Problem 7To solve this problem you needed to know that:

1. square has two pairs of parallel sides that intersect at 90° angle;2. real image is formed by converging lens if the object´s distance is

greater than focal length (from Newton´s equation: f2=x1x2);

3. rays intersecting first focal point will be parallel to optical axis after passing through the lens;

4. 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;

5. 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.