Following formula is known as “Gaussian Lens Formula”
1/O+1/I=1/f1O+1I=1f
where OO is object distance, II is image distance and ff 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 + yO=f+xandI=f+y into the Gaussian Lens Formula. Here, x and yxandy are the distances of the object and image respectively from the focal points. We get
1/(f + x)+1/( f + y)=1/f1f+x+1f+y=1f
=>((f+y)+(f+x))/((f + x)( f + y))=1/f⇒(f+y)+(f+x)(f+x)(f+y)=1f
After simplifying and Cross-multiplying we get
f(2f+x+y)=(f + x)( f + y)f(2f+x+y)=(f+x)(f+y)
=>2f^2+xf+yf=f^2 + xf+ fy + xy⇒2f2+xf+yf=f2+xf+fy+xy
=>f^2= xy⇒f2=xy
(In calculations ff is taken as negative for a diverging "concave" lens).
-.-.-.-.-.-.-.-.-.-.-.
Example
Q. An object is located at 15 cm15cm from a diverging lens which has a focal length of -10 cm−10cm. Where is the image formed?
A.
By definition of xx
x=O-fx=O−f
=>x=15-(-10)=25cm⇒x=15−(−10)=25cm
Using the formula
f^2=xyf2=xy
=>y=f^2/x=(-10)^2/25=4cm⇒y=f2x=(−10)225=4cm
Now the image distance I=f+yI=f+y
=>I= (-10)+4 = -6 cm⇒I=(−10)+4=−6cm to right of lens, which is 6 cm6cm to left of lens.
