Effective focal length of thin lenses placed in contact
Three lenses in contact have a combined focal length of 12cm. When the third lens is removed the combined focal length becomes 760cm. What is the focal length of the third lens?
Let f1, f2 and f3 be three focal lengths of the lenses. Combined Focal length: f=12cm So, f1=f11+f21+f31 When the third lens is removed, f′′1=f11+f21, where f′′=760 This means, 121=7601+f31 ⇒f3=−30cm Since f3 is negative, it is a diverging lens having focal length 30cm.
Combination of thin lenses in contact
Consider two lenses A and B of focal length f1 and f2 placed in contact with each other. An object is placed at O on the common principal axis. The lens A produces an image at I1 and this image acts as the object for the second lens B. The final image is produced at I as shown in figure. PO = u, object distance for the first lens (A), PI = v, final image distance and PI1=v1, image distance for the first lens (A) and also object distance for second lens (B). For the image I1 produced by the first lens A, v11−u1=f11 .... (1) For the final image I, produced by the second lens B, v1−v11=f21 ... (2) Adding equations (1) and (2), v1−u1=f11+f21 ... (3) If the combination is replaced by a single lens of focal length F such that it forms the image of O at the same position I, then v1−u1=F1 ... (4) From equations (3) and (4), F1=f11+f21...... (5) This F is the focal length of the equivalent lens for the combination.