For a system of lenses the net power of the system will be

P_{eq} = P_{1} + P_{2} + P_{3} + P_{4} + ……..

Provided all the thin lens all in close contact. There focal length of the net system can be written as

$ \displaystyle \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} + ……. $

f should be taken with proper sign.

When a convex and a concave lens of equal focal length are placed in contact, the equivalent focal length is equal to infinite. Therefore the power becomes zero.

If the lenses are kept at a separation d, then the effective focal length f is given as

$ \displaystyle \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} – \frac{d}{f_1 f_2} $

If the lens is converging put +ve for its focal length & -ve for the diverging lens.

⇒ $ \displaystyle P = P_1 + P_2 – d (P_1 P_2) $

Note: The overall magnification M of the system is given as the product of individual magnification m; M = m_{1} m_{2} m_{3} …….

**Silvering of lenses**

If any surface of a lens is silvered, it will ultimately behave as a mirror and the power of mirror thus formed will be equal to the sum of powers of the optical lenses and the mirrors in between .

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