Convex Mirrors: Videos & Practice Problems
Convex Mirrors are mirrors that bulge outward and make reflected light diverge. In a ray diagram, the parallel ray reflects away from the focal point, the F-ray reflects parallel to the axis, and the center-of-curvature ray reflects back on itself. Because the reflected rays spread out, the image is found by tracing them backward behind the mirror.
A convex mirror always forms the same kind of image: virtual, upright, and reduced. The image appears behind the mirror and lies between the mirror and the focal point. The same mirror equation is used as for other spherical mirrors, \( \frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{f} \) , with a negative focal length for convex mirrors. The focal length and radius of curvature are related by \( f=\frac{1}{2}R \) , so the radius is also negative by the same sign convention. Magnification connects size and position through \( \frac{h_i}{h_o}=-\frac{d_i}{d_o} \) .
Ray Diagrams for Convex Mirrors
The sideview mirrors on the sides of vehicles are convex mirrors. If a car is from your passenger-side mirror, but produces an image behind the mirror, what is its radius of curvature?
Ray Diagrams for Convex Mirrors Example 1
Ray Diagrams for Convex Mirrors Example 2
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Convex mirrors always produce images with specific characteristics. The image formed is virtual, meaning the reflected rays diverge and the image appears to be behind the mirror. It is upright, so the image maintains the same orientation as the object. Additionally, the image is reduced in size compared to the actual object. This occurs because the reflected rays spread out, and when traced backward, they converge at a point between the mirror and the focal point behind the mirror. These properties make convex mirrors useful in applications like vehicle side mirrors and security mirrors, where a wide field of view is needed but the image size is smaller.
To draw ray diagrams for convex mirrors, use the three principal rays: the parallel ray (P-ray), the focal ray (F-ray), and the center of curvature ray (C-ray). The P-ray travels parallel to the principal axis and reflects as if it came from the focal point behind the mirror. The F-ray is drawn toward the focal point behind the mirror but reflects parallel to the principal axis. The C-ray heads toward the center of curvature behind the mirror and reflects back on itself. Since the reflected rays diverge, the image is located by tracing these rays backward behind the mirror, where they appear to converge. This method helps visualize the virtual, upright, and reduced image formed by convex mirrors.
The mirror equation for convex mirrors is the same as for other spherical mirrors: , where is the object distance, is the image distance, and is the focal length. For convex mirrors, the focal length is negative because the focal point is behind the mirror. The radius of curvature is related to the focal length by , and is also negative for convex mirrors. By substituting known values and using the correct sign conventions, you can solve for the image distance and determine the image's position.
The focal length of a convex mirror is negative because the focal point lies behind the mirror, opposite the side where the object and incoming light rays are located. In the sign convention used for spherical mirrors, distances measured in the direction of the incoming light (toward the mirror) are positive, while distances measured opposite to this direction (behind the mirror) are negative. Since the focal point of a convex mirror is virtual and located behind the mirror, its focal length is assigned a negative value. This negative focal length affects calculations in the mirror equation and helps distinguish convex mirrors from concave mirrors, which have positive focal lengths.
Magnification in convex mirrors is calculated using the formula , where and are the heights of the image and object respectively, and and are the image and object distances. Since the image formed by a convex mirror is virtual and reduced, the magnification is positive but less than one, indicating the image is upright and smaller than the object. This formula helps quantify how much smaller the image appears compared to the actual object.
