An object 4.0 mm high is placed 18 cm from a convex mirror of radius of curvature 18 cm. Show by ray tracing that the image is virtual, and estimate the image distance.

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Understand the problem: A convex mirror is given with a radius of curvature \( R = 18 \; \text{cm} \). The object height is \( h_o = 4.0 \; \text{mm} \) and the object distance is \( d_o = 18 \; \text{cm} \). We need to determine the image distance \( d_i \) and show that the image is virtual using ray tracing.

Recall the mirror equation: \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length. For a convex mirror, the focal length is related to the radius of curvature by \( f = \frac{R}{2} \). Since the mirror is convex, the focal length is positive, so \( f = +9 \; \text{cm} \).

Substitute the known values into the mirror equation: \( \frac{1}{9} = \frac{1}{18} + \frac{1}{d_i} \). Rearrange to solve for \( \frac{1}{d_i} \): \( \frac{1}{d_i} = \frac{1}{9} - \frac{1}{18} \). Simplify the fractions to find \( d_i \).

Interpret the result: The calculated \( d_i \) will be positive, indicating that the image is virtual and located on the same side as the object. Virtual images in convex mirrors are always upright and reduced in size.

To confirm this visually, perform ray tracing: Draw a convex mirror and an object in front of it. Trace at least two rays: (1) A ray parallel to the principal axis reflects as if it came from the focal point, and (2) a ray directed toward the center of curvature reflects back on itself. The reflected rays appear to diverge, and their extensions meet behind the mirror, forming a virtual image.

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주요 개념

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Convex Mirror Properties

A convex mirror is a curved mirror that bulges outward, causing light rays to diverge. This type of mirror always produces virtual images, which are upright and smaller than the object. The focal point of a convex mirror is located behind the mirror, and its radius of curvature is positive, affecting how images are formed.

Ray Tracing Technique

Ray tracing is a method used to determine the position and characteristics of an image formed by a mirror or lens. For convex mirrors, two principal rays are typically drawn: one parallel to the principal axis that reflects through the focal point, and another directed towards the focal point that reflects parallel to the axis. The intersection of these reflected rays indicates the location of the virtual image.

Image Distance and Magnification

Image distance refers to the distance from the mirror to the image formed. In the case of convex mirrors, this distance is considered negative, as virtual images appear behind the mirror. Magnification is the ratio of the height of the image to the height of the object, which helps in understanding how the size of the image relates to the object, typically resulting in a smaller, upright image for convex mirrors.