A dentist uses a curved mirror to view the back side of...

A dentist uses a curved mirror to view the back side of teeth on the upper jaw. Suppose she wants an erect image with a magnification of 2.0 when the mirror is $1.2 \mathrm{cm}$ from a tooth. (Treat this problem as though the object and image lie along a straight line.) Use ray tracing to decide whether a converging or diverging mirror is needed, and to estimate its focal length.

Key Concepts

Magnification

Magnification in mirror problems describes the ratio of the image size to the object size, and it can be given by the relation m = -di/do. The sign and the magnitude of m indicate the orientation (inverted or erect) and the size of the image relative to the object. A magnification greater than one means the image is larger than the object.

Mirror Equation

This is the fundamental relation in geometrical optics that connects the object distance, image distance, and the focal length of a mirror (or lens). It is typically written as 1/f = 1/do + 1/di, where f is the focal length, do is the object distance, and di is the image distance. It is used to determine where an image will form for a given mirror and object placement.

Ray Tracing

Ray tracing is a graphical method used to determine the path of light rays reflected by mirrors. It involves drawing key rays from a point on the object to determine where they converge (or appear to diverge from) after reflection. This method helps in predicting the nature (real or virtual), orientation, and size of the image.

Converging vs Diverging Mirrors

Converging mirrors (concave mirrors) and diverging mirrors (convex mirrors) form images in different ways. A converging mirror can produce a virtual, erect, and magnified image when the object is placed within the focal length, while diverging mirrors typically produce virtual and diminished images. Understanding which type is used depends on the desired image characteristics such as size and orientation.

Sign Conventions

In mirror problems, sign conventions are crucial for correctly interpreting distances and image orientation. Distances measured in the same direction as incident light are typically positive, and those measured in the opposite direction are negative. The sign of the focal length also distinguishes between converging (positive focal length) and diverging (negative focal length) mirrors.

Transcript

00:01 This question is about a dentist using a curved mirror to view the back side of the teeth.

00:08 So she wants to see an erect image with a magnification of 2 .0 when the mirror is 1 .2 cm from the tooth.

00:18 Okay, we want to find out through ray tracing and see whether converging or diverging mirror is needed and estimate is focal length.

00:29 Okay, so one thing we need to note is that diverging mirror always produce smaller image.

00:40 So for magnified images, only converging mirror can do that.

00:48 So just to note, only converging mirror can produce magnified.

01:06 Image okay so which means that can only be done using converging mirror in for this situation okay and so you'll be looking at the rules for commogee mirror by when it comes to drawing the rays and if you set up the problem problem using a proper greed and scale this is how the ray diagram looks like okay so there are three rays that we draw right so ray number one is here no ray number one is the blue one okay the light blue one here the ray parallel to the axis reflects through the focal point and then ray number two is this dark blue one here an incoming ray that passes through the focal point k reflects back parallel to the axis and then the third ray is a ray that strikes the center of the mirror reflects at an equal angle on the opposite opposite of the optical axis.

02:13 So you can see that the three rays that are reflected of the mirror.

02:17 They don't converge to a point on the left side.

02:20 So you need to extend the line back.

02:23 Okay.

02:24 Until you see, you get a point that the three rays converge.

02:32 Okay...

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