An office copier uses a lens to place an image of a document onto a rotating drum. The copy is made from this image. (a) What kind of lens is used? Should it be 1 - converging, or 2 - diverging?

Suppose the lens has a focal length of magnitude 13.0 cm. If the document and its copy are to have the same size, but are inverted with respect to one another, (b) how far from the document is the lens located and (c) how far from the lens is the image located?

(a) A converging lens is used for this purpose.

(b) To form an image of the same size as the object, the object should be at 2f (twice the focal length) from the lens. In this case, the focal length is 13.0 cm, so the object (document) should be 2 * 13.0 cm = 26.0 cm away from the lens.

(c) The image will also be at 2f (twice the focal length) from the lens, but on the other side. So the image will be 26.0 cm away from the lens as well.

(a) The lens used in the office copier should be a 1 - converging lens.

(b) To have the document and its copy at the same size and inverted with respect to one another, we can use the lens equation:

1/f = 1/di + 1/do

Where:
f = focal length of the lens
di = image distance
do = object distance (distance from the lens to the document)

Rearranging the equation, we can solve for do:

1/do = 1/f - 1/di

Since the document and the image are at the same size, we can assume the magnification, M, is 1. This means the image distance is equal to the object distance but with opposite signs:

di = -do

Substituting this into the equation, we get:

1/do = 1/f + 1/(-do)

Simplifying:

1/do = 1/f - 1/do

2/do = 1/f

do = f/2

Given that the focal length, f, is 13.0 cm, we can substitute it into the equation:

do = 13.0 cm / 2

do = 6.5 cm

Therefore, the lens is located 6.5 cm from the document.

(c) The image distance, di, is the same as the object distance, but with an opposite sign. Since the object distance, do, is 6.5 cm, the image distance, di, is also -6.5 cm. Therefore, the image is located 6.5 cm from the lens.

To determine the type of lens used in the office copier, we need to understand the basic principles of how it forms an image on the rotating drum.

(a) The lens used in the office copier is a converging lens (1). A converging lens is also known as a convex lens, which is thicker at the center and thinner at the edges. This type of lens brings light rays that are initially parallel together to a focal point.

(b) To determine the distance from the document to the lens, we can use the lens formula:

1/f = 1/do + 1/di

where:
f = focal length
do = object distance (distance from the document to the lens)
di = image distance (distance from the lens to the image)

In this case, the document and its copy are of the same size but are inverted with respect to each other. This means the image is formed on the opposite side of the lens compared to the document. The image distance (di) would be negative.

Given that the focal length (f) is 13.0 cm, and assuming the image distance (di) is -13.0 cm (since it is inverted), we can rearrange the lens formula to solve for the object distance (do):

1/f = 1/do + 1/di

Substituting the given values:

1/13.0 = 1/do + 1/-13.0

Simplifying the equation:

1/do = 1/13.0 - 1/13.0

1/do = 0

Therefore, the object distance (do) is at infinity or very far from the lens. In practical terms, this means the document should be placed far enough from the lens for it to be considered at an infinite distance.

(c) Since the object distance (do) is at infinity, the image distance (di) will be the distance from the lens to the image. In this case, the image distance (di) is -13.0 cm, as it was assumed to be inverted.

Hence, the image in the copier is located 13.0 cm from the lens.