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Magnification of a Lens

Lens magnification and image orientation

How it is calculated

The tool uses the thin-lens relation m = −v / u, where v is the image distance and u is the object distance, both in the same unit (cm, mm, any). The unit cancels, so m has no unit. The sign of m gives the orientation: negative is inverted, positive is upright. The size is |m|: above 1 the image is larger than the object, below 1 it is smaller. The result is rounded to two decimals. Neither distance may be 0, since that returns no result. Negative distances are accepted and flip the sign, so follow one sign convention for both fields.

Worked example

An object is 40 cm from the lens and its image forms 12 cm away. Substitute into m = −v / u: m = −12 / 40 = −0.30, shown as -0.30x. The negative sign means the image is inverted, and since 0.30 is below 1 the image is about 30 percent of the object's size, so it is a reduced, upside-down image.

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Common mistakes

  • Mixing units, such as the object distance in metres and the image distance in centimetres. The ratio is only valid when both are in the same unit.
  • Dropping the minus sign in the formula. A real image on the far side with positive v and u gives a negative m, which means inverted, not smaller.
  • Entering the focal length instead of the image distance. This tool needs v and u; it does not compute v from f.
  • Reading -2.00x as a shrunk image. The magnitude 2 means twice the size; the minus only says it is upside down.
  • Entering 0 for either distance. There is no image or no object then, so the tool shows nothing.

About this calculator

For a thin lens, magnification is m = −v/u, where v is the image distance and u the object distance (using the real-is-positive sign convention: u is positive for a real object, and v is positive for a real image formed on the far side of the lens and negative for a virtual image on the same side as the object). The magnitude of m tells you the size ratio between image and object, while its sign tells you the orientation: negative means inverted, positive means upright.

Optics students use it to predict whether a lens produces a magnified or reduced, real or virtual image before ever setting up an experiment, and camera, telescope, and microscope designers use the same relationship to work out how a chosen lens-to-sensor or lens-to-eyepiece distance will actually magnify the final image.

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