Equation of diverging (concave) lens

Before deriving the equation of the concave lens, first understood the sign rules of the concave lens.

Sign rules of the concave lens

The following are the sign rules of the concave lens.

The object distance (do)

If the object is on the side of the lens that is the same as the direction of the beam of light, then the object distance is positive.

The image distance (di)

If a beam of light passes the image, then the image distance is positive (real image). If the image does not pass through the beam of light, the image distance is negative (virtual image).

The focal length (f)

If the focal point of the lens is passed through a beam of light, the focal length of the lens is positive. Conversely, if the lens’s focal point is not passed by light, the lens’s focal length is negative. The focal point of the concave lens is not passed by light, so the focal length of the concave lens is negative.

The height of the object (ho)

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

The height of the image (hi)

If the image is above the principal axis, the image height is positive (image is upright). If the image is below the principal axis, the image height is negative (image is inverted).

The magnification of image (m)

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

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The equation of the concave lens

Based on the figure below, two beams of light are drawn towards the concave lens, and the concave lens refracts the light beam.

Equation of diverging (concave) lens 1

s = do = the object distance, s’ = di = the image distance, h = P P’ = the height of object, h’ = Q Q’ = the height of image, F1 and F2 = the focal point of the concave lens.

The P’AP triangle is similar to the Q’AQ triangle. Therefore :

Equation of diverging (concave) lens 2

The BF2A triangle is similar to the Q’F2Q triangle where the distance of AB = the height of the object (h) and the distance of F2A = the focal length (f) of the concave lens. Therefore :

Equation of diverging (concave) lens 3

Equation of diverging (concave) lens 4

Based on the the the sign rules of the concave lens, this equation can be changed to like the equation of curved mirror, if the image distance (di) is given a negative sign because the beam of light does not pass the image and the focal length (f) is also given a negative sign because the focal point of the concave lens is not passed by light (compare with the figure of the image formation above). According to this statement, the equation of the concave lens changes to:

Equation of diverging (concave) lens 5

do = the object distance, di = the image distance, f = the focal length

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The magnification of image (m)

Observe the figure of the image formation above. The P’AP and Q’AQ triangles are similar so that we can derive the relationship between the object distance and the image distance with the object height and the image height:

Equation of diverging (concave) lens 6

This equation is written again as below by adding m:

Equation of diverging (concave) lens 7

m = the magnification of the image

ho = the object height (positive if it is above the principal axis or the object is upright)

hi = the image height (positive if it is above the principal axis or the image is upright)

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

di = the image distance (positive if the beam of light pass through the image or image is real)

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