Optics · 17.1
Curved Mirrors
A concave mirror gathers parallel rays to a focal point; a convex mirror spreads them from one. The mirror equation predicts where every image forms and how large it is.
Build the model
Connect the measurement to the mechanism.
For small curvatures the focal length is half the radius, f = R/2. The mirror equation 1/f = 1/dₒ + 1/dᵢ then locates the image, and m = −dᵢ/dₒ gives its size and orientation. Concave mirrors make real or virtual images depending on where the object sits; convex mirrors only ever make small upright virtual ones.
- Simple definition
- A curved mirror focuses or spreads reflected rays; the mirror equation relates object distance, image distance, and focal length.
- Example
- A make-up mirror is concave: hold your face inside its focal length and you see a magnified upright reflection.
The focus sits halfway to the centre of curvature — parallel rays meet there (or appear to come from there).
Concave: f > 0. Convex: f < 0
Given the object's distance and the focal length, solve for where the image forms — signs tell you which side and whether it is real.
dᵢ > 0 real, dᵢ < 0 virtual
How many times taller the image is than the object; a minus sign means it hangs upside down.
Negative m = inverted image
Three principal rays
Parallel ray reflects through the focus; focal ray reflects parallel; centre-of-curvature ray returns on itself. Any two locate the image.
Concave cases
Object beyond C: real, inverted, smaller. Between C and F: real, inverted, larger. Inside F: virtual, upright, magnified — the shaving-mirror setting.
Convex always
Diverging mirrors give small, upright, virtual images with a wide field of view — hence shop-security and car door mirrors.
Change one variable at a time
Make the relationship visible.
Slide the object through F and watch the image flip from real-inverted to virtual-upright.
Image distance dᵢ15 cm
Typereal, inverted
Magnification m-0.5×
Catch the common trap
Explain before calculating.
An object sits 30 cm from a concave mirror of focal length 10 cm. Where is the image?
Choose an answer to test the model.
Worked examples
State the rule, substitute, then check units.
EasyA concave mirror has radius of curvature 40 cm. Find its focal length.
- f = R/2 = 40 ÷ 2.
- f = 20 cm.
Answerf = 20 cm
MediumA convex mirror with f = −20 cm views a car 10 m away. Find the image position and magnification.
- 1/dᵢ = 1/f − 1/dₒ = −1/20 − 1/1000 (work in cm: −0.05 − 0.001).
- dᵢ ≈ −19.6 cm — virtual, just behind the mirror.
- m = −dᵢ/dₒ = 19.6/1000 ≈ 0.02 — tiny and upright, so 'objects are closer than they appear'.
Answer≈ 19.6 cm behind the mirror, m ≈ +0.02
HardAn object 4.0 cm tall stands 15 cm from a concave mirror (f = 10 cm). Find the image position, size, and nature.
- 1/dᵢ = 1/10 − 1/15 = 1/30 → dᵢ = 30 cm in front (real).
- m = −30/15 = −2.0.
- Image: 8.0 cm tall, inverted, real — a projector geometry.
Answer30 cm in front; 8.0 cm, inverted, real
ChallengingA dentist's concave mirror (f = 2.0 cm) is held 1.5 cm from a tooth. Find the image and explain why this working distance is chosen.
- 1/dᵢ = 1/2.0 − 1/1.5 = −1/6 → dᵢ = −6.0 cm (behind the mirror: virtual).
- m = −(−6.0)/1.5 = +4.0 — upright and four times life size.
- Inside the focal length is the ONLY concave-mirror regime giving an upright magnified image — pull back past 2 cm and the view flips upside down.
AnswerVirtual, upright, ×4 at 6.0 cm behind the mirror — must stay inside f