Following formula is known as “Gaussian Lens Formula”
#1/O+1/I=1/f#
where #O# is object distance, #I# is image distance and #f# focal length of lens.
An alternate lens formula is known as the Newtonian Lens Formula
which can be obtained by substituting #O = f + x and I = f + y# into the Gaussian Lens Formula. Here, #x and y# are the distances of the object and image respectively from the focal points. We get
#1/(f + x)+1/( f + y)=1/f#
#=>((f+y)+(f+x))/((f + x)( f + y))=1/f#
After simplifying and Cross-multiplying we get
#f(2f+x+y)=(f + x)( f + y)#
#=>2f^2+xf+yf=f^2 + xf+ fy + xy#
#=>f^2= xy#
(In calculations #f# is taken as negative for a diverging "concave" lens).
-.-.-.-.-.-.-.-.-.-.-.
Example
Q. An object is located at #15 cm# from a diverging lens which has a focal length of #-10 cm#. Where is the image formed?
A.
By definition of #x#
#x=O-f#
#=>x=15-(-10)=25cm#
Using the formula
#f^2=xy#
#=>y=f^2/x=(-10)^2/25=4cm#
Now the image distance #I=f+y#
#=>I= (-10)+4 = -6 cm# to right of lens, which is #6 cm# to left of lens.
