The concave mirror equation

Before deriving the equation of the concave mirror, first read some sign rules for the concave mirror below.

The sign rules for the concave mirror

Object distance (do)

If the object is in the front of the mirror surface which reflects light, the object distance (do) is positive.

Image distance (di)

If the image is in the front of the mirror surface which reflecting light, where light passes through the image, then the image distance (di) is positive (real image). If the image is behind the mirror surface which reflecting light, where light does not pass through the image, then the image distance is negative (virtual image).

The radius of curvature (R)

The center of the curvature of the concave mirror is in the front of the mirror surface which reflects light. Therefore, the radius of curvature of the concave mirror is positive. The radius of curvature is positive, so the focal length (f) is also positive.

Object height (h)

If the object is above the principal axis of the concave mirror, the height of the object (h) is positive (object is upright). Conversely, if the object is below the principal axis of the concave mirror, the height of the object is negative (object is inverted).

Image height (h’)

If the image is above the principal axis of the concave mirror, the image height (h ‘) is positive (image is upright). If the image is below the principal axis of the concave mirror, the image height is negative (image is inverted).

Magnification of image (m)

If the magnification of image > 1 then the size of the image is greater than the size of the object. If the magnification of image = 1, the size of the image is the same as the size of the object. If the magnification of the image is < 1, the image size is smaller than the object size.

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The equation of concave mirror

On the figure below, two beams of light are drawn to the concave mirror, and the concave mirror reflects the beam of light.

Equation of concave mirror 1

s = do = object distance, s’ = di = image distance, h = P P’ = object height, h’ = Q Q’ = image height, F = the focal point of the concave mirror.

The figure above shows two rays, P’BFQ ‘and P’AQ’. The P’AQ ray fulfills the law of reflection of light. Hence the P’AP triangle is similar to Q’AQ. Therefore :

Equation of concave mirror 2

On the P’BFQ ’ ray, the BFA triangle is similar to QFQ’ where the distance of AB = object height (h) and the distance of FA = the focal length (f) of the concave mirror. Therefore :

Equation of concave mirror 3

The left and right of the equations 1 and 2 are the same, so the right equation is equalized:

Equation of concave mirror 4

Multiply the two equations by the image distance (di):

Equation of concave mirror 5

do = the object distance (positive if the object is in the front of the surface of the concave mirror that reflects light)

di = the image distance (positive if the image in the front of the surface of the concave mirror that reflects light. Image is real)

f = focal length (positive because the focal point of the concave mirror is located in the front of the surface of the concave mirror that reflects light)

Always remember the sign rules of the concave mirror when using this equation to solve the problems of the concave mirror

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Magnification of image (m)

Observe the figure of the image formation above. Similar to the PAP ‘and QAQ’ triangles, we can reduce the relationship between the object distance and the image distance to the object height and the image height:

Equation of concave mirror 6

h = object height (positive if the object is above the principal axis of the concave mirror. The object is upright. Negative if the object is inverted)

h’ = image height (positive if the image is above the principal axis of the concave mirror. The image is upright. Negative when the image is inverted)

do = object distance (positive if the light beam pass through the object)

di = image distance (positive if the light beam passes through the image. The image is real. Negative if the light beam passes through the image. The image is virtual).

The equation above can be written again as below by adding the symbol of m:

Equation of concave mirror 7

m = the magnification of the image

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