Sign Convention & Critical Rules
Master this = Master 70% of the chapter
Why This Page Is CRITICAL
70% of students lose marks not because they don't know formulas, but because they mess up signs. This is the difference between scoring 30% and 95%.
Cartesian Sign Convention (NEW CONVENTION)
The Golden Rules
Rule 1: Origin & Direction
- Origin: Pole of mirror/optical center of lens
- Positive direction: Left to right (direction of incident light)
- Negative direction: Right to left (opposite to incident light)
Rule 2: Measurements
- Heights above axis: Positive (+)
- Heights below axis: Negative (-)
- Distances measured against incident light: Negative (-)
- Distances measured with incident light: Positive (+)
Sign Convention Summary Table
| Quantity | When Positive (+) | When Negative (-) |
|---|---|---|
| Object Distance (u) | Never (object always on left) | Always negative in standard setup |
| Image Distance (v) | Real image (on right/same side as reflected/refracted light) | Virtual image (on left/opposite side) |
| Focal Length (f) | Converging: Concave mirror, Convex lens | Diverging: Convex mirror, Concave lens |
| Radius (R) | Center of curvature on right of pole | Center of curvature on left of pole |
| Height (h) | Above principal axis | Below principal axis |
| Magnification (m) | Erect image | Inverted image |
Most Common Question: "Why is u always negative?" Because object is always placed on the left (against the positive direction) in standard setup.
Mirrors - Sign Convention
Concave Mirror:
- f = negative (converging)
- R = negative
- u = always negative
- v = negative (real image)
- v = positive (virtual image, object between F and P)
Convex Mirror:
- f = positive (diverging)
- R = positive
- u = always negative
- v = always positive (always virtual)
Remember: Concave mirror can form both real and virtual images. Convex mirror only virtual.
Lenses - Sign Convention
Convex Lens:
- f = positive (converging)
- u = always negative
- v = positive (real image)
- v = negative (virtual image, object between F and lens)
Concave Lens:
- f = negative (diverging)
- u = always negative
- v = always negative (always virtual)
Don't confuse: Concave mirror = converging (f < 0), but Concave lens = diverging (f < 0). Convex mirror = diverging (f > 0), Convex lens = converging (f > 0).
Real vs Virtual Images
Real Image
- Formed by actual intersection of rays
- Can be projected on screen
- Usually inverted (m < 0)
- For mirrors: v < 0
- For lenses: v > 0
Virtual Image
- Formed by apparent intersection of rays
- Cannot be projected on screen
- Usually erect (m > 0)
- For mirrors: v > 0
- For lenses: v < 0
Quick Check: If you can catch the image on a paper/screen, it's real. If you can only see it by looking at the mirror/lens, it's virtual.
Top 10 Sign Convention Mistakes
Mistake 1: Forgetting to put negative sign for u
Wrong: If object is 20 cm from mirror, u = 20 cm
Right: u = -20 cm (always negative in standard setup)
Mistake 2: Using mirror formula for lens
Mirror: 1/f = 1/v + 1/u (PLUS sign)
Lens: 1/f = 1/v - 1/u (MINUS sign)
This single error fails the entire problem.
Mistake 3: Wrong focal length sign
Converging (Concave mirror, Convex lens): f > 0? NO!
Correct: Concave mirror f < 0, Convex lens f > 0
Remember: Sign depends on where focus is, not on converging/diverging nature alone.
Mistake 4: Magnification sign confusion
Mirror: m = -v/u (negative sign present)
Lens: m = v/u (NO negative sign)
If m < 0 → inverted image, if m > 0 → erect image (both cases)
Mistake 5: Confusing R and f signs
R and f always have the same sign. If f = -15 cm, then R = -30 cm.
Relationship: f = R/2 (with same sign)
Sign Convention Practice
Quick Fire Questions (Answer in your head)
Q1: Object 30 cm from concave mirror. What is u?
Answer: u = -30 cm (negative because on left side)
Q2: Convex lens forms image 40 cm on opposite side. What is v?
Answer: v = +40 cm (positive because real image on right side)
Q3: Convex mirror focal length 20 cm. What is f?
Answer: f = +20 cm (positive for convex mirror)
Q4: Concave lens forms image on same side as object, 15 cm away. What is v?
Answer: v = -15 cm (negative because virtual image on left side)
Small Angle Approximation (Paraxial Approximation)
When angles are very small (< 10°):
cos θ ≈ 1
Why this matters:
- All mirror and lens formulas are derived using this approximation
- Rays must be close to principal axis (paraxial rays)
- For large angles, aberrations occur and formulas break down
Unless stated otherwise, always assume paraxial approximation in exams. If question mentions "wide beam" or "large angle," then formulas may not apply directly.