What is reflection? The two laws of reflection
When light falls on a smooth shiny surface and bounces back into the same medium, we call it reflection. The ray that arrives is the incident ray, the ray that leaves is the reflected ray, and the line drawn at 90° to the surface at the point where the ray hits is the normal.
- First law: the angle of incidence equals the angle of reflection (∠i = ∠r). Both angles are measured from the normal, not from the mirror.
- Second law: the incident ray, the normal at that point and the reflected ray all lie in the same plane.
These laws hold for every reflecting surface: flat mirrors, curved mirrors, even water. For a curved mirror the normal at a point is the line joining that point to the centre of curvature.
Image formed by a plane mirror
A plane (flat) mirror always gives an image that is:
- virtual and erect: reflected rays never really meet; they only appear to come from a point behind the mirror, so the image cannot be caught on a screen;
- the same size as the object (magnification = +1);
- as far behind the mirror as the object is in front;
- laterally inverted: your right hand looks like a left hand. That is why AMBULANCE is painted mirror-wise on the front of ambulances.
Spherical mirrors: concave and convex
A spherical mirror is a piece cut from a hollow shiny sphere. If the shiny side is the inner (caved-in) side, it is a concave mirror; if the shiny side bulges out, it is a convex mirror.
Important terms
- Pole (P): the centre of the mirror's surface.
- Centre of curvature (C): the centre of the sphere the mirror was cut from. It is in front of a concave mirror and behind a convex mirror.
- Radius of curvature (R): the distance PC.
- Principal axis: the straight line through P and C.
- Principal focus (F): the point where rays parallel to the axis meet after reflection (concave), or seem to come from after reflection (convex).
- Focal length (f): the distance PF.
- Aperture: the width of the reflecting surface.
For mirrors with a small aperture, R = 2f, so F lies halfway between P and C.
Rules for drawing ray diagrams
To locate an image, take any two of these rays from the tip of the object. Where they meet (or seem to meet) is the tip of the image.
- A ray parallel to the principal axis passes through F after reflection (concave) or seems to come from F (convex).
- A ray passing through F (concave) or heading toward F (convex) comes back parallel to the axis.
- A ray through C (concave) or heading toward C (convex) hits the mirror along the normal and comes straight back on itself.
- A ray hitting the pole bounces back at the same angle on the other side of the axis (the axis is the normal at P).
In the 3D bench the orange ray uses rule 1 and the teal ray uses rule 4. Dotted lines show where reflected rays only seem to come from: that is a virtual image.
Image formation by a concave mirror (every object position)
| Object position | Image position | Size | Nature |
|---|---|---|---|
| At infinity | At F | Point-sized, highly diminished | Real, inverted |
| Beyond C | Between F and C | Diminished | Real, inverted |
| At C | At C | Same size | Real, inverted |
| Between C and F | Beyond C | Enlarged | Real, inverted |
| At F | At infinity | Highly enlarged | Real, inverted |
| Between P and F | Behind the mirror | Enlarged | Virtual, erect |
Pattern to notice: as the object comes closer from infinity to F, the real image moves away from F toward infinity and keeps growing. Inside F the image becomes virtual and upright. Walk through all six cases in story step 4 or by dragging the object.
Image formation by a convex mirror
| Object position | Image position | Size | Nature |
|---|---|---|---|
| At infinity | At F, behind the mirror | Point-sized | Virtual, erect |
| Anywhere between infinity and P | Between P and F, behind the mirror | Diminished | Virtual, erect |
The image of a convex mirror never crosses F and is never real. The closer the object, the bigger (but still smaller than the object) the image.
Uses of concave and convex mirrors
- Concave: torches, car headlights and searchlights (bulb at F gives a strong parallel beam); shaving and make-up mirrors (face inside F gives a big upright image); dentists' mirrors; solar cookers and solar furnaces (sunlight focused at F).
- Convex: rear-view mirrors of cars and scooters (always an upright image and a much wider field of view); mirrors at sharp turns and in shops for security.
New Cartesian sign convention for spherical mirrors
- Put the pole P at the origin and the principal axis along the x-axis.
- Always place the object on the left, so light travels left to right.
- Distances measured in the direction of the incident light (to the right of P) are positive; against it (to the left) are negative. So u is always negative.
- Heights above the axis are positive; below are negative.
Results: a concave mirror has negative f; a convex mirror has positive f. A real image (in front) has negative v; a virtual image (behind) has positive v.
Mirror formula and magnification
The object distance u, image distance v and focal length f are linked by the mirror formula:
1/v + 1/u = 1/f
Magnification tells how many times bigger the image is and whether it is upright:
m = h′/h = −v/u
- m negative → real and inverted; m positive → virtual and erect.
- |m| > 1 → enlarged; |m| < 1 → diminished.
How to solve: write the known values with signs, put them into the formula, solve for the unknown, then read the sign of the answer to decide the nature. Board papers usually ask one ray diagram (2–3 marks) and one numerical on the mirror formula (3 marks) from this part.
Key formulas and definitions
- ∠i = ∠r (laws of reflection)
- R = 2f
- Mirror formula: 1/v + 1/u = 1/f
- Magnification: m = h′/h = −v/u
- Concave mirror: f negative; convex mirror: f positive; u always negative
Worked examples
1. A concave mirror has a radius of curvature of 30 cm. What is its focal length (with sign)?
f = R/2 = 30/2 = 15 cm. F is in front of a concave mirror, against the incident light, so f = −15 cm.
2. A convex mirror has a focal length of 12 cm. Find its radius of curvature.
R = 2f = 2 × 12 = 24 cm. The centre of curvature is behind the mirror, so R = +24 cm.
3. An object is placed 30 cm in front of a concave mirror of focal length 20 cm. Find the image position, magnification and nature.
u = −30 cm, f = −20 cm. 1/v = 1/f − 1/u = −1/20 + 1/30 = (−3 + 2)/60 = −1/60, so v = −60 cm. m = −v/u = −(−60)/(−30) = −2. The image is 60 cm in front of the mirror: real, inverted, twice as big (object between C and F, image beyond C).
4. An object 4 cm tall stands 10 cm in front of a concave mirror of focal length 15 cm. Find the image position and height.
u = −10, f = −15. 1/v = −1/15 + 1/10 = (−2 + 3)/30 = 1/30, so v = +30 cm (behind the mirror). m = −v/u = −30/(−10) = +3. h′ = m × h = 3 × 4 = 12 cm. Virtual, erect, 3 times enlarged: this is how a shaving mirror works.
5. An object is 20 cm in front of a convex mirror of focal length 20 cm. Find the image.
u = −20, f = +20. 1/v = 1/20 − 1/(−20) = 1/20 + 1/20 = 1/10, so v = +10 cm. m = −10/(−20) = +0.5. Image 10 cm behind the mirror, virtual, erect, half the size.
6. A concave mirror forms a real image, three times the size of the object, on a screen 60 cm from the mirror. Where is the object and what is the focal length?
Real image on a screen: v = −60 cm and m = −3 (real images are inverted). m = −v/u → −3 = 60/u → u = −20 cm. 1/f = 1/v + 1/u = −1/60 − 1/20 = (−1 − 3)/60 = −4/60, so f = −15 cm. Object 20 cm in front; focal length 15 cm.
7. A car's convex rear-view mirror has R = 3 m. A bus is 5 m behind it. Find the position and size ratio of the bus's image.
f = R/2 = +1.5 m, u = −5 m. 1/v = 1/1.5 + 1/5 = 0.667 + 0.200 = 0.867, so v ≈ +1.15 m. m = −v/u = 1.15/5 ≈ 0.23. The image is 1.15 m behind the mirror, virtual, erect and about a quarter of the size, so a large area fits in a small mirror.
Common mistakes
- Measuring the angle of incidence from the mirror surface instead of from the normal.
- Forgetting signs: putting u = +30 instead of −30. The object is always on the left, so u is always negative.
- Giving a concave mirror a positive focal length. Concave f is negative; convex f is positive.
- Saying a virtual image can be caught on a screen. Only real images (where rays actually meet) can be.