Combination of Thin Lenses in Contact
For first lens :
`1/(v^') -1/u = 1/f_1`
For second lens :
`1/v -1/v^' = 1/f_2`
Combining, we get
`1/v - 1/u = 1/(f_1) + 1/(f_2)`
`=>` `1/(f_(eq)) = 1/(f_1) + 1/f_2`
i.e., `P_(eq)=P_1 + P_2`
Similarly for n thin lenses in contact
`1/(f_(eq)) = 1/(f_1) + 1/(f_2)+1/(f_3)+...........+(1/f_n)`
`P_(eq)=P_1 + P_2+P_3.......P_n`
`text(Note :)`
(i) If two thin lenses of equal focal but of opposite nature (i.e. one convergent and other divergent) are put in contact, the resultant focal length of the combination will be.
`1/(f_(eq)) = 1/f + 1/(-f) = 0`
i.e., `f_(eq)=oo` and `P_(eq)=0`
i.e., the system will be have as a plane glass plate.
(ii) If two lenses of same nature are put in contact, then as (`f_1` and `f_2` are magnitude of focal lengths)
`1/(f_(eq)) = 1/(f_1) + 1/f_2`
`1/(f_(eq)) > 1/f_1` and `1/(f_(eq)) > 1/f_2`
The resultant focal length will be lesser than individual.
(iii) If two thin lenses of opposite nature with different focal lengths are put in contact, the resultant focal length will be of same nature as that of the lens of shorter focal length but its magnitude will be more than that of shorter focal length.
(iv) If a lens of focal length f is divided into two equal parts as shown in figure and each part has a focal length f' then each part will have focal length 2 times. Now if these part are put in contact as in figure (a), (b) or (c) the resultant focal length of the combination will be equal to initial value.
`1/(v^') -1/u = 1/f_1`
For second lens :
`1/v -1/v^' = 1/f_2`
Combining, we get
`1/v - 1/u = 1/(f_1) + 1/(f_2)`
`=>` `1/(f_(eq)) = 1/(f_1) + 1/f_2`
i.e., `P_(eq)=P_1 + P_2`
Similarly for n thin lenses in contact
`1/(f_(eq)) = 1/(f_1) + 1/(f_2)+1/(f_3)+...........+(1/f_n)`
`P_(eq)=P_1 + P_2+P_3.......P_n`
`text(Note :)`
(i) If two thin lenses of equal focal but of opposite nature (i.e. one convergent and other divergent) are put in contact, the resultant focal length of the combination will be.
`1/(f_(eq)) = 1/f + 1/(-f) = 0`
i.e., `f_(eq)=oo` and `P_(eq)=0`
i.e., the system will be have as a plane glass plate.
(ii) If two lenses of same nature are put in contact, then as (`f_1` and `f_2` are magnitude of focal lengths)
`1/(f_(eq)) = 1/(f_1) + 1/f_2`
`1/(f_(eq)) > 1/f_1` and `1/(f_(eq)) > 1/f_2`
The resultant focal length will be lesser than individual.
(iii) If two thin lenses of opposite nature with different focal lengths are put in contact, the resultant focal length will be of same nature as that of the lens of shorter focal length but its magnitude will be more than that of shorter focal length.
(iv) If a lens of focal length f is divided into two equal parts as shown in figure and each part has a focal length f' then each part will have focal length 2 times. Now if these part are put in contact as in figure (a), (b) or (c) the resultant focal length of the combination will be equal to initial value.