if an 80 meter high object placed 2 meter from a convex lens forms an image whose distance from the object is 2 meter; show with the aid of a diagram with appropriate scale the height of the image formed and confirm your answer using mirror law formula?
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To solve this problem, we need to analyze the situation and use the lens formula to calculate the height of the image formed. Let's follow the steps below:
Step 1: Draw a diagram with appropriate scale.
Start by drawing a convex lens as a symmetrical curved line. Mark the optical center of the lens (O). Label the object (O') at a distance of 2 meters from the lens with a height of 80 meters. Now, mark the image (I) at a distance of 2 meters on the opposite side of the lens. Label the height of the image as 'h'.
O' O I
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Step 2: Applying the lens formula.
The lens formula is given by:
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.
From the given data, we have:
u = -2 meters (since the object is on the left side of the lens)
v = -2 meters (since the image is on the left side of the lens)
Step 3: Calculate the focal length.
To calculate the focal length (f), we can rearrange the lens formula as follows:
1/f = 1/v - 1/u
1/f = 1/-2 - 1/-2
1/f = -1/2 + 1/2
1/f = 0
Since the focal length is 0, it means the lens is a plane mirror. In this case, the image formed will be a virtual image with the same size and distance as the object.
Step 4: Confirming with the mirror formula.
The mirror formula is given by:
1/f = 1/v + 1/u
Since we obtained f = 0, this lens acts as a mirror with a focal length of zero. Now, let's use the mirror formula to confirm that the image formed is virtual and has the same size as the object.
1/0 = 1/v + 1/-2
This equation simplifies to:
0 = 1/v - 1/2
Simplifying further, we multiply both sides by (2v) to eliminate fractions:
0 = (2v)/v - (2v)/2
0 = 2 - v/2
v/2 = 2
v = 4 meters
This calculation confirms that the image formed is 4 meters away from the lens, on the same side as the object, and has the same size (height of 80 meters) as the object.
Therefore, the height of the image formed is also 80 meters.