How does the curvature affect the focal length?

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From: Christian Döllinger, Monika v. Aufschnaiter

As of March 14, 2020

First, watch the video for the following questions:

  • How does a converging lens influence an incident light beam?
  • How can one explain the term focal point for a converging lens?
  • How are the focal length and the curvature of the lens related?
  • How does the focal length of the converging lens affect the captured image?

Below you will find the answers to the questions and more information.

Collecting lenses: formulas

Lens formula: 1 / b + 1 / g = 1 / f
Image scale: B / G = b / g

Image distance b
Image size B
Focal length f
Object distance g
Item size G

The focal length of a converging lens depends on the radii of curvature r1 and r2 its spherical shell-shaped boundary surfaces and from the material (e.g. glass) from which the lens is made. The optical properties of transparent materials are determined, among other things, by their refractive index n. The refractive index is a measure of how strongly the light is refracted when air passes into the substance in question (e.g. glass) at a certain angle of incidence. For glass, the refractive index n depends on the type of glass used; the values ​​range from about 1.3 to 1.8. The focal length f of a converging lens made of a material with the refractive index n and the radii of curvature r1 and r2 is given by the relationship

1 / f = (n - 1) * (1 / r1 - 1 / r2)                

It should be noted that the radius of convex, i.e. outwardly curved lens surfaces is assigned a positive sign, while the concave surface is assigned a negative sign. In the case of a bi-convex lens, the two lens surfaces are curved differently; accordingly, the radii of curvature of such a lens have different signs.

Illustration of the four lenses to which the tasks relate.

a) For the focal length of the Bi-convex lens with n = 1.5 as well as
r1 = 10 cm and
r2 = -10 cm results from the relationship 1 / f = (n - 1) · (1 / r1 - 1 / r2):

1 / f = (1.5 - 1) · [1/10 cm - (-1/10 cm)]
1 / f = 0.5 x 2/10 cm
1 / f = 1/10 cm
f = 10 cm

Their refractive power is the reciprocal of the focal length converted into m:

f = 10 cm = 0.1 m

The refractive power of the lens is therefore 10 diopters.

b) Lens b is a Plan convex lensbecause one of the two lens surfaces is flat. The flat lens surface can be given an infinitely large radius of curvature r2 assign so that for 1 / r2 the value is 0. For the focal length of this lens we get:

1 / f = (1.5 - 1) · [1/10 cm - 0]
1 / f = 0.5 x 0.1 1 / cm
1 / f = 0.05 1 / cm
f = 20 cm

The focal length of the plan-convex lens is twice as large as that of the bi-convex lens with the same radii of curvature. Their refractive power is therefore only half as great (5 diopters)

c) This lens has the same focal length as lens b) because it is the same lens; It does not matter which of the two lens surfaces the light hits, it is deflected by the lens in the same way

d) With this glass body, both boundary surfaces have the same radius of curvature because of the matching curvature, so r applies1 = r2 = 10 cm and thus

1 / r1 - 1 / r2 = 1/10 cm - 1/10 cm = 0, that is, the focal length of this curved glass body is equal to 0, so it has no collecting (or dispersing) effect on incident light; the light therefore passes through the glass body without changing direction, but is shifted in parallel - as with a plane-parallel plate.

This sounds paradoxical at first - and yet it is possible: You can make a converging lens from clear ice if you put distilled water in a vessel that consists of two spherical shell-shaped parts and place the entire arrangement in a freezer compartment. To remove the lenticular ice body, the vessel must be warmed up slightly. The lens made of ice in this way acts like a glass lens. If it has a sufficiently large diameter, the ice lens can concentrate enough sunlight in the focal point. So much that - if the focal length is not too great - newspaper that is in the focal point of the ice lens actually ignites.

To define the image with a converging lens, one must know the course of at least two rays: where the two marked rays intersect after passing through the lens, all other rays of the incident light beam emanating from a certain object point also run through this image point. For 3 rays, the so-called main rays, the further ray path can be specified: the Parallel beam - it runs behind the lens through the focal point and thus becomes one Focus beam, the Center ray remains center ray: a beam of light passing through the focal point of the lens, after passing through the lens, parallel to the optical axis.

How a lens with a longer focal length works

With a telephoto lens, the viewer sees objects that are far away sharply. On the one hand, this works because the focal length of the lens in the telephoto lens is larger. On the other hand, the distance between the lens and the image is greater. The larger this is, the larger the image width. This is why the telephoto lens of a camera is longer than a normal lens, with which one can take sharp pictures of close objects.