Combination of Thin Lenses in Contact Equivalent Focal length, Power and Magnification
(i) Equivalent focal length of the combination of Lenses :
Consider two thin lenses of focal lengths $f_1$ and $f_2$ respectively, placed in contact with each other . Let $O$ be the point object placed on the principal axis of the lenses in contact. If the first lens forms an image $I_1$ of the object $O$ at a distance $v_1$ from it.
$\frac{1}{v_1} - \frac{1}{u} = \frac{1}{f_1} \quad \text{(Lens formula)} \quad \dots (i)$
Since the second lens is in contact with the first, so $I_1$ acts as a virtual object for the second lens which forms the image $I$ at a distance $v$ from it.
$\frac{1}{v} - \frac{1}{v_1} = \frac{1}{f_2} \quad \dots (ii)$
Adding eqs. (i) and (ii), we get
$\frac{1}{v_1} - \frac{1}{u} +\frac{1}{v} - \frac{1}{v_1}$
$\frac{1}{v} - \frac{1}{u} = \frac{1}{f_1} + \frac{1}{f_2}$
or
$\frac{1}{F} = $\frac{1}{v} - \frac{1}{u} \quad \dots (iii)$
Where $\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \quad \dots (iv)$
here $F$ is the effective focal length or equivalent focal length of the combination of two lenses in contact.
Thus, the two thin lenses in contact behave as a single lens known as equivalent lens with focal length $F$ called equivalent focal length.
If more than two lenses are in contact, then the equivalent focal length of the combination is given by
$\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} + \dots \quad \dots (v)$
(ii) Power of the Equivalent Lens :
Power of a lens is given by,
$P = \frac{1}{f} \quad \dots (i)
where $f$ is the focal length of the lens.
If two lenses are in contact, then the focal length of the equivalent lens formed by these two lenses is given by
$\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \quad \dots (ii)$
Using eqn. (i), we get
$P = P_1 + P_2 \quad \dots (iii)
If more than two lenses are in contact, then the power of the equivalent lens is given by
$P = P_1 + P_2 + P_3 + \dots \quad \dots (iv)$
Thus, power of an equivalent lens is the algebraic sum of the powers of individual lenses in contact.
(iii) Magnification of the Equivalent Lens :
Consider two lenses $L_1$ and $L_2$ in contact having linear magnifications $m_1$ and $m_2$ respectively. Then the net linear magnification of an equivalent lens formed by two lenses in contact is given by
m = m_1 \times m_2 \quad \dots (i)
If equivalent lens is formed by more than two lenses, then the net linear magnification is given by
$m = m_1 \times m_2 \times m_3 \times \dots \quad \dots (ii)$
Thus, the combination of lenses magnifies the final image.
(iv) Uses of Equivalent Lens :
Equivalent lens (i.e., combination of lenses) is used in optical instruments like telescope, microscope, camera and binoculars to get sharp images of the objects.