Combinations Of Thin Lenses In Contact

In a combination of thin lenses, we use two thin lenses, a convex lens and a concave lens. Lenses formed after combinations of two or more lenses are termed compound lenses. When two lenses, such as a convex and a concave lens combine, the first lens forms an image that serves as an object for the second lens. Compound lenses are useful in the formation of instruments such as telescopes, microscopes, binoculars, etc. 

Lens

A lens is a piece of glass that has two opposite regular spherical surfaces. Lenses are optical devices and act as refracting mediums. There are two types of lenses, convex and concave, based on the property to converge or diverge rays of light that pass through them. These lenses can be combined in different ways to obtain desired focal lengths to perform specific functions.

Focus and Focal Length of a Lens

Rays starting from a fixed point on the principal axis of a lens or appearing to go towards a fixed point on the axis, after refraction through the lens, become parallel to the principal axis. This point is called the first focus of the lens.

The distance of the first focus from the optical centre of the lens is called the first focal length of the lens.

The rays travelling parallel to the axis of the lens, after refraction through the lens, either go towards a fixed point on the axis or appear to come from a point. This point is called the ‘second focus’ or ‘principal focus’ of the lens.

The distance of the second focus from the optical centre of the lens is called the second focal length of the lens.

Power of a Lens

The power of a lens is the measure of convergence or divergence that the lens introduces in the light falling on it. The power P of a lens is defined as the tangent of the angle by which it converges or diverges a beam of light falling at a unit distant from the optical centre.

The unit of power is Dioptre (D).

The power of the lens is equal to the reciprocal of its focal length.

P=1/f

Combined Focal Length of Two Thin Lenses in Contact

Taking both convex lenses into consideration:

Let’s assume that two thin convex lenses L1 and L2 of focal lenses F1 and F2 are placed in contact in the air with a common principal axis. A point object O is placed on the principal axis at a distance u from the first lens L1.

Its image would be formed by the lines L1 alone at I’ distant v’ from L1.

From the lens formula, we get

1/v’-1/u = 1/f1………………(1)

It serves as a virtual object for the second lens L2 which forms a final image I at a distance v from it. 

Then, we have,

1/v-1/v’=1/f2……………..(2)

Adding the first and second equations,

1/v-1/u=1/f1+1/f2…………….(3)

Let the two lenses be replaced by a single lens of focal length F, which forms an image I at distance v, of an object at distance u from the lens. For this lens,

1/v-1/u=1/f ………………(4)

From the third and fourth equations, we get

1/f= 1/f1+1/f2……………………………….(5)

Taking one convex and one concave lens: 

Assume that the focal length of the convex lens is f1, and that of the concave lens is f2. If f is the focal length of equivalent lens. Then,

1/f=1/f1+ 1/-f2

= 1/f1-1/f2

f=f1f2/f2-f1

Combined Power of Two Lenses Placed in Contact

If the powers of two thin lenses placed in contact are P1 and P2, then from the fifth equation, the power of the combined lens is

P= P1+ P2

Magnification by Combinations of Lenses

When lenses are used in combination, the magnification M of the final image is given by the product of the magnifications by the individual lenses, that is

M = m1 x m2 x m3……

The combination of lenses is generally used in designing objects such as microscopes, cameras, telescopes, and other optical instruments.

Conclusion

Lenses are mainly of two types, convex and concave. These lenses can be combined in different ways to attain different focal lengths and perform different functions. Combining lenses(compound lenses) reaps the advantage of performing tasks that cannot be performed by a single lens. Lenses can be recognised by knowing their focal length. The power of a lens is the extent to which it converges or diverges a ray of light from its original path. Lenses also help in the magnification of light.