Convex: positive power. Concave: negative. Combining lenses.

Power of a Lens — P = 1/f in Diopters

Question

A convex lens has a focal length of 25 cm. Find its power. If this lens is placed in contact with a concave lens of focal length 50 cm, find the power of the combination and the effective focal length.

(CBSE 2024 Board Exam — 3 marks)

Solution — Step by Step

Convert focal length to metres

Power uses SI units — focal length must be in metres, not centimetres. This is where most marks get dropped.

Convex lens: $f_1 = +25\,\text{cm} = +0.25\,\text{m}$

Concave lens: $f_2 = -50\,\text{cm} = -0.50\,\text{m}$

The sign matters: convex lenses have positive focal length, concave lenses have negative.

Calculate individual powers

Power is simply the reciprocal of focal length in metres:

$P = \frac{1}{f(\text{in metres})}$

$P_1 = \frac{1}{+0.25} = +4\,\text{D}$

$P_2 = \frac{1}{-0.50} = -2\,\text{D}$

Find power of the combination

When lenses are placed in contact, powers add algebraically. This is the big advantage of the power formula over the lens formula.

$P = P_1 + P_2 = +4 + (-2) = +2\,\text{D}$

Find effective focal length

Now reverse the power formula:

$f = \frac{1}{P} = \frac{1}{+2} = +0.5\,\text{m} = +50\,\text{cm}$

Positive power → the combination behaves as a convex lens with focal length 50 cm.

Why This Works

The unit dioptre (D) exists precisely because optometrists and physicists needed a quantity that adds simply. Focal lengths don't add — a 25 cm lens and a 50 cm lens in contact do NOT give a 75 cm lens. But their powers do add, because power is proportional to the bending ability of the lens.

Think of it this way: each lens bends the rays by a certain amount. When you stack two lenses, the total bending is the sum of the individual bendings. Power measures bending ability directly, which is why it follows this additive rule.

The sign convention carries the physics: positive power means the lens converges light (convex), negative means it diverges (concave). The combination here has $P = +2\,\text{D}$ , so it's net converging — the stronger convex wins.

Alternative Method

We can verify using the combined focal length formula directly:

$\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}$

$\frac{1}{f} = \frac{1}{+25} + \frac{1}{-50} = \frac{2}{50} - \frac{1}{50} = \frac{1}{50}$

So $f = +50\,\text{cm}$ , which matches.

The power route is faster in exams — especially when three or more lenses are combined, or when the question gives power directly (as optician prescriptions do).

💡 Expert Tip

Optician prescriptions are written in dioptres: "–1.5 D" means a concave lens with $f = –66.7\,\text{cm}$ . NEET occasionally frames questions this way to test whether you can work backwards.

Common Mistake

⚠️ Common Mistake

Using focal length in centimetres directly in the formula.

Writing $P = \frac{1}{25} = 0.04\,\text{D}$ instead of $\frac{1}{0.25} = 4\,\text{D}$ is the single most common error in this topic — it appeared as a distractor in CBSE 2022 as well. The formula $P = 1/f$ only works when $f$ is in metres. If you leave it in centimetres, your answer is off by a factor of 100.

Always write the unit conversion as a separate step so the examiner can award partial marks even if you make an arithmetic error later.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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