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Mirror Equation Calculator

Determine unknown optical properties of curved and plane mirrors using the fundamental mirror equation. Whether you're solving for object or image position, focal length, radius of curvature, or magnification, this tool handles all three mirror types with automatic Cartesian sign convention application. Essential for optics students and practitioners working with reflective optical systems.

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Creators Steven Wooding, BSc Physics
3,993 people find this calculator helpful

The mirror equation relates three fundamental quantities in spherical mirror optics. For both concave and convex mirrors, the relationship between object distance, image distance, and focal length follows a single mathematical form:

1/f = 1/u + 1/v

r = 2f

m_linear = −(v/u)

m_areal = (v/u)²

  • f — Focal length of the mirror, measured from the pole to the principal focus
  • u — Object distance from the mirror's pole; negative for real objects in standard convention
  • v — Image distance from the pole; sign indicates image position relative to the mirror
  • r — Radius of curvature, the radius of the sphere from which the mirror section is cut
  • m_linear — Linear magnification, the ratio of image height to object height
  • m_areal — Areal magnification, the ratio of image area to object area

Understanding Mirror Types and Sign Conventions

Mirrors follow the Cartesian sign convention to maintain consistency across calculations. For concave mirrors, the focal length and radius of curvature are negative, placing the optical centre ahead of the reflecting surface. Real objects always have negative distance values. Image distance remains negative for real images (in front of the mirror) and positive for virtual images (behind the mirror).

With convex mirrors, focal length and radius of curvature become positive. Object distance stays negative, but image distance is always positive because the virtual image forms behind the curved surface. Convex mirrors cannot produce real images—reflected rays diverge and appear to originate from a point behind the mirror.

Plane mirrors represent a limiting case where both focal length and radius of curvature approach infinity. The image distance equals the object distance in magnitude but opposite in sign (v = −u), meaning the image appears as far behind the mirror as the object sits in front. Linear and areal magnifications both equal unity, so the image matches the object's size exactly.

Magnification in Mirror Systems

Linear magnification describes the scaling factor between object and image heights. A magnification of −2 means the image is twice as tall but inverted relative to the object. The negative sign indicates image inversion for real images formed by concave mirrors; virtual images (including all convex mirror and plane mirror images) show positive magnification values.

Areal magnification represents the square of linear magnification and directly compares the surface areas. If linear magnification is −0.5, areal magnification becomes 0.25, meaning the image occupies one-quarter the area of the original object. This relationship applies universally across mirror types and holds even when magnification values are fractional or exceed unity.

The magnification equations reveal a critical insight: both linear and areal magnification depend solely on the ratio of image to object distances, making them independent of the mirror's focal length once those distances are established.

Common Pitfalls and Practical Considerations

Mastering mirror calculations requires awareness of these frequent errors and real-world constraints.

  1. Sign convention errors dominate mistakes — The Cartesian convention assigns negative values to object distances and focal lengths of concave mirrors. Forgetting the negative sign on object distance or mixing conventions between mirror types leads to completely incorrect image positions. Always verify your sign assignments before substituting into the mirror equation.
  2. Confusing focal length with radius of curvature — The relationship f = r/2 is exact and non-negotiable. Many students incorrectly assume they're identical. Since radius of curvature is twice the focal length, a mirror with r = 20 cm has f = 10 cm. Using the wrong value cascades errors through all subsequent calculations.
  3. Virtual images behind the mirror are perfectly real calculations — Convex mirrors always produce virtual images, yet the mathematics applies identically. A positive image distance simply means the image sits behind the reflecting surface. Virtual images cannot be projected onto a screen, but the equations describing their position, size, and magnification remain mathematically rigorous.
  4. Plane mirror focal length isn't a number—it's a concept — Treating plane mirror focal length as 'very large' rather than truly infinite can introduce computational errors. Plane mirrors are mathematically idealized as sections of spheres with infinite radius. Their magnification is always exactly 1.0, not approximately 1.0, because the physics prevents convergence or divergence of reflected rays.

Image Formation in Curved Mirrors

Concave mirrors exhibit diverse image behavior depending on object placement. Place an object beyond the centre of curvature (2f), and a real, inverted, diminished image forms between f and 2f. Position the object between f and the mirror, and a virtual, upright, magnified image appears behind the reflecting surface. At the focal point itself, reflected rays become parallel and form no image. These transitions underpin optical devices from telescopes to makeup mirrors.

Convex mirrors maintain consistent behavior: regardless of object location, the image always appears virtual, upright, and smaller than the object, positioned between the pole and the focal point behind the mirror. This predictability makes convex mirrors ideal for safety applications, rearview mirrors, and security monitoring where a wide field of view is essential.

Plane mirrors produce images that are always virtual, upright, and life-sized, located symmetrically behind the mirror. The perpendicular distance from object to mirror plane equals the perpendicular distance from mirror to image, with no magnification or reduction involved.

Frequently Asked Questions

How does Cartesian sign convention apply differently to concave and convex mirrors?

Cartesian convention treats mirror type as defining which quantities are positive or negative. For concave mirrors, focal length and radius are negative values, while object distance is conventionally negative for real objects. For convex mirrors, focal length and radius become positive, but object distance remains negative. This systematic approach ensures the mirror equation produces correct image positions without requiring separate formulas. The key difference lies in what emerges from the calculation: concave mirrors can yield either positive image distances (virtual images behind) or negative image distances (real images in front), while convex mirrors always produce positive image distances for all object positions.

Why does a convex mirror always form virtual images?

Convex mirror geometry inherently diverges incident light rays outward. When reflected rays cannot physically intersect in front of the mirror, they must diverge as if originating from a point behind the surface. This point of apparent origin is the virtual image. Real images require actual convergence of light rays at a physical location where a screen could display the image. The mathematics confirms this physics: for any object distance placed before a convex mirror, the mirror equation always yields a positive image distance, placing the image behind the reflecting surface where light rays don't actually converge. This makes convex mirrors ideal for applications requiring broad viewing angles without the ability to project real images.

What is the relationship between focal length and radius of curvature?

Focal length equals half the radius of curvature in all spherical mirrors: f = r/2. This relationship emerges from the geometry of spherical surfaces and the principles of reflection. When parallel light rays strike a spherical mirror, they converge (or appear to converge) at the focal point, which sits halfway between the pole and the centre of curvature. Since the centre of curvature is located at distance r from the pole, the focal point must be at distance r/2. Therefore, a mirror with radius of curvature of 30 cm has a focal length of 15 cm, regardless of whether it's concave or convex.

How do linear and areal magnifications differ in their meaning?

Linear magnification measures height scaling between object and image, expressing how many times taller or shorter the image appears. A linear magnification of −2 indicates the image is twice as tall and inverted. Areal magnification compares total surface areas and always equals the square of linear magnification. If linear magnification is −2, areal magnification is 4, meaning the image occupies four times the area. This squared relationship is crucial for applications involving energy density or light intensity distribution: if magnification stretches an image to twice the height and twice the width, the area increases by a factor of four, concentrating or dispersing incident light accordingly.

Can the mirror equation apply to plane mirrors, and if so, how?

Yes, the mirror equation applies universally to plane mirrors as a limiting case where focal length approaches infinity. Substituting infinity into the equation: 1/∞ = 1/u + 1/v becomes 0 = 1/u + 1/v, which simplifies to v = −u. This confirms that images in plane mirrors appear equidistant behind the mirror as objects sit in front. The radius of curvature of a plane mirror is also infinite, since a plane is mathematically equivalent to a sphere with infinite radius. Although plane mirrors don't have a finite focal point, the equation still governs their optical behavior perfectly.

What happens to the image when an object is placed exactly at the focal point of a concave mirror?

When an object sits precisely at the focal point of a concave mirror, the mirror equation produces an undefined result: 1/f = 1/u + 1/v becomes 1/u + 1/v = 1/u, leaving 1/v = 0, or v = ∞. Physically, reflected rays from an object at the focal point emerge parallel to the principal axis. These parallel rays never converge, so no image forms at any finite distance. This scenario is the limiting case between real and virtual image formation. In practical applications, even slight displacement from the focal point determines whether a real image forms in front of the mirror or a virtual image behind it.

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