Transcript Slide 1

Problem 7
To solve this problem you needed to know that:
1.
2.
3.
4.
5.
square has two pairs of parallel sides that intersect at 90° angle;
real image is formed by converging lens if the object´s distance is
greater than focal length (from Newton´s equation: f2=x1x2);
rays intersecting first focal point will be parallel to optical axis after
passing through the lens;
two parallel rays at an angle relative to optical axis will intersect at
point lying in the second focal plane of the lens – this point is
found by using the above (hint 3.) and;
basic trigonometry
• Opposite sides of this
quadrilateral are formed
by initialy parallel sides of
the square.
Points at which opposite
sides intersect should be
lying in the second focal
plane of the lens
For ideal thin lens, focal
planes are not curved, so
we simply connect points
F´ and F´´
To find the solution, let
us assume that the focal
lenght is i.e. 3 cm
Using the hints 3. and 4. in
reverse, two rays (shown in
yellow) should be parallel to
sides of the square, and
therefore meet at 90°angle.
Also, they intersect at the
first focal point
To find focal length and
optical axis offset (a1 or a2)
we mark angles of rays
relative to optical axis
(sides of the square!)
Let us mark the distance F´F´´ as
d. This distance is a indenpendent
from the focal lenght and offset so,
by using trigonometry we have:
(1)
Using hints 3. and 4. again
we can find the position of
the “square”
Let us mark the “square”
side as s´ and translate it
to form a triangle with
sides s´(red), d´(brown)
and angles equal to α,β
and 90°.We repeat the
same for side s´´.
This gives us:
(2)
To find distances d´and
d´´ , we will use angles
Δ, Δ´, δ, δ´. Note that
these angles are
indenpendent from the
focal length and offset,
so we have:
(3)
(4)
Using all four equation,
we have:
From this picture; we can see
that angles Δ, Δ´, δ, δ´ are
indeed indenpendent. Their
values are:
Using these data, we obtain:
The complete solution
occupies only one page
To avoid taking the
derivatives, uncertanities are
found by directly inputing the
data, and calculating the
final uncertanity.