W9_parallel_resonance.eps

Week 12: Lenses and Mirrors 399

α β

θ

δ

Figure 159:α+θ=β.

easily see thatα+θ+δ=π. But we canalsosee thatδ+β=π. Therefore:

α+θ=β (971)

and similarly (considering the other triangle involvingβandθ)

β+θ=γ (972)

If we eliminateθ, we get:

α+γ= 2β (973)

Finally, if we substitute in all of the small angle approximations and cancell, we get:

1

s

+^1

s

=^2

r

(974)

As we move the object back farther and farther from the mirror (lets→ ∞) we note

that the image distance approachesr/2. Rays coming from an infinitely distant object

arrive at the mirrorparalleland convergeats′=r/2. Wedefinethe point where a lens or

mirror focusesparallel, paraxial raysto be thefocal pointof the lens or mirror. Thus:

f=

r

2

(975)

and

1

s+

1

s=

1

f (976)

This is avery important result!It is the equation we will use to analyzeall images formed

by curved mirrors and thin lenses(after we derive the same formula for the latter) so be

sure that you have learned it and understand it.

The focal lengthfof a mirror (or lens) is the point where incident parallel rays are

focusedto(for positive focal lengths) or appear to be defocusedfrom(for negative

focal lengths). fis typically measured in meters (SI) or centimeters (for convenience).

However, the strength oflensesis usually given indiopters, where:

d=^1

f

(977)

withfin meters. This a one diopter (1.00d) lens has a focal length of 1 meter. A 10.00d

lens has a focal length of 0.1 meter. A diverging lens with a focal length of one centimeter

is -100.00d.

It is possible to use the same inverse length units to write the thin lens/mirror equation

above. If we definex= 1/s,x′= 1/s′, then:

x+x′=d (978)

is thedirect(instead of reciprocal) rule. Note well that the ranges ofx,x′, anddhave a

very different meaning.d= 0 means a focal length of±∞, a flat mirror (or non-focusing

lens). x= 0 is similarlys=±∞, generally +∞. Here it is quite easy to see how and

whenxandx′change sign if either one of them is larger thand.