A concave shaving mirror has a radius of curvature of 25.0 cm . For each of the following cases,

A concave shaving mirror has a radius of curvature of 25.0...

Key Concepts

Upright vs. Inverted Image

The orientation of an image—whether upright or inverted—is determined by the way light is reflected by the mirror. Real images, formed by the actual convergence of light rays, are generally inverted relative to the object, while virtual images, which the eye perceives by tracing rays backward, remain upright.

Concave Mirror

A concave mirror is a spherical mirror that curves inward, resembling a portion of the interior of a sphere. It is capable of converging light rays to a focal point, and its properties are widely used in optical instruments such as shaving mirrors, telescopes, and headlights.

Radius of Curvature

The radius of curvature is the distance from the mirror’s surface to the center of the sphere from which the mirror segment is taken. It is a fundamental parameter in determining the mirror’s focal length and overall image-forming properties.

Focal Length

The focal length of a concave mirror is the distance from the mirror to its focal point and is determined by the radius of curvature. It is given by the relationship f = R/2, where R is the radius of curvature. This value is crucial for applying the mirror equation to predict image properties.

Mirror Equation

The mirror equation, 1/f = 1/do + 1/di, relates the object distance (do), the image distance (di), and the focal length (f) of a mirror. This fundamental formula is used to determine the position and nature (real or virtual) of the image produced by the mirror.

Magnification

Magnification in mirror optics is defined as the ratio of the image height to the object height and is also expressed as m = -di/do, where di is the image distance and do is the object distance. It provides insights into the size, orientation, and nature (upright or inverted) of the resulting image.

Real vs. Virtual Image

In mirror optics, a real image is formed when light rays actually converge at a point, allowing the image to be projected onto a screen. Conversely, a virtual image is formed when light rays appear to diverge from a point behind the mirror. Real images are typically inverted, while virtual images are usually upright.

Transcript

00:01 In this question, we've got a concave mirror with a radius of curvature of 25 centimeters.

00:08 And we are asked to determine the magnification, as well as whether the image that's formed is real, virtual upright or inverted.

00:20 And we're given a variety of image distances to use.

00:24 So first of all, we're using a concave mirror, which should be shaped something like this.

00:30 And a concave mirror is what we call a converging optic.

00:34 So what that means is that two rays that come in that are parallel to the mirror will be reflected such that they will move towards one another and eventually converge upon one another.

00:50 So this is what's called a converging optic.

00:54 And for converging optics, our focal length is always positive.

00:59 So focal length or f is always positive.

01:08 Now for this optic, we weren't given the focal length, but we can easily calculate it from the radius of curvature.

01:15 So the focal length will be r over two.

01:20 So the radius of curvature is 25 centimeters, and that means that our focal length is 12 .5 centimeters, and it's positive because this is a converging optic.

01:32 A concave mirror.

01:36 So in part a, we're asked to find the magnification of a pencil that is 45 centimeters from the mirror.

01:46 So that 45 centimeters is what we call the object distance.

01:50 It's the distance between the object and the mirror.

01:56 So that's going to be 45 centimeters.

02:00 And in the textbook that is denoted as p.

02:07 And then what we need to find is the image distance.

02:10 So the distance between the mirror and where the image is formed.

02:16 And that's going to be denoted as q.

02:20 The reason we need to find the image distance is because the formula for magnification utilizes that image distance.

02:28 So if we have focal length and object distance, we can use this formula here.

02:34 To calculate the third thing, which is the image distance.

02:39 And so we'll just rearrange this for q.

02:43 So we've got 1 over q is 1 over f minus 1 over p.

02:50 And so 1 over q will be equal to 1 over 12 .5 minus 1 over 45.

03:00 And i'm just keeping everything in centimeters here.

03:03 And that means we'll get an answer in centimeters.

03:05 So once you put that into your calculator, you're going to get 0 .055.

03:11 And just be careful because that's not q.

03:14 It's still one over q.

03:16 So we need to do one divided by this answer to get what q actually is.

03:26 So that gives a value of 17 .3 centimeters.

03:32 So once you have both the image distance and the object distance you can very easily calculate the magnification using the formula m is equal to negative q over p and magnification is quite literally you know how big does the image appear relative to its original object so subbing in what we have here we get negative 17 over 45 and that will give a magnification of negative 0 .38.

04:20 The negative here will indicate whether the image is inverted or uprights.

04:28 If it's negative, that means that the image is inverted...

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