Curved Mirrors, the Mirror Equation and Real Images
For paraxial rays, a spherical mirror has focal-length magnitude equal to half its radius of curvature. Ray construction and the mirror equation locate images. Signed image distance and magnification describe whether an image is real or virtual, upright or inverted.
What a learner can do afterwards
- Locates an image by ray construction for concave and convex mirrors and checks it against the mirror equation
- Relates paraxial focal length to radius of curvature and distinguishes spherical aberration in an axial beam from off-axis coma
- Explains why a shaving mirror magnifies while a car wing mirror shrinks the view and widens it
1 · Read
A concave spherical mirror reflects from a sphere's inside; a convex mirror reflects from its outside. The optical axis joins the vertex V to the centre of curvature C. For rays close to this axis at small angles, |f| = R/2, where R is the radius's magnitude. This is the paraxial approximation. The focus F is in front of a concave mirror and behind a convex mirror.
Trace two rays from the same object tip. For a concave mirror, an axis-parallel ray reflects through the focus F; a ray along the focal line reflects parallel to the axis. A ray along the line through the centre of curvature C retraces its path. For a convex mirror, F and C lie behind the surface. Reflected rays or their backward extensions follow these rules. Real images use ray intersections; virtual images use backward-extension intersections.
For a real object inside a concave mirror's focal length, the image is virtual, upright and enlarged, as in a shaving mirror. Beyond the focus, it is real and inverted. At the focus, reflected rays are parallel and no finite image forms. A convex mirror gives a real object a virtual, upright, reduced image. Compared with an equal-sized plane mirror, its curved surface shows a wider field.
Use 1/f = 1/do + 1/di and m = -di/do, measuring distances from the vertex. For a real object, do is positive. Take f positive for concave mirrors and negative for convex mirrors. Positive di means a real image in front; negative di means a virtual image behind. Positive m means upright; negative m means inverted. The magnitude |m| is the image-to-object height ratio.
Let a concave mirror have f = 12 cm and do = 18 cm. Then 1/di = 1/12 - 1/18 = 1/36 cm^-1, so di = +36 cm. Next, m = -36/18 = -2. The image is real, inverted and twice the object's height. Compare this result with ray construction. A disagreement calls for checking signs, arithmetic and ray paths; agreement is a useful check, not proof.
Spherical aberration occurs even for an axial parallel beam: rays at different heights on a spherical mirror focus at different distances. A wider opening can admit more edge rays and increase this blur. An ideal paraboloid brings an axial parallel beam to one focus, removing spherical aberration. For light tilted to the axis it can still produce coma, a comet-like image blur. Telescope primaries need not all be paraboloids.
For a paraxial paper diagram, put the vertex at (0,0). Coordinates are in cm: x increases in front of the mirror and y measures height. Plot the object tip (do,h) and focus (f,0), using a small h. Its axis-parallel ray strikes near (0,h), then reflects along the line through (f,0). Its ray to the vertex reflects along the line through (0,0) and (do,-h), with equal angles to the axis. Extend reflected lines behind the mirror when locating virtual images.
The line shows the signed x coordinate only. Draw the rays separately on paper.
Concave example, in cm: f = 15, do = 20, h = 0.2. Draw the reflected parallel-ray line through (0,0.2) and (15,0). Draw the reflected vertex-ray line through (0,0) and (20,-0.2). Their forward intersection is (60,-0.6): the image is 60 cm in front and inverted. Check 1/di = 1/15 - 1/20 = 1/60, giving di = +60 cm. Also m = -60/20 = -3 agrees with the height -0.6 cm. A paper drawing gives an approximate intersection to compare with the equation.
Convex example, in cm: f = -18, do = 36, h = 0.3. Draw the reflected parallel-ray line through (0,0.3) and (-18,0). Draw the reflected vertex-ray line through (0,0) and (36,-0.3). Their backward extensions intersect at (-12,0.1): a virtual image 12 cm behind the mirror. Check 1/di = -1/18 - 1/36 = -1/12, giving di = -12 cm. The magnification m = -(-12)/36 = 1/3 agrees with the upright image height 0.1 cm.
This sequence checks the arithmetic. It does not draw the reflected rays.
Use paraxial ray rules and signed equations together. Real images can reach a screen. A paraboloid corrects axial spherical aberration, not every kind of blur.
Lesson, questions and flashcards written for this dataset. OpenStax texts were consulted for physics facts; textbook passages are not reproduced.
Readings and flashcards written for this dataset, with OpenStax texts consulted for physics facts. Linked videos are identified as Khan Academy material in the source records; their licenses are unspecified.
More to read
Reading a mirror result
Suppose a concave mirror has focal length 15 cm and a real object is 45 cm from its vertex. Use the paraxial model and positive distances for real objects and images. The equation gives 1/di = 1/15 - 1/45 = 2/45 cm^-1. Taking the reciprocal gives di = 22.5 cm. This positive image distance puts the real image in front of the mirror. Now calculate m = -22.5/45 = -0.5. The sign says inverted. The magnitude says the image is half the object's height. Image distance and magnification answer different questions, so interpret both. For a construction check, start two principal rays at the same object tip and follow their reflections. Their intersection locates the real image tip. For a virtual image, extend the reflected rays backward instead. Those extensions describe an apparent origin; they are not paths travelled by light behind the mirror.
Comparing close and wide views
Two mirrors can both give virtual images while changing apparent size in different ways. Place a real object 3 cm in front of a concave mirror with f = 9 cm. Then 1/di = 1/9 - 1/3 = -2/9 cm^-1, giving di = -4.5 cm. Its magnification is -(-4.5)/3 = +1.5. The image is behind the mirror, upright and enlarged. This is the arrangement used for a close view in a concave shaving mirror. For comparison, take a convex mirror with f = -9 cm and a real object at 18 cm. Now 1/di = -1/9 - 1/18 = -1/6 cm^-1, so di = -6 cm. Its magnification is +1/3: virtual and upright again, but reduced. A convex mirror can show a wider field than a plane mirror of the same size. This makes it useful for watching traffic or a shop aisle. Neither virtual image can be caught directly on a screen at its apparent position.
Axial beams and tilted beams
Keep a parallel beam aligned with a spherical mirror's axis. Rays hitting near the edge do not focus at the same distance as rays hitting near the centre. The resulting spherical aberration can blur the image even though the beam is aligned with the axis. Reducing the opening excludes some edge rays. That can reduce this blur, but it also reduces the light collected. The paraxial focal length describes the central rays: its magnitude is half the spherical radius. An ideal paraboloid brings an axial parallel beam to one focus. This corrects spherical aberration for that beam. Now tilt the incoming beam relative to the axis. A paraboloid can still produce coma, a blur with a comet-like shape. A statement that a mirror corrects spherical aberration therefore does not mean it gives sharp images at every viewing angle. Surface shape, the direction of the incoming light and any additional optical elements all matter.
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