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Asked by adjacentcalliber10 | 14 May, 2019, 10:28: AM
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Let x cm is the distance of object in front of lens-1 of focal length 3 cm.
 
if v is the image distance, we have  begin mathsize 12px style 1 over v plus 1 over x equals 1 third space space space o r space space space v space equals space fraction numerator 3 x over denominator x minus 3 end fraction end style    ..............(1)
hence magnification m1 of first lense is given by,   m1 = -v/u  = 3/(x-3)   ....................(2)
 
If L is the distance between lenses , then object distance for second lens-2 of focal length 5 cm is given by
 
begin mathsize 12px style L minus v space equals space L space minus space fraction numerator 3 x over denominator left parenthesis x minus 3 right parenthesis end fraction space equals space fraction numerator open parentheses L minus 3 close parentheses x space minus 3 L over denominator open parentheses x minus 3 close parentheses end fraction end style  ..............................(3)
If v' is the image distance for second lens,  then we have 
 
begin mathsize 12px style fraction numerator 1 over denominator v space apostrophe end fraction space equals space 1 fifth minus fraction numerator open parentheses x space minus space 3 close parentheses over denominator open parentheses L minus 3 close parentheses x minus 3 L end fraction space equals space fraction numerator begin display style open parentheses L minus 3 close parentheses x minus 3 L space minus 5 x plus 15 end style over denominator 5 space open square brackets open parentheses L minus 3 close parentheses x minus 3 L close square brackets end fraction space equals space fraction numerator open parentheses L minus 8 close parentheses begin display style x end style begin display style space end style begin display style plus end style begin display style space end style begin display style open parentheses 15 minus 3 L close parentheses end style over denominator 5 space open square brackets open parentheses L minus 3 close parentheses x minus 3 L close square brackets end fraction end style
Hence    begin mathsize 12px style v space apostrophe space equals space fraction numerator 5 space open square brackets open parentheses L minus 3 close parentheses x minus 3 L close square brackets over denominator open parentheses L minus 8 close parentheses x space plus space open parentheses 15 minus 3 L close parentheses end fraction end style  .................................(4)
magnification m2 of the second lens is given by,  m2 = v' /(L - v' )  = begin mathsize 12px style space fraction numerator 5 space left parenthesis x minus 3 right parenthesis over denominator open parentheses L minus 8 close parentheses x space plus space open parentheses 15 minus 3 L close parentheses end fraction end style  ...................(5)
Overall magnification M = m1m2 = begin mathsize 12px style space fraction numerator 15 space over denominator open parentheses L minus 8 close parentheses x space plus space open parentheses 15 minus 3 L close parentheses end fraction end style  ........................(6)
To make the overall magnification M independent of x, we put  (dM/dx) = 0
 
hence begin mathsize 12px style fraction numerator d M over denominator d t end fraction space equals space space fraction numerator negative 15 left parenthesis L minus 8 right parenthesis space over denominator open square brackets open parentheses L minus 8 close parentheses x space plus space open parentheses 15 minus 3 L close parentheses close square brackets squared end fraction space space equals space 0 space space space space space space o r space L space equals space 8 space c m end style
Overall magnififcation M is obtained using eqn.(6) with L = 8 cm and it is -5/3 
Answered by Thiyagarajan K | 16 May, 2019, 16:46: PM

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