A child holds a candy bar 15.5 cm in front of the convex side-view mirror of an automobile. The

A child holds a candy bar 15.5 cm in front of the convex...

Key Concepts

Mirror Equation

The mirror equation, given by 1/f = 1/do + 1/di, relates the focal length (f) of a spherical mirror to the object distance (do) and the image distance (di). This equation is fundamental in geometrical optics as it allows us to determine one of these quantities when the other two are known.

Magnification

Magnification (m) is defined as the ratio of the image height (hi) to the object height (ho), and for mirrors, it is given by m = -di/do. This relation not only provides the scale change between the object and image but also indicates whether the image is upright or inverted. A positive magnification means the image is upright, which is typical for virtual images in convex mirrors.

Relationship Between Focal Length and Radius of Curvature

The focal length of a spherical mirror is directly related to its radius of curvature by the formula f = R/2. This concept is crucial because once the focal length is determined from image/object distances and magnification, the radius of curvature can be readily calculated, linking the geometrical shape of the mirror to its optical properties.

Convex Mirror Sign Conventions

For convex mirrors, which produce virtual, upright, and reduced images, special sign conventions are used: the object distance is positive, while the image distance is negative. Recognizing these conventions is essential for correctly applying the mirror and magnification equations, as it ensures that the computed focal length and radius of curvature are interpreted correctly within the context of convex mirror behavior.

Transcript

00:01 In this question, we have a convex spherical mirror with a child holding a candy bar 15 .5 centimeters in front of the convex side.

00:10 The image height is told to be reduced by one half.

00:14 We would like to find the radius of curvature for this mirror.

00:20 To start, we will need to know which equations we should use.

00:26 Since we're calculating the radius of curvature, it will be helpful to have this formula that the focus of a spherical mirror is a equal to negative radius of curvature divided by 2.

00:39 Now we note that we have p, and we can probably calculate q.

00:45 If we have p, q, and f, this tells us to use the lens equation.

00:50 1 over f equals 1 over p plus 1 over q.

00:59 All that is left to do is find q.

01:04 If we find q, then we can use this equation to calculate f, and then this equation to calculate r.

01:15 Notice that i've drawn already the rays, the principal rays for this object and optical setup.

01:24 With this geometry, we see that an image will be formed virtually behind the mirror at approximately this location.

01:33 Notice that this is not drawn to scale.

01:40 Okay.

01:41 Since the image height is reduced by one half, this gives us some additional information about the system.

01:48 Notably, it's magnification.

01:50 The magnification is equal to the ratio of image heights, which we are told is one -half...

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