Combination of thin lenses
Consider two thin convex lenses L1 and L2 of focal length f1 and f2 placed coaxially in contact to each other. A point object O is placed in the principal axis at a distance u from the lens L1. In the absence of lens L2, as the rays of light incident on the lens L1, this lens L1 converges the rays, and thus, image is formed at point I1. So, for lens L1, we have object distance = u and image distance = v1
So, from lens formula,
1/f1 = 1/u + 1/v1 .........(i)
When lens L2 is placed in front of the image, I1 acts as virtual objects for lens L2 since the converging beam is incident on L2 as shown in figure.
For lens L2,
Object distance = -v1
Image distance = v
1/f2 = 1/-v1 + 1/v
1/f2 = 1/v – 1/v1 ……… (ii)
Adding equation (i) and (ii)
1/f2 + 1/f1 = 1/u + 1/v1 + 1/v – 1/v1
1/f2 + 1/f1 = 1/u + 1/v …….. (iii)
If F is the combined focal length of two thin lenses placed in contact having object distance u and image distance v, then
1/ F = 1/u + 1/v ……… (iv)
From equation (iii) and (iv)
1/f1 + 1/f2 = 1/ F
This formula is applicable for any two lenses, both concave and convex or convex and concave in contact. The proper sign of the focal length must be inserted in the formula.
Power of a lens
The power of a lens is the measure of its ability to converge or diverge the light rays falling onto it.
Mathematically, power of a lens P = 1/ f
If P = 1 diopter, then f = 1m
Power of lens in combination,
P = 1/f1 + 1/f2
If there are more than two lenses then,
P = 1/f1 + 1/f2 + ………
Magnification
The linear magnification of a lens is defined as the ratio of the size of the image and the size of the object.
Magnification, m = size of image / size of object
From the figure, Magnification m = IQ/OP
Here, ΔOPC ~ ΔIQC
IQ/OP = CI/OC
m = v/ u
Where v is the image distance and, u is the object distance.
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