A lens of focal length 12cm forms an image three times the size of the real object, what is the distance between the object and the image
To solve this problem, we will use the lens formula:
1/f = 1/v - 1/u
Where:
f = focal length
v = image distance
u = object distance
Given:
f = 12 cm (focal length)
The image is three times the size of the real object.
Let's assume the size of the real object is h, and the size of the image is 3h.
Using the magnification formula:
magnification (m) = -v/u = height of image/height of object
We can substitute the given values:
- v/u = 3h/h
Simplifying the equation, we get:
- v/u = 3
Since h cancels out, we know that magnification is only dependent on the ratio of v/u.
Now, let's substitute the values into the lens formula:
1/12 = 1/v - 1/u
Since magnification (m) = -v/u = 3, we can replace -v/u with -3:
1/12 = 1/v + 1/(3u)
To simplify the equation, let's find a common denominator:
1/12 = (1/(3u) + 1/v)((3u)(v)/(3u)(v))
1/12 = (v + 3u)/(3uv)
Cross-multiplying, we get:
1 = 4uv + 36u
Rearranging the equation:
4uv + 36u - 1 = 0
Solving this quadratic equation for u, we find two roots. However, we are only interested in the positive root, as the distances must be positive values.
u ≈ 0.004 cm
Now, substituting this value into the lens formula:
1/12 = 1/v - 1/0.004
Solving for v, we find:
v ≈ 0.00412 cm
Therefore, the approximate distance between the object and the image is 0.00412 cm.
To find the distance between the object and the image formed by the lens, we can use the lens formula:
1/f = 1/v - 1/u
Where:
f is the focal length of the lens,
v is the distance of the image from the lens, and
u is the distance of the object from the lens.
Given that the focal length of the lens is 12 cm, and the image formed is three times the size of the real object, we can determine the relationship between v and u.
Let the distance of the object from the lens be u.
Let the distance of the image from the lens be v.
Given that the image formed is three times the size of the real object, we can write:
v/u = 3/1
Simplifying the above equation:
v = 3u
Now, substituting the values into the lens formula:
1/12 = 1/3u - 1/u
Multiplying through by 12u to clear the fractions:
u = 12u/3 - 12
Rearranging the equation:
12 = 12u/3 - u
12 = (12 - 3u)/3
Multiplying through by 3 to eliminate the denominator:
36 = 12 - 3u
Moving the terms around:
3u = 12 - 36
3u = -24
Dividing both sides by 3:
u = -8
Since distance cannot be negative, we will ignore the negative solution. Therefore, the distance between the object and the image formed by the lens is 8 cm.