Figure 34-37 gives the latversus the object distance p from...

Figure $34-37$ gives the latversus the object distance $p$ from a spherical mirror as the object is moved along the mirror's central axis through a range of values for $p$. The horizontal scale is set by $p_{s}=10.0 \mathrm{~cm} .$ What is the magnification of the object when the object is $21 \mathrm{~cm}$ from the mirror?

Key Concepts

Mirror Equation

The mirror equation is a fundamental relationship in geometrical optics that relates the object distance, the image distance, and the focal length of a spherical mirror. It is expressed as 1/p + 1/q = 1/f, where p is the object distance, q is the image distance, and f is the focal length of the mirror. This equation is essential in determining one of these variables when the other two are known.

Magnification

Magnification in the context of spherical mirrors describes the ratio of the image size to the object size and also indicates image orientation. It is given by m = -q/p, where q is the image distance and p is the object distance. The negative sign indicates that the image is inverted relative to the object when the magnification is negative.

Focal Length of a Spherical Mirror

The focal length of a spherical mirror is the distance from the mirror's surface to its focal point, where parallel rays of light converge (or appear to diverge from in the case of a convex mirror). The focal length is a critical parameter in the mirror equation, as it influences both the image location and the magnification.

Sign Conventions

Sign conventions in the study of spherical mirrors are crucial for correctly applying the mirror equation and determining magnification. These conventions dictate the positive and negative assignments for object distance, image distance, and focal length based on the geometry of the mirror and the direction of the incident light, thereby affecting the interpretation of the image's orientation and relative size.

Transcript

0:00 Hi there.

00:01 So for this problem, we are given the graph of the lateral magnification of an object versus the object distance from a spherical mirror, as the object is moved along the mirror's central axis through a range of values for p.

00:21 Now, the horizontal scale is set by the following value, that is p .s.

00:29 This value right.

00:30 Here, it is 10 centimeters and what we need to obtain is what is the magnification of the object when the object is at a distance 21 centimeters away from the mirror.

00:58 Now we note in the graph that we are given that there is a singularity in this graph like which tells us that there is no point where the object distance is equal to the focal distance now since the value for the object distance is greater than zero then this means that the focal length is not positive so we know it is not a flat mirror seems the curves shown in this graph doesn't decrease with the object, with the, sorry, as you can see in the picture, the curve decreased with the object distance.

01:55 So we concluded that it is a combats mirror.

02:01 Now, we'd summing that the point at this value right here, the point at 0 .5 of magnetic, it's equal to the point in here.

02:20 Now, combining the equations that we know for the magnification, we know that for the magnification, we have that that is equal to minus the ratio between the image distance and the object distance that also can be written as the focal distance minus the focal distance over the different.

02:47 Between these two expressions.

02:51 Now, if we solve for the focal distance, we're going to solve for that, so we will have the following.

03:06 We pass this to the other side...

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