For a fixed distance L between the object and the screen, there are

two positions (1 and 2) of a converging lens which produce a reel image
on the screen. These two positions are D apart.

a) Derive the expression for the focal length f of the lens in terms of L
and D.

b) Express the height of the object h in terms of the heights of the two
images h1 and h2.

a) To derive the expression for the focal length f in terms of L and D, we can use the lens formula:

1/f = 1/v - 1/u,

where f is the focal length, v is the image distance, and u is the object distance.

In the first position (1), the object distance u1 is L, and the image distance v1 is L + D.

Similarly, in the second position (2), the object distance u2 is L, and the image distance v2 is L - D.

Plugging these values into the lens formula for each position, we get:

1/f = 1/(L + D) - 1/L (equation 1)
1/f = 1/(L - D) - 1/L (equation 2)

Now, we can manipulate these equations to solve for f in terms of L and D.

Starting with equation 1:
1/f = 1/(L + D) - 1/L

Multiplying both sides of the equation by L(L + D), we get:
L(L + D)/f = L + D - (L + D)(L + D)/L

Simplifying further:
L(L + D)/f = L + D - (L^2 + 2LD + D^2)/L

Combining like terms:
L(L + D)/f = (L^2 + LD - L^2 - 2LD - D^2)/L

Further simplification:
L(L + D)/f = (-LD - D^2)/L

Cancelling out L from both sides of the equation:
L + D = -D - D^2/f

Rearranging the equation to solve for f:
1/f = -(L + D)/(D^2 + LD)

Taking the reciprocal of both sides:
f = -1/[(L + D)/(D^2 + LD)]

Simplifying further:
f = -(D^2 + LD)/(L + D)

Therefore, the expression for the focal length f of the lens in terms of L and D is f = -(D^2 + LD)/(L + D).

b) To express the height of the object h in terms of the heights of the two images h1 and h2, we can use the magnification formula:

m = h'/h,

where m is the magnification, h' is the image height, and h is the object height.

In position 1, the magnification is given by:
m1 = h1/h

In position 2, the magnification is given by:
m2 = h2/h

Since the magnification is the same for both positions (as the images are reel), we can equate m1 and m2:

h1/h = h2/h

Cross-multiplying:
h1 = h2

Therefore, the height of the object h is equal to the heights of the two images h1 and h2.