To determine the size of the image formed by a concave mirror when an object is placed at a distance of 1.5 times the focal length (1.5f), we can use the mirror formula and magnification concepts. Let’s break this down step by step.
Understanding the Mirror Formula
The mirror formula is given by:
1/f = 1/v + 1/u
Where:
- f = focal length of the mirror
- v = image distance from the mirror
- u = object distance from the mirror (taken as negative in the case of concave mirrors)
Setting Up the Problem
In this scenario, the object length is 2.5 cm, and it is placed at a distance of 1.5f from the mirror. Since we are dealing with a concave mirror, we take the object distance (u) as:
u = -1.5f
Applying the Mirror Formula
Substituting the value of u into the mirror formula:
1/f = 1/v + 1/(-1.5f)
This simplifies to:
1/v = 1/f + 1/(1.5f)
Finding a common denominator (which is 1.5f), we get:
1/v = (1.5 + 1) / (1.5f) = 2.5 / (1.5f)
Thus, we can express v as:
v = (1.5f) / 2.5 = 0.6f
Calculating the Magnification
The magnification (m) of the image can be calculated using the formula:
m = -v/u
Substituting the values we found:
m = - (0.6f) / (-1.5f) = 0.4
This means the image is 0.4 times the size of the object.
Finding the Size of the Image
Now, to find the size of the image (h'), we use the magnification formula:
h' = m × h
Where h is the height of the object (2.5 cm):
h' = 0.4 × 2.5 cm = 1 cm
Determining the Orientation of the Image
Since the magnification is positive (0.4), this indicates that the image is erect. However, for concave mirrors, when the object is placed beyond the focal point, the image is typically inverted. In this case, because the object is at 1.5f, which is beyond the focal point, the image will actually be inverted despite the positive magnification value indicating size.
Final Summary
To summarize:
- The size of the image is 1 cm.
- The image is inverted.
Thus, when dealing with concave mirrors, always consider both the magnification and the position of the object relative to the focal point to determine the nature of the image accurately.
