A convex lens forms a real and inverted image of a needle at a distance of 50 cm from it. Where is the needle placed in front of the convex lens if the image is equal to the size of the object? Also, find the power of the lens.

The situation you’ve described involves a convex lens forming a real and inverted image of a needle at a certain distance. In this case, since the image is real and inverted, the object distance (u) is negative, and the image distance (v) is also negative.

Given:
v = -50 cm
You’ve mentioned that the image is equal in size to the object (ℎi = ℎ ho). In such a case, for a convex lens, the magnification (m) is -1. The magnification is given by the formula:
m = -v/u
Since m = − 1, we can write:
-1 = -v/u
Solving for u:
u = v
so, u = -50 cm

The object distance (u) is the distance from the object to the lens. Since it is negative, it means the object (the needle) is placed 50 cm to the left of the convex lens.

Now, to find the power of the lens (P), you can use the lens formula:
1/f = 1/v – 1/u
Substitute the given values:
1/f = 1/-50 – 1/-50
1/f = -2/50
1/f = -1/25
f = 25 cm

The negative sign for the focal length indicates that the lens is a converging lens (convex lens). The power (P) of a lens in dioptres is given by the reciprocal of the focal length in meters:
P = 1/f
P = 1/-0.25
P = -4 Dioptres.

So, the needle is placed 50 cm to the left of the convex lens, and the power of the lens is − 4 Dioptres