COMBINATION OF THIN LENSES IN CONTACT

Consider two lenses A and B of focal length f1 and f2 placed in contact with each other.

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The image of point object O will be at I1 formed by lens A, which act as a virtual object for second lens B producing the final image at I. since lenses are thin, therefore we assume the optical centres of the lenses to be coincident. Let this central point is P.

For lens A

\begin{array}{l}
\frac{1}{{{{\rm{v}}_1}}}\, - \frac{1}{u}\, = \,\frac{1}{{{f_1}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,.....................\,\left( 1 \right)\\
{\rm{for lens B}}\\
\frac{1}{{\rm{v}}}\, - \,\frac{1}{{{{\rm{v}}_1}}}\, = \,\frac{1}{{{f_2}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...................\,\left( 2 \right)\\
{\rm{Adding}}\,{\rm{equation}}\,\left( 1 \right)\,{\rm{and}}\left( 2 \right)\,\\
\frac{1}{{\rm{v}}}\, - \,\frac{1}{u}\, = \,\frac{1}{{{f_1}}}\, + \,\frac{1}{{{f_2}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...................\left( 3 \right)
\end{array}

Considering equivalent lens of focal length f, then

\begin{array}{l}
\frac{1}{{\rm{v}}}\, - \,\frac{1}{u}\, = \,\frac{1}{f}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,....................\left( 4 \right)\\
Comparing\,\left( 3 \right)\,and\,\left( 4 \right)\\
\frac{1}{f}\, = \,\frac{1}{{{f_1}}}\, + \,\frac{1}{{{f_2}}}\,\,
\end{array}

 Purpose of combination of lens.

(a) To meet desired magnification m\, = {m_1} \times {m_2} \times {m_3} \times \,... \times \,{m_n}

(b) It also enhance sharpness of the image.

(c) To make final image erect.

(d) To remove certain defects in the lens.

Such a system of combination of lenses is commonly used in designing lenses for cameras, microscopes, telescopes and other optical instruments.

Note:

(1) If several thin lenses of focal length {f_1},\,{f_2},\,{f_3},\,.................are in contact, then effective focal length is

\frac{1}{f} = \,\frac{1}{{{f_1}}}\, + \,\frac{1}{{{f_2}}}\, + \,\frac{1}{{{f_3}}}\, + \,\,...\,\,\, = \,\sum\limits_{i\, = 1}^n {\frac{1}{{{f_i}}}}

(2) Equivalent power of the combination is an algebraic sum of individual powers.

P\, = \,{P_1}\, + \,{P_2}\, + \,{P_3}\, + \,\,...

Post Author: E-Physics

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