a concave mirror produces a real image 1 cm tall of an object 2.5 mm tall placed 5 cm from the mirror.find the position of the image and the focal length of the mirror.
To find the position of the image and the focal length of the mirror, we can use the mirror formula:
1/f = 1/v - 1/u
Where:
f = focal length of the mirror
v = image distance from the mirror (positive for real image)
u = object distance from the mirror (positive for real object)
Given:
Object height, h = 2.5 mm = 0.25 cm
Image height, h' = 1 cm
Object distance, u = -5 cm (negative since it's in front of the mirror)
Image distance, v = ? (we need to find this)
We can start by finding the focal length using the magnification formula:
magnification (m) = h'/h = -v/u
Substituting the given values, we have:
0.25/1 = -v/(-5)
0.25 = 5v
Solving for v, we get:
v = 0.25/5
v = 0.05 cm
Now, we can find the focal length using the mirror formula:
1/f = 1/v - 1/u
1/f = 1/0.05 - 1/(-5)
1/f = 20 + 1/5
1/f = 101/5
Cross multiplying, we have:
f = 5/101
f ≈ 0.04 cm
Therefore, the position of the image is 0.05 cm from the mirror (positive since it's on the other side of the mirror) and the focal length of the mirror is approximately 0.04 cm.
To find the position of the image and the focal length of the concave mirror, we can use the mirror equation:
1/f = 1/o + 1/i
Where:
f = focal length of the mirror (unknown)
o = object distance from the mirror = -5 cm (negative since the object is placed in front of the mirror)
i = image distance from the mirror (unknown)
Given that the real image produced by the concave mirror is 1 cm tall and the object height is 2.5 mm, we can use the magnification equation:
m = -i/o
Where:
m = magnification = -1 cm / 2.5 mm = -10
Substituting the magnification value into the mirror equation, we have:
1/f = 1/-5 + 1/i
Simplifying the equation, we get:
1/f = (i - 5) / (-5i)
Cross multiplying, we have:
-f(5i - 5) = i
Expanding and rearranging the equation, we get:
f = -5i / (-5i + 5)
Since the image height is positive and the object height is negative (since it's an inverted image), the magnification should also be negative. So, we can use the value of m = -10 to find the image distance, i.
-m = i/o
-10 = i / -5
i = 50 cm
Substituting the value of i into the equation for f:
f = -5 * 50 / (-5 * 50 + 5)
f = -250 / (-250 + 5)
f = -250 / -245
f ≈ 1.02 cm
Therefore, the position of the image is 50 cm from the mirror and the focal length of the mirror is approximately 1.02 cm.