A dedicated sports-car enthusiast polishes the inside and...

A dedicated sports-car enthusiast polishes the inside and outside surfaces of a hubcap that is a section of a sphere. When he looks into one side of the hubcap, he sees an image of his face $30.0 \mathrm{cm}$ behind the hubcap. He then turns the hubcap over and sees another image of his face $10.0 \mathrm{cm}$ behind the hubcap. a. How far is his face from the hubcap? b. What is the radius of curvature of the hubcap? c. What is the magnification for each image? d. Are the images real or virtual? e. Are the images upright or inverted?

A dedicated sports-car enthusiast polishes the inside and outside surfaces of a hubcap that is a section of a sphere. When he looks into one side of the hubcap, he sees an image of his face $30.0 \mathrm{cm}$ behind the hubcap. He then turns the hubcap over and sees another image of his face $10.0 \mathrm{cm}$ behind the hubcap.
a. How far is his face from the hubcap?
b. What is the radius of curvature of the hubcap?
c. What is the magnification for each image?
d. Are the images real or virtual?
e. Are the images upright or inverted?

Key Concepts

Mirror Equation

The mirror equation, expressed as 1/f = 1/d_o + 1/d_i, relates the object distance d_o, image distance d_i, and the focal length f of a spherical mirror. This equation is fundamental in geometrical optics to determine where an image will form and its characteristics when an object is placed in front of a mirror.

Focal Length and Radius of Curvature

The focal length of a spherical mirror is directly related to its radius of curvature by the relation f = R/2. This relationship helps in linking the physical shape of the mirror to its optical properties, determining how strongly it converges or diverges light.

Magnification

Magnification in mirror systems is given by m = -d_i/d_o, which indicates the ratio of the image height to the object height and also provides information about the orientation of the image. A negative magnification signifies an inverted image, while a positive magnification indicates an upright image.

Real Versus Virtual Images

In optics, real images are formed when light rays physically converge and can be projected on a screen, whereas virtual images occur when the rays appear to diverge from a point behind the mirror and cannot be projected. The classification is essential to understand the nature of the image produced by reflecting surfaces.

Upright Versus Inverted Images

The orientation of an image—whether it is upright or inverted—is determined by the magnification and the geometry of the mirror. In mirror systems, an inverted image (which is indicated by a negative magnification) typically corresponds to a real image, while an upright image (positive magnification) usually characterizes a virtual image.

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