Light Reflection and Refraction Class 10: Sign Convention

In Light Reflection and Refraction (Class 10 Science), most wrong answers come from one thing: a wrong sign. This post explains the New Cartesian sign convention step by step, then shows how to use it in the mirror formula, the lens formula, magnification and power of a lens. Every worked example is solved with signs shown at each step.

The sign convention in four rules

Think of the mirror or lens as sitting on a graph.

1. The origin is the pole P (for a mirror) or the optical centre O (for a lens). The principal axis is the x-axis.
2. Light travels from left to right, and the object is placed on the left. So the object distance u is negative.
3. Distances to the right of the origin are positive; distances to the left are negative.
4. Heights above the axis are positive; heights below are negative.

From these rules come the signs of focal length:

A concave mirror has a real focus: parallel rays actually meet there. A convex mirror has a virtual focus behind it: reflected rays only appear to come from it.

Also remember: for a spherical mirror with a small aperture, R = 2f. A mirror with R = 20 cm has f = 20 / 2 = 10 cm.

Mirror formula and magnification

Mirror formula: 1/v + 1/u = 1/f

Magnification: m = h′ / h = -v / u

How to read m:

• m negative → real and inverted image.
• m positive → virtual and erect image.
• Size of m greater than 1 → enlarged; equal to 1 → same size; less than 1 → diminished.

So m = -2 means a real, inverted image twice the size of the object.

Solved mirror examples

Example 1: concave mirror, object at C. f = -15 cm, u = -30 cm.

• 1/v = 1/f - 1/u = -1/15 - (-1/30) = -1/15 + 1/30
• = -2/30 + 1/30 = -1/30
• v = -30 cm (in front of the mirror, so real)
• m = -v/u = -(-30)/(-30) = -1 → real, inverted, same size.

This matches the rule: an object at C forms an image at C of the same size.

Example 2: concave mirror, object between P and F. f = -20 cm, u = -10 cm.

• 1/v = 1/f - 1/u = -1/20 + 1/10 = -1/20 + 2/20 = 1/20
• v = +20 cm (positive, so behind the mirror)
• m = -v/u = -(20)/(-10) = +2 → virtual, erect, enlarged.

This is why a concave mirror works as a shaving mirror or a dentist's mirror.

Example 3: convex mirror. R = 3 m, so f = +1.5 m. Object at u = -5.0 m.

• 1/v = 1/f - 1/u = 1/1.5 + 1/5
• = (5 + 1.5)/7.5 = 6.5/7.5
• v = 7.5/6.5 ≈ +1.15 m → behind the mirror, virtual.

A convex mirror always gives a virtual, erect, diminished image, which is why it is used as a rear-view mirror with a wide field of view.

Lens formula and magnification

The lens formula has a minus sign where the mirror formula has a plus:

Lens formula: 1/v - 1/u = 1/f

Magnification for a lens: m = h′ / h = v / u (no minus sign)

Solved lens examples

Example 4: convex lens. f = +10 cm, u = -15 cm.

• 1/v = 1/f + 1/u = 1/10 + 1/(-15) = 1/10 - 1/15
• = 3/30 - 2/30 = 1/30
• v = +30 cm (other side of the lens, real)
• m = v/u = 30/(-15) = -2 → real, inverted, twice the size.

The object here is between F₁ and 2F₁, and the image is beyond 2F₂ and enlarged, just as the image table says.

Example 5: concave lens. f = -15 cm, u = -30 cm.

• 1/v = 1/f + 1/u = -1/15 - 1/30
• = -2/30 - 1/30 = -3/30 = -1/10
• v = -10 cm (same side as the object, virtual)
• m = v/u = (-10)/(-30) = +1/3 → virtual, erect, diminished.

A concave lens always gives such an image, between its focus and the optical centre.

Power of a lens

P = 1/f, with f in metres. The unit is the dioptre (D). A lens of focal length 1 m has power 1 D.

• Convex lens: positive power. Concave lens: negative power.
• Thin lenses in contact: P = P₁ + P₂ + P₃ ...

Example 6: a concave lens has focal length 2 m. f = -2 m, so P = 1/(-2) = -0.5 D.

Example 7: a lens of +2.0 D. f = 1/P = 1/2.0 = 0.5 m (50 cm). Positive, so it is a convex (converging) lens.

Where refraction fits in

The sign convention handles the maths, but remember the idea behind refraction: light bends because its speed changes. Going from air into glass, it slows down and bends towards the normal; coming out, it speeds up and bends away. The refractive index is n = c / v. If glass has n = 1.5, the speed of light in it is (3 × 10⁸ m/s) / 1.5 = 2 × 10⁸ m/s.

Common mistakes to avoid

• Putting u as positive. The object is on the left, so u is negative.
• Using the mirror formula for a lens (or the other way round). Mirror: plus sign. Lens: minus sign.
• Using m = -v/u for a lens. For a lens, m = v/u.
• Forgetting f in metres for power. 50 cm must become 0.5 m before P = 1/f.
• Giving a concave mirror a positive f. Concave mirror and concave lens both have negative f.
• Dropping the sign of the answer. The sign of v tells you where the image is; the sign of m tells you whether it is real or virtual.

Quick check questions

1. What is the sign of f for a convex mirror? Answer: Positive.
2. A mirror has R = 20 cm. Find f. Answer: f = R/2 = 10 cm.
3. An image has m = +3. Describe it. Answer: Virtual, erect and three times the size of the object.
4. Which formula has the minus sign, 1/v - 1/u = 1/f? Answer: The lens formula.
5. A lens has power -4 D. Find its focal length and type. Answer: f = 1/(-4) = -0.25 m (-25 cm), a concave lens.
6. Where must an object be placed in front of a convex lens to get a real image of the same size? Answer: At 2F₁.

Want to practise more sign-convention problems until they feel easy? Study this chapter with Joy on Learnijoy.