈ 0.0155
do ≈ 1/0.0155
do ≈ 64.5 cm
Therefore, the object distance (do) is approximately 64.5 cm.
Now, we can substitute the obtained object distance (do) into the mirror formula to find the image distance (di):
1/27.0 = 1/64.5 + 1/di
Simplifying the equation:
1/di = 1/27.0 - 1/64.5
1/di = (64.5 - 27.0)/(27.0 x 64.5)
1/di = 37.5/(27.0 x 64.5)
1/di = 37.5/1741.5
1/di ≈ 0.0215
di ≈ 1/0.0215
di ≈ 46.5 cm
Therefore, the image distance (di) is approximately 46.5 cm.
To summarize, for the case where the object lies beyond the center of curvature:
do ≈ 64.5 cm
di ≈ 46.5 cm
(b) If the object lies within the focal point, the object distance (do) is less than the focal length (f).
Given:
focal length (f) = 27.0 cm
distance between object and image (di) = 46.5 cm
In this case, the image distance (di) is negative.
Using the mirror formula:
1/f = 1/do + 1/di
Since the object distance (do) is less than the focal length (f), we can assume that the image distance (di) is negative.
1/27.0 = 1/do + -1/46.5
Simplifying the equation:
1/do = 1/27.0 - 1/46.5
1/do = (46.5 - 27.0)/(27.0 x 46.5)
1/do = 19.5/(27.0 x 46.5)
1/do = 19.5/1255.5
1/do ≈ 0.0155
do ≈ 1/0.0155
do ≈ 64.5 cm
Therefore, the object distance (do) is approximately 64.5 cm.
Now, we can substitute the obtained object distance (do) into the mirror formula to find the image distance (di):
1/27.0 = 1/64.5 - 1/di
Simplifying the equation:
1/di = 1/27.0 - 1/64.5
1/di = (64.5 - 27.0)/(27.0 x 64.5)
1/di = 37.5/(27.0 x 64.5)
1/di = 37.5/1741.5
1/di ≈ 0.0215
di ≈ 1/0.0215
di ≈ -46.5 cm
Therefore, the image distance (di) is approximately -46.5 cm.
To summarize, for the case where the object lies within the focal point:
do ≈ 64.5 cm
di ≈ -46.5 cm
To find the object and image distances for a concave mirror, you can 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 of the object from the mirror),
di is the image distance (distance of the image from the mirror).
(a) Object lies beyond the center of curvature:
In this case, the object distance (do) will be greater than the focal length (f).
Given:
f = 27.0 cm
di - do = 46.5 cm
First, let's find the object distance (do):
1/f = 1/do + 1/di
Substituting the given values:
1/27 = 1/do + 1/46.5
We can rearrange the equation to solve for do:
1/do = 1/27 - 1/46.5
Now, let's calculate the value of 1/do:
1/do = 0.037 - 0.022
1/do = 0.015
Taking the reciprocal of both sides:
do = 1/0.015
do ≈ 66.7 cm
Therefore, the object distance (do) ≈ 66.7 cm.
To find the image distance (di), we can use the relation di - do = 46.5 cm:
di = do + 46.5
di ≈ 66.7 + 46.5
di ≈ 113.2 cm
Therefore, the image distance (di) ≈ 113.2 cm.
(b) Object lies within the focal point:
In this case, the object distance (do) will be smaller than the focal length (f).
Given:
f = 27.0 cm
di - do = 46.5 cm
Using the same mirror equation: 1/f = 1/do + 1/di
Since the object lies within the focal point, let's assign a negative value to do:
do = -x (where x is a positive value)
Now, substituting the given values into the equation:
1/27 = 1/-x + 1/46.5
1/-x = 1/27 - 1/46.5
1/-x = 0.037 - 0.022
1/-x = 0.015
Taking the reciprocal of both sides:
-x = 1/0.015
x ≈ -66.7 cm
Therefore, the object distance (do) ≈ -66.7 cm.
To find the image distance (di), we can use the relation di - do = 46.5 cm:
di = do + 46.5
di ≈ -66.7 + 46.5
di ≈ -20.2 cm
Therefore, the image distance (di) ≈ -20.2 cm.
Note: The negative sign in the image distance indicates that the image is formed on the same side as the object, which is a virtual image.