an object is placed 15 cm in front of a convex lens of focal length 10cm. Find the nature and position of the image formed. where should a concave mirror of radius of curvature 20cm be placed so that the final image is formed at the position of the object itself.
To determine the nature and position of the image formed by a convex lens, we can use the lens formula:
1/f = 1/v - 1/u
Where:
f = focal length of the lens
v = image distance from the lens (positive if the image is on the opposite side of the lens from the object)
u = object distance from the lens (positive if the object is on the same side as the incident light)
Given:
focal length (f) = 10 cm
object distance (u) = -15 cm (negative since it is in front of the lens)
Substituting the given values into the lens formula:
1/10 = 1/v - 1/(-15)
Simplifying the equation:
1/10 = 1/v + 1/15
To add the fractions, find the common denominator of 10 and 15, which is 30:
3/30 = 2/30 + 1/30
Combining the fractions:
3/30 = 3/30
Therefore, the image is formed at a distance of 30 cm from the lens on the same side as the object. The negative sign indicates that the image is virtual.
Now, to determine the position of the concave mirror, we need to place it in a way that the final image is formed at the position of the object itself. This can be achieved using mirror formula:
1/f = 1/v - 1/u
Where:
f = radius of curvature of the mirror (half of the radius, R)
v = image distance from the mirror (positive if the image is on the opposite side of the mirror from the object)
u = object distance from the mirror (positive if the object is on the same side as the incident light)
Given:
radius of curvature (R) = 20 cm
object distance (u) = -15 cm (negative since it is in front of the mirror)
v = -u (image formed at the position of the object itself)
Substituting the given values into the mirror formula:
1/20 = 1/(-15) - 1/(-15)
Simplifying the equation:
1/20 = 2/(-15)
To solve for the image distance (v), cross-multiply:
15 = 2 * 20
15 = 40
This equation does not hold true. Therefore, it is not possible to place a concave mirror of radius of curvature 20 cm to obtain a final image at the position of the object itself.