A dentist uses a curved mirror to view teeth on the upper side of the mouth. Suppose she wants a

A dentist uses a curved mirror to view teeth on the upper...

Key Concepts

Ray Diagrams

Ray diagrams are graphical tools in optics used to predict and analyze the behavior of light as it interacts with mirrors or lenses. They involve drawing specific rays—such as those parallel to the principal axis and those passing through or directed towards the focal point—to determine the location, orientation, and size of the resulting image. Ray diagrams serve as a powerful visual aid in understanding the geometric relationships in image formation.

Concave Mirrors

Concave mirrors are spherical mirrors that converge incoming light to a focal point. They have the unique ability to produce both real and virtual images depending on the object's position relative to their focal length. When the object is positioned closer to the mirror than the focal point, a concave mirror forms an upright, virtual, and magnified image, which is essential in applications where enlarged views are desired.

Virtual Image Formation

Virtual images are formed when the outgoing rays from a mirror or lens appear to diverge from a point behind the mirror. In the context of curved mirrors, placing an object within the focal length of a concave mirror results in an image that cannot be projected on a screen because the light rays do not actually converge but only appear to do so. This property is critical when an upright and magnified view is necessary.

Mirror Equation

The mirror equation, 1/f = 1/do + 1/di, where f is the focal length, do is the object distance, and di is the image distance, is fundamental in geometrical optics. This relationship helps in determining the critical parameters that govern the image formation by spherical mirrors, including the position and size of the image. It offers a quantitative way to predict how changes in the object distance and focal length affect the image.

Magnification

Magnification in mirror systems is defined by the ratio m = -di/do, where m represents the magnification factor, di is the image distance, and do is the object distance. The sign of the magnification indicates the orientation of the image: a positive value corresponds to an upright image, while a negative value indicates an inverted image. Understanding magnification is essential to ensure that the resulting image meets the desired size and orientation.

Focal Length and Radius of Curvature

The focal length of a spherical mirror is directly related to its radius of curvature through the relation f = R/2. This connection allows one to easily determine one parameter if the other is known. In the design and analysis of optical instruments, knowing the focal length is imperative as it influences the image formation characteristics such as size, distance, and clarity of the image produced.

Transcript

00:02 In the first part of the given problem, the dentist is using a spherical mirror to get magnified and still erect image of the tooth.

00:35 So it is clear that that mirror, that is spherical mirror, should be on cave mirror only.

01:02 Because convex spherical mirrors although they make a direct image but that image is always inverted sorry that image is always diminished in size means smaller in size than that of the object means a convex spherical mirror cannot form a magnified image.

02:03 So the dentist should use a spherical, a concave spherical mirror only.

02:09 Now, in the second part of the problem, the magnification, the lateral magnification to be obtained by the dentist is plus 2 .0 as the image is erect.

02:20 The distance of the teeth means the object distance is given as s is equal to my mind.

02:27 1 .25 centimeter here negative sign as per the sign convention and the sign convention which we are using to solve such problems is that all the distances measured in front of the mirror will be taken as negative and all other distances measured behind the mirror are taken as positive so as the object is kept in front of the mirror that's why it will be taken as negative now using the expression for lateral magnification in terms of focal length and object distance m is equal to f by f minus s there s is the object distance and f is the focal length so plugging in all known values for m this is 2 .0 is equal to focal length which is missing divided by f minus minus 1 .25 means it becomes f f upon f plus 1 .25.

03:27 So finally, making a cross multiplication, we get 2f plus 2 .5 is equal to f.

03:37 Or finally, this f is equal to minus 2 .5 centimeter.

03:45 F focal length of the concave mirror and its radius of curvature r as radius of curvature is twice of the focal length so this is r is equal to two times of 2 .5 which comes out to be 5 .0 centimeter taking its magnitude only ignoring the negative sign associated with it this is the answer for the second part of this problem now in the third part of the problem we have to show this by array tracing so, first of all, we draw a principal axis.

04:30 Then over this principal axis we draw a concave spherical mirror...

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