An object is placed 50.0 cm from a concave spherical mirror with focal length of magnitude 20.0

An object is placed 50.0 cm from a concave spherical mirror...

Key Concepts

Image Orientation

Image orientation refers to whether the image produced by a mirror is upright or inverted relative to the object. In spherical mirror problems, this is determined using calculations from the mirror equation and the magnification formula. If the magnification is negative, the image is inverted, while a positive magnification means the image is upright. Understanding the orientation helps in visualizing the image structure in optical setups.

Real vs Virtual Images

The distinction between real and virtual images is a key concept in optics. A real image is formed when light rays actually converge at a point after reflection or refraction, and it can be projected onto a screen. In contrast, a virtual image is formed when light rays appear to diverge from a point behind the mirror or lens; such images cannot be projected. This concept is critical when analyzing how mirrors form images under different conditions.

Sign Conventions

Sign conventions in optics are essential for correctly applying equations, such as the mirror equation. They determine the positive and negative directions for object distances, image distances, and focal lengths. For example, in many conventions, distances measured in the direction of the incoming light are taken as positive, and for concave mirrors, the focal length is typically positive. Proper use of these conventions ensures accurate prediction of image properties.

Focal Length

Focal length is the distance between the mirror’s surface and its focal point, where parallel rays incident on the mirror converge (or appear to diverge from) after reflection. In the context of spherical mirrors, the magnitude and sign of the focal length are crucial in determining the type of image formed. For a concave mirror, the focal length is positive when using the appropriate sign convention.

Mirror Equation

The mirror equation is a fundamental relation in geometric optics that relates the object distance, the image distance, and the focal length of a mirror. It is usually written as 1/f = 1/do + 1/di, where f is the focal length, do is the distance from the object to the mirror, and di is the distance from the image to the mirror. This equation is used to predict where an image will form and is central to solving problems involving spherical mirrors.

Magnification

Magnification in mirror optics quantifies the ratio of the image height to the object height, and is given by the formula m = -di/do, where di is the image distance and do is the object distance. The negative sign indicates the inversion of the image relative to the object. Magnification provides insight into the size and orientation of the image formed by the mirror.

Transcript

00:01 An object is placed 50 centimeters in front of a concave mirror, and we're asked about the location, the magnification, and the orientation of our image.

00:13 So to solve these optics problems, we want to first draw out our diagram, identify the components, and orient ourselves to solve the problem.

00:25 So what we have here is we've drawn out our diagram, our ray diagram, and this point, bar here is our principal axis.

00:35 This is our concave mirror.

00:38 We have our object located here, and we have our image formed by the rays.

00:44 So what i'll do is i'll label this our object.

00:48 I'll label this m for our concave mirror.

00:54 I'll label this as i for our image.

00:58 And you can see here that once we know where our object is we can draw our rays, one ray at least being parallel to the principal axis and then once it hits the mirror it reflects toward this point which is the focus point i'll label that here as f and then once that ray passes the focus point it keeps going down and our second one and our second ray at least we have to have is the opposite we have the ray going to our focus point hitting the the mirror and then turning out and reflecting parallel to the principal axis.

01:48 And once you intersect those two rays, you can form your image.

01:52 So i'll just label the first one, the first ray, as one, and our second ray is two.

02:06 I'll put these arrows here to indicate the direction.

02:17 So we have drawn out our ray diagram.

02:21 Now, in this textbook, we're supposed to have anything to the front of our mirror be positive.

02:30 And anything to the back of our mirror be negative.

02:33 What does that mean? it means that, well, we can have our mirror.

02:38 Imagine if we're having our mirror set at a number line.

02:43 Our mirror would be zero.

02:45 And the length of the object and the image and the focal length and that occurs to the left of the mirror is going to be positive.

02:59 So just for an example, i'll have this number line here.

03:03 And this is our positive direction.

03:09 Anything you measure on this side of the mirror will be positive.

03:13 Anything on the other side is going to be negative according to the conventions on this textbook.

03:26 So sometimes when you do these problems it will be in reverse, but for the rest of this textbook this is how we should do our orientation.

03:37 Positive will be on the left and negative will always be on the right.

03:49 So now that we oriented ourselves, let's also indicate our object length or our length and also try to solve this problem using the mirror equation.

04:09 So after we draw, okay, so now we just have to use mirrors equation.

04:46 So what is our mirrors equation? the mirror equation is 1 over p plus 1 over q equals 1 over f.

05:05 Where p is our object length, q is our image length, and f is our focal length.

05:09 So we are given that p is 50, since it's 50 centimeters in front of our mirror, and since it's to the front, since it's to the left of our mirror, or to the front of our mirror, we're going to give it a positive length.

05:29 We're given our focal length, and the focal length is 20 centimeters...

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