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Other equations for calculating the new focal number after placing the additional lens

We have noticed from the application of the equation F` = F x P`` to P that we first have to calculate P`` (that is, the focal length of the sum of the original and additional lenses). However, there are two other equations (1) that are suitable for calculating the new focal number without first going through the step of calculating (p), which is the focal length of the two lenses.

The first equation is suitable for application in the case of placing the additional positive lens.

The second equation is valid for application in the case of placing the additional negative lens.

First: If the additional lens is positive, then the new focal number is estimated according to the following:
P` = P X P` on P` + P

Second: If the additional lens is negative, then the new focal number is estimated according to the following:

P` = P × P` on P` - P

Assuming that the new focal number = f

And that the first f-number (before putting on the extra lens) = f

And the focal length of the original lens = p

And that the focal length of the additional lens = p`

Important notes regarding the use of supplemental lenses in general:

It is noted that these new focal numbers do not fully indicate the intensity of the image's illumination. Rather, it is only an approximation for the following reasons:

(a) The previous equations overlooked an important fact, which is that it is often not possible to place the additional lens so that it is almost adjacent to the original lens, but there may be a space between them depending on the degree of protrusion of the front part of the cylinder.
metal surrounding the original lens.

(b) That the previous equations have ignored the amount of what we lose absorbed or reflected by the additional lens, or the amount of radiation lost as a result of the distortion caused by these lenses.

(c) The focal length number may sometimes decrease numerically (as in Example No. 1 pg. 177,176), which indicates a lens that is able to form a more luminous image, but this increase in luminance may be lost due to an increase in the distance of the film from the lens, and as a result of the previous factors.

The amount of increase in exposure when increasing the extension of the bellows:
We aim for two purposes behind knowing the new focal number when placing an additional lens or when increasing the extension of the camera bellows, and these two purposes are:

(a) Depth of field suitable for the new focal number (and we will talk about depth of field in the next section).

(b) The amount of increase or decrease in exposure if the extension of the bellows is increased or decreased in length. The increase or decrease in exposure depends on the ratio between:
First f-number: To the new f-number.

If the first f-number = 2 f
The new f-number = 4 f

The exposure must be increased by the ratio between these two focal numbers. (We have previously explained how to calculate this ratio on page 58 under the heading Comparing a focal number with another) This ratio is called the exposure factor Exp sure factor.

There is another direct and easy way to find out the extent of the increase in exposure required after increasing the length of the bellows (i.e. to know the exposure factor) without going through the step of calculating the new focal number, by applying the following equation:

Exposure factor = (blower length)2 over (focal length)2

If we assume, for example, that the focal length of the original lens = 12 cm, and that the bellows has been extended to 24 cm, then the exposure factor becomes = 24 x 24 divided by 12 x 12 = 4 times.

The previous equation is of great importance in the case of macrophotography, where it is required to greatly increase the length of the bellows.