Mirror Equation Calculator

Geometrical optics relies on the standard mirror equation to relate the focal length (f), object distance from the mirror (u), and image distance from the mirror (v). For spherical mirrors, the foundational formula is expressed as:

1 / f = (1 / u) + (1 / v)

Additionally, the radius of curvature (r) is always twice the focal length, defined as r = 2f. Magnification describes how much larger or smaller an image appears compared to the original object. Linear magnification is calculated using the ratio of image distance to object distance: m = -(v / u). For two-dimensional surface areas, areal magnification is found by squaring the linear magnification ratio: m_area = (v / u)^2. Plane mirrors represent a special case where the focal length approaches infinity, resulting in an image distance equal in magnitude but opposite in sign to the object distance (v = -u).

Let us walk through a practical physics problem using a concave mirror. Suppose we have a concave mirror with a focal length (f) of 15 cm. An object is placed at a distance (u) of 45 cm in front of the mirror. We want to find the image distance (v) and the linear magnification.

1. Start with the mirror equation: 1 / f = (1 / u) + (1 / v)

2. Substitute our known values: 1 / 15 = (1 / 45) + (1 / v)

3. Isolate the image distance term: 1 / v = (1 / 15) - (1 / 45)

4. Find a common denominator: 1 / v = (3 / 45) - (1 / 45) = 2 / 45

5. Solve for v by taking the reciprocal: v = 45 / 2 = 22.5 cm. The positive sign indicates a real, inverted image formed on the same side as the object.

6. Calculate linear magnification: m = -(v / u) = -(22.5 / 45) = -0.5. This means the image is real, inverted, and reduced to half the size of the object.

Mastering sign conventions is crucial for optical calculations. By standard modern convention (Cartesian sign convention), distances measured in the direction of incoming light are negative, while distances measured against incoming light are positive. For real objects placed in front of any standard mirror, the object distance (u) is always negative. Always verify your mirror type before running calculations, as concave mirrors can yield both real and virtual images, whereas convex mirrors exclusively produce virtual, diminished images behind the reflective surface.