Concave and Convex Mirror Simulator

What Is the Concave and Convex Mirror Simulator?

Reading the Mirror Equation on a Ray Bench

Every curved mirror obeys one relationship, 1/f = 1/v + 1/u, with the focal length equal to half the radius of curvature, f = R/2. Drag the Object distance slider and the bench traces the standard rays and solves for the image distance v and the magnification m = -v/u. A positive v means a real image formed in front of the mirror; a negative v means a virtual image that only appears to sit behind it.

On a concave mirror the image changes character as the object moves. Beyond the centre of curvature it is real, inverted and shrunk; between the focus and the centre it is real, inverted and magnified; and once the object slips inside the focal point it flips to virtual, upright and enlarged — the make-up-mirror case. Slide past the focus and watch the readout swing from “Real, inverted” to “Virtual, upright,” with the reflected rays turning parallel at the focal point itself, where no image forms.

A convex mirror is simpler and never changes its mind: for any object it returns a virtual, upright, diminished image with a magnification between zero and one, which is exactly why it is used for wide-angle car and security mirrors. Put precise numbers on any case with the lens and mirror calculator, compare it with its refracting cousin in the concave vs convex lens simulator, or explore more of our interactive physics labs.

Where the ideal mirror equation breaks down

1/f = 1/v + 1/u and f = R/2 are both paraxial results, derived on the assumption that every ray stays close to the axis and meets the mirror at a small angle. A real spherical mirror does not oblige. Rays striking near the rim cross the axis closer to the mirror than rays near the centre, so there is no single focal point — only a blurred caustic. The bench above draws every ray meeting neatly at one image point, which is the paraxial fiction. That error is spherical aberration, and it is why serious telescopes use a paraboloid: it brings parallel rays to one true focus, and a sphere only nearly does.

Second, the equation returns positions that no mirror can usefully deliver. Keep the mirror concave at f = 15 cm and nudge the object in to u = 16 cm: the readouts give v = 240.0 cm and m = -15.00, an image fifteen times life size and over two metres away. The algebra is right and the optics is hopeless. The same gathered light is now spread over roughly 225 times the area, so the image dims in proportion, every aberration is magnified with it, and the mirror must be large enough to catch rays that steep. Magnification is never free, and the equation never mentions the cost.

Third, the “No image” state is partly a modelling convenience. Set u = 15 cm with f = 15 cm and the panel reports No image — reflected rays are parallel, which is correct at exactly the focal point. But the simulator declares it across a small band around f, because a nearly-zero (u - f) would otherwise fling the image out to absurd distances. Real optics has no such band: v grows smoothly, and the answer stops being meaningful long before it stops being computable.

Finally, the model assumes a perfectly reflecting mathematical surface. A household mirror is silvered on the back of a glass sheet, so the front surface throws a faint second reflection out of step with the main one — the ghost you see at a shallow angle, and the reason precision instruments use front-surface mirrors. Coatings are imperfect too: a good aluminium film returns roughly 90 per cent of visible light, so a design that bounces a beam several times loses a serious share before any image forms.

Where this is actually used

The passenger-side car mirror. Switch to Convex and set f = 80 cm — the panel confirms R = 160 cm — with the object at u = 200 cm. The readouts give v = -57.1 cm and m = +0.29: a virtual, upright image just over half a metre behind the glass at 29 per cent of life size. That shrinking is what buys the wide field of view, and it is why the glass is stamped “objects in mirror are closer than they appear”: your eye reads a small car as a distant car.

A shaving or make-up mirror. Go back to Concave, set f = 25 cm, and bring the object inside the focus at u = 15 cm. The panel flips to Virtual, upright with v = -37.5 cm and m = +2.50, turning a 4 cm object into a 10.0 cm image. That is the entire design brief: the user has to sit inside the focal length, which is why such mirrors are ground with a long radius (here R = 50 cm) and why your reflection inverts and shrinks if you lean back past the focus.

Reflecting telescopes. Push the object far off and the image collapses onto the focal plane: at u = 200 cm with f = 5 cm the readout gives v = 5.1 cm, within a few per cent of f. Astronomical objects are effectively at infinity, so a telescope mirror always forms its image at f, and the image size is just the focal length times the object's angular width. A 1000 mm mirror renders the Moon's half-degree disc about 8.7 mm across at the sensor, which is why long focal length, not high magnification, puts detail on an astrophotograph.