# Determining the focal length of a concave mirror by locating centre of curvature

Discussion introduction Curved mirrors come in two basic types: those that converge parallel incident rays of light and those that diverge parallel incident rays of light. Now, imagine a mirror with the opposite curvature — one where the reflecting surface is on the "outside", like looking into a spoon that's been flipped upside down from its useful orientation, a convex mirror.

Imagine a set of rays parallel to the principal axis incident on a spherical mirror paraxial rays as they are sometimes called.

### Plane mirror equation

Imagine a set of rays parallel to the principal axis incident on a spherical mirror paraxial rays as they are sometimes called. In the demonstration, the pencil is placed at the focal point and the concave mirror is pointed upwards towards the sun. Why just read about it and when you could be interacting with it? Note also that the image height is a positive value, meaning an upright image. Remember to convert all distances to metres! Interact - that's exactly what you do when you use one of The Physics Classroom's Interactives. I won't try this proof. The distance from the vertex to the center of curvature is known as the radius of curvature represented by R. Discussion introduction Curved mirrors come in two basic types: those that converge parallel incident rays of light and those that diverge parallel incident rays of light. It's not until we encounter situations requiring extreme precision that we'll deal with this aberration as it is literally called. Extend it to infinity in both directions. The vergence is negative so the rays are diverging thus forming only a virtual image. Rays of light parallel to the principal axis of a concave mirror will appear to converge on a point in front of the mirror somewhere between the mirror's pole and its center of curvature. Calculate the final vergence. If you've mastered this then there are further problems of a similar nature in the on-line examples.

This diagram shows the parallel, central and radial rays. If you've mastered this then there are further problems of a similar nature in the on-line examples. Real telescopes are made with parabolic or hyperbolic mirrors, but as I said earlier, we'll deal with this later.

Interact - that's exactly what you do when you use one of The Physics Classroom's Interactives. Perhaps you would like to take some time to try the problems in the Check Your Understanding section below.

### How to find the focal point of a concave mirror

Start by tracing a line from the center of curvature of the sphere through the geometric center of the spherical cap. Do the numbers make sense?? Calculate the final vergence. Any line through the center of curvature of a sphere is an axis of symmetry for the sphere, but only one of these is a line of symmetry for the spherical cap. The focal length of a spherical mirror is then approximately half its radius of curvature. The distance from the pole to the center of curvature is called no surprise, I hope the radius of curvature r. Let's now talk about how they're used.

Real objects and images have positive distances while virtual objects and images have negative distances. Again, a negative or positive sign in front of the numerical value for a physical quantity represents information about direction.

## How to find radius of curvature of a mirror

For many mundane applications, it's close enough to the truth that we won't care. To determine the image distance, the mirror equation will have to be used. If the mirror causes the light to converge then a real image is formed. We will assume it to be exactly true until becomes a problem. Next identify the unknown quantities that you wish to solve for. The centre of curvature C is the centre of the circle sphere of which the mirror is an arc. A common Physics demonstration involves using a large demonstration mirror to set a pencil aflame in a matter of seconds. In the animation the first two rays from the object are examples of the first two characteristic rays described above. Whatever rays of light from the sun that hit the mirror are focused at the point where the pencil is located.

Draw a diagram like above except with the mirror curved in the other direction. Calculate the final vergence. Rays passing through the focal point are reflected parallel to the principal axis - concave for convex mirrors a ray that would have passed through the focal point is reflected parallel to the axis.

Note however that when working with vergences we treat object and image distances as just positive numbers - there are no signs associated with these values.

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