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isaac physics quick help

i would like some help with this question and a explanation would be approved?

Work out a general formula for the smallest possible distance between an object and its real image formed with a convex lens of focal length f. What is the value of this distance when f=15.0cm?

Reply 1

Joinedup

simple form of the thin lens equation is
1/f=1/i + 1/o

f is the focal length, a constant (so 1/f is also constant)

we can already see a few things
if 1/i increases then 1/o must decrease - and there's symmetry between i and o, if 1/o increases then 1/i must decrease in the same way.
if i = f then 1/o must equal zero - which is the same as saying that o is infinity - and the same reasoning applies if it's o=f

I would suggest the most obvious starting point is the symmetrical 1/i = 1/o = 1/2f
so i=o=2f
if f were 100 then i=o=200 would work and the distance between image and object would be 400

but can you get less than 400 by making a small change?

well if i was 200.10000 then o would have to be 199.9001 for 400.0001 total distance image to object so a slight change to i makes the overall distance slightly greater... pretty clearly the symmetry of the equation means that reducing i to slightly less than 200 will also increase the total distance.

so slight changes from 2f make the distance greater and gross changes that take i or o close to f make the total distance tend to infinity... i'd suggest that the closest distance from image to object is 4f