Expert: Bobby Soltani - 1/19/2006

Question
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Followup To
Question -
Hello Bobby,

I'm trying to solve this alebra problem.  Any help appreciated. This is not a homework question - I'm a photographer working with different lenses.

F = focal length
S1 = object to lens distance
S2 = lens to image distance
M = magnification

From Wikipedia I found the following expression.

M = F/ (F - S1)

I know that 1/F = 1/S1 + 1/S2

so I substituted S1 in the Wikipedia equation with 1(1/F - 1/S2) but got in a tangle as my maths isn't good enough.

I'm trying to find an equation that relates M, F and S2 (not S1) together.

Any help appreciated!
Regards,
Usuff

Answer -
Hi Usuff,

I get the following relationship.

M = (F - S2)/F

This can also be written as

M = 1 - (S2/F)

Let me know if you have any questions.

Bobby
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Bobby,
Thanks for that.  I put figures in the solution and it gave the right answer. Would it be possible to walk me through how you got the answer? I was able to express S1 as 1/((1/F) - (1/S2)) and put it into the magnification formula.  But I couldn't simplify any further after that.
Thanks.
Regards,
Usuff

Answer
Hi,

I wasn't sure if you needed all the steps.  Here they are.

M = F/ (F - S1)

I know that 1/F = 1/S1 + 1/S2

solve this second equation for s1.

1/s1 = 1/f - 1/s2
get a common denominator on the right side by multiplying the first term by s2/s2, and the second by f/f

1/s1 = s2/(fs2) - f/(fs2)
combine the terms on the right since the denominator is the same

1/s1 = (s2 - f)/(fs2)
flip both sides of the equation

s1 = (fs2)/(s2 - f)

plug this in for s1 in the first equation.

M = f / (f - (fs2)/(s2-f))
multiply the f in the denominator by (s2-f)/(s2-f) to get a common denominator.

M = f / (f(s2-f)/(s2-f) - (fs2)(s2-f))
combine the terms in the denominator

M = f / ((f(s2-f)-fs2)/(s2-f))
since the s2-f is in the denominator of the denominator, we can move it up to the numerator. (e.g. 1/(1/6) = 6)

M = f(s2-f) / (f(s2-f) - fs2)
factor out an f from the two terms in the denominator

M = f(s2 - f) / (f((s2-f) - s2)
cancel the f in the numerator with the f in the denominator.

M = (s2 - f) / (s2 - f - s2)
do the subtraction in the denominator

M = (s2 - f)/(-f)

multiply by -1/-1 to get the negative out of the denominator.

M = (f - s2)/f

Let me know if you have any questions.

Bobby