Now we will talk about geometric optics. In this section, a lot of time is devoted to such an object as a lens. After all, it can be different. At the same time, the formula of a thin lens is the same for all cases. You just need to know how to apply it correctly.
Types of Lenses
It is always a body transparent to light rays , which has a special shape. The appearance of the object is dictated by two spherical surfaces. One of them can be replaced with a flat one.
Moreover, the lens may be thicker middle or edge. In the first case, it will be called convex, in the second - concave. Moreover, depending on how concave, convex and flat surfaces are combined, the lenses can also be different. Namely: biconvex and biconcave, plano-convex and plano-concave, convex-concave and concave-convex.
Under normal conditions, these objects are used in the air. They are made from a substance whose optical density is greater than that of air. Therefore, a convex lens will be collecting, and a concave lens will be scattering.
General characteristics
Before talking about the formula of a thin lens , you need to decide on the basic concepts. They must be known. Since they will constantly be addressed by various tasks.
The main optical axis is straight. It is drawn through the centers of both spherical surfaces and determines the location of the center of the lens. There are additional optical axes. They are drawn through a point that is the center of the lens, but do not contain the centers of spherical surfaces.
In the formula of a thin lens there is a quantity that determines its focal length. So, the focus is a point on the main optical axis. In it, rays crossing parallel to the specified axis intersect.
Moreover, the tricks of each thin lens are always two. They are located on both sides of its surfaces. Both focuses on the collector are valid. In scattering - imaginary.
The distance from the lens to the focal point is the focal length (letter F ) . Moreover, its value can be positive (in the case of collecting) or negative (for scattering).
Another characteristic is associated with focal length - optical power. It is customary to denote D. Its value is always the reciprocal of the focus, that is, D = 1 / F. The optical power in diopters is measured (abbreviated diopters).
What other designations are in the formula of a thin lens
In addition to the already specified focal length, you will need to know several distances and sizes. For all types of lenses they are the same and are presented in the table.
| Designation | Title |
| d | distance to subject |
| h | height of the studied subject |
| f | distance to image |
| H | height of the resulting image |
All indicated distances and heights are usually measured in meters.
In physics, the notion of magnification is also associated with the formula of a thin lens. It is defined as the ratio of the size of the image to the height of the object, that is, H / h . It can be denoted by the letter G.
What you need to build an image in a thin lens
This is necessary to know in order to obtain the formula of a thin lens collecting or scattering. The drawing begins with the fact that both lenses have their own schematic image. Both of them look like a cut. Only in the collecting at its ends the arrows are directed outward, and in the scattering - inward of this segment.
Now to this segment it is necessary to draw a perpendicular to its middle. This will show the main optical axis. On it, on both sides of the lens at the same distance, it is supposed to note tricks.
The object whose image you want to build is drawn in the form of an arrow. It shows where the top of the item is. In general, an object is placed parallel to the lens.
How to build an image in a thin lens
In order to build an image of an object, it is enough to find the points of the ends of the image, and then connect them. Each of these two points can result from the intersection of two rays. The simplest to build are two of them.
Coming from a specified point parallel to the main optical axis. After contact with the lens, he goes through the main focus. If we are talking about a collecting lens, then this focus is behind the lens and the beam goes through it. When scattering is considered, the beam must be drawn so that its continuation passes through the focus in front of the lens.
Going directly through the optical center of the lens. He does not change his direction after her.
There are situations when an object is placed perpendicular to the main optical axis and ends on it. Then it is enough to build an image of a point that corresponds to the edge of the arrow, not lying on the axis. And then draw from it a perpendicular to the axis. This will be the image of the subject.
The intersection of the constructed points gives an image. In a thin collecting lens, a real image is obtained. That is, it is obtained directly at the intersection of the rays. An exception is the situation when the subject is placed between the lens and the focus (as in a magnifying glass), then the image turns out to be imaginary. In scattering, it always turns out to be imaginary. After all, it is obtained at the intersection not of the rays themselves, but of their extensions.
The actual image is usually drawn with a solid line. But the imaginary - dotted line. This is due to the fact that the first is actually present there, and the second is only seen.
Derivation of the thin lens formula
This is conveniently done on the basis of a drawing illustrating the construction of a real image in a collecting lens. The designation of the segments is indicated on the drawing.
The optics section is not in vain called geometric. It will take knowledge from this section of mathematics. First you need to consider the triangles AOW and A 1 OB 1 . They are similar, because they have two equal angles (straight and vertical). From their similarity it follows that the modules of the segments A 1 B 1 and AB belong as modules of the segments OB 1 and OB.
Similar (based on the same principle at two angles) are two more triangles: COF and A 1 FB 1 . They already have equal ratios of such moduli of segments: A 1 B 1 with CO and FB 1 with OF. Based on the construction, the segments AB and CO will be equal. Therefore, the left-hand sides of the indicated equalities of relations are the same. Therefore, the right are equal. That is, OB 1 / OB is equal to FB 1 / OF.
In the indicated equality, the segments indicated by dots can be replaced by the corresponding physical concepts. So OB 1 is the distance from the lens to the image. OB is the distance from the subject to the lens. OF is the focal length. And the segment FB 1 is equal to the difference between the distance to the image and focus. Therefore, it can be rewritten in another way:
f / d = ( f - F ) / F or Ff = df - dF.
To derive the thin lens formula, the last equality must be divided by dfF. Then it turns out:
1 / d + 1 / f = 1 / F.
This one has the formula of a thin collecting lens. A scattering focal length is negative. This leads to a change in equality. True, it is insignificant. Just in the formula of a thin scattering lens, there is a minus in front of the 1 / F ratio . That is:
1 / d + 1 / f = - 1 / F.
The problem of finding magnification lenses
Condition. The focal length of the collecting lens is 0.26 m. It is required to calculate its increase if the subject is at a distance of 30 cm.
Decision. It starts with the introduction of notation and the translation of units in C. So, d = 30 cm = 0.3 m and F = 0.26 m are known . Now you need to select the formulas, the main one is that indicated for magnification, the second for a thin collecting lens.
They need to be somehow combined. To do this, you will have to consider the drawing of the image construction in the collecting lens. From these triangles it is clear that Ξ = H / h = f / d. That is, in order to find the increase, you have to calculate the ratio of the distance to the image to the distance to the subject.
The second is known. But the distance to the image is supposed to be derived from the formula indicated earlier. It turns out that
f = dF / ( d - F ).
Now these two formulas need to be combined.
Ξ = dF / ( d ( d - F )) = F / ( d - F ).
At this point, solving the problem of the thin lens formula comes down to elementary calculations. It remains to substitute the known quantities:
G = 0.26 / (0.3 - 0.26) = 0.26 / 0.04 = 6.5.
Answer: the lens gives a magnification of 6.5 times.
The task in which you need to find focus
Condition. The lamp is located one meter from the collecting lens. An image of its spiral is obtained on a screen 25 cm away from the lens. Calculate the focal length of the specified lens.
Decision. It is supposed to write the following values ββinto the data: d = 1 m and f = 25 cm = 0.25 m. This information is enough to calculate the focal length from the formula of a thin lens.
So 1 / F = 1/1 + 1 / 0.25 = 1 + 4 = 5. But in the problem you need to know the focus, and not the optical power. Therefore, it remains only to divide 1 by 5, and get the focal length:
F = 1/5 = 0.2 m.
Answer: the focal length of the collecting lens is 0.2 m.
The problem of finding the distance to the image
Condition . The candle was placed at a distance of 15 cm from the collecting lens. Its optical power is 10 diopters. The screen behind the lens is set so that it produces a clear image of the candle. What is this distance equal to?
Decision. In a short record it is supposed to write such data: d = 15 cm = 0.15 m, D = 10 diopters. The formula above should be written with a slight change. Namely, on the right-hand side of the equality, put D instead of 1 / F.
After several transformations, we obtain the following formula for the distance from the lens to the image:
f = d / ( dD - 1).
Now you need to substitute all the numbers and count. It turns out this value for f: 0.3 m
Answer: the distance from the lens to the screen is 0.3 m.
The problem of the distance between an object and its image
Condition. The object and its image are 11 cm apart. A collecting lens gives a 3-fold increase. Find her focal length.
Decision. The distance between the object and its image is conveniently indicated by the letter L = 72 cm = 0.72 m. Increase G = 3.
Two situations are possible here. First - the subject is behind the focus, that is, the image is real. In the second, an object between the focus and the lens. Then the image is on the same side as the subject, and imaginary.
Consider the first situation. The subject and image are on opposite sides of the collecting lens. Here you can write the following formula: L = d + f. The second equation is supposed to write: G = f / d. It is necessary to solve the system of these equations with two unknowns. To do this, replace L with 0.72 m, and G with 3.
From the second equation it turns out that f = 3 d. Then the first is transformed as follows: 0.72 = 4 d. From it it is easy to count d = 0, 18 (m). Now it is easy to determine f = 0.54 (m).
It remains to use the thin lens formula to calculate the focal length. F = (0.18 * 0.54) / (0.18 + 0.54) = 0.135 (m). This is the answer for the first case.
In the second situation, the image is imaginary, and the formula for L will be different: L = f - d. The second equation for the system will be the same. Reasoning in the same way, we get that d = 0, 36 (m), and f = 1,08 (m). A similar calculation of the focal length will give the following result: 0.54 (m).
Answer: the focal length of the lens is 0.135 m or 0.54 m.
Instead of a conclusion
The path of the rays in a thin lens is an important practical application of geometric optics. After all, they are used in many devices from a simple magnifier to precision microscopes and telescopes. Therefore, you need to know about them.
The derived formula of a thin lens allows us to solve many problems. Moreover, it allows you to draw conclusions about which image give different types of lenses. In this case, it is enough to know its focal length and the distance to the subject.