Mirror Equation Calculator
Mirror Equation Calculator
Calculate image distance using the mirror formula.
1/f = 1/do + 1/di
Understanding the Mirror Equation
Spherical mirrors, including concave mirrors and convex mirrors, form images by reflecting light rays. The mirror equation provides a mathematical link between an object’s distance from the mirror, the distance of the image formed, and the focal length of the mirror. It is an essential principle in optics, helping students and professionals understand how curved mirrors manipulate light to produce images.
The Formula
1/f = 1/u + 1/v
Where:
- f = focal length of the mirror
- u = object distance (measured from the mirror along the principal axis)
- v = image distance (measured from the mirror along the principal axis)
Concave vs. Convex Mirrors
A concave mirror (converging mirror) focuses parallel rays inward and can produce real or virtual images depending on object position. A convex mirror (diverging mirror) spreads rays outward and always forms diminished, virtual images. The same mirror equation applies, but f is taken as positive for concave and negative for convex mirrors, following the sign convention.
Derivation
The mirror equation is derived using geometry and similar triangles, considering incident rays, reflected rays, and the focal point of the spherical mirror. It mathematically expresses the law of reflection and allows prediction of image properties without physically tracing rays.
Magnification
Along with the mirror equation, magnification is an important aspect of image analysis:
M = -v/u
If M is negative, the image is real and inverted. If positive, the image is virtual and upright.
Applications of the Mirror Equation
- Telescope mirrors: Concave mirrors form sharp images of distant celestial objects.
- Shaving/Makeup mirrors: Concave mirrors enlarge close objects for better viewing.
- Vehicle side mirrors: Convex mirrors provide a wide field of view, showing more area behind.
- Solar concentrators: Concave mirrors focus sunlight to produce heat energy.
- Medical instruments: Mirrors help focus light in examination tools.
Worked Example
Suppose an object is placed 30 cm from a concave mirror with focal length 15 cm. Using the mirror equation:
1/f = 1/u + 1/v
1/15 = 1/30 + 1/v → 1/v = 1/15 - 1/30 = 1/30 → v = 30 cm.
Thus, the image is real, inverted, and of the same size as the object.
Sign Conventions
The mirror equation requires careful use of the Cartesian sign convention:
- Distances measured against incident light (towards the mirror) are negative.
- Distances measured along the direction of reflected light are positive.
- Focal length f is positive for concave and negative for convex mirrors.
Historical Perspective
Curved mirrors have been studied since antiquity. The Greeks used polished bronze concave mirrors to focus sunlight. With the advent of modern science, astronomers like Newton built reflecting telescopes using concave mirrors. Today, mirror equations guide not only scientific instruments but also everyday devices such as car mirrors and optical systems.
Conclusion
The mirror equation is a cornerstone of optics. By connecting focal length, object distance, and image distance, it simplifies the analysis of image formation in spherical mirrors. Its wide-ranging applications in astronomy, daily life, and engineering highlight the importance of mastering this principle for students and practitioners of physics.