Combining two thin lenses in contact: finding the effective focal length with 1/f = 1/f1 + 1/f2

Two lenses, one focus: a simple shortcut for a powerful idea

Let’s start with a tiny mental image. You’ve got two thin lenses sitting right up against each other, doing their bit to bend light. Think of them as a tag team, each lens adding a little extra push to the rays headed toward the image. When they’re in close contact, their combined effect is neat and predictable: you just add their powers. In math terms, the effective focal length f of the pair satisfies

1/f = 1/f1 + 1/f2

Here f1 and f2 are the focal lengths of the individual lenses. This isn’t just a clever trick for exams or problem sets; it’s a handy rule that pops up in cameras, magnifiers, and many lab experiments. So, let’s unpack what this means and how to use it with confidence.

Why this formula makes sense (without getting lost in the algebra)

A quick way to see why powers add is to switch the perspective a bit. A lens doesn’t “move” light in a single, fixed way; it changes the direction of light rays, and that change is proportional to how strongly the lens can bend the light—its power. The more curved a surface, the stronger the bend. If you have two lenses, each one bending a ray a bit, the total bend is the sum of the two bends, provided they are very close together. When the lenses are thin and touch, the distance between them becomes negligible, and the total bending behaves like a single lens with a new, combined strength.

That combined strength is what we call the optical power, P, defined as P = 1/f. For the two lenses in contact, the separate powers add:

P = P1 + P2, where P1 = 1/f1 and P2 = 1/f2.

Rearranging gives the tidy relation you’ll memorize: 1/f = 1/f1 + 1/f2. It’s the physics version of “two small pushes become one bigger push,” but with light.

What happens when you mix signs? A quick reality check

Not all focal lengths are positive. A positive focal length means a converging lens (like a typical convex lens), while a negative focal length means a diverging lens (like a concave lens). Mix two positives, or two negatives, and you’ll see a predictable outcome. Mix a positive and a negative, and the sign of the resulting focal length can flip depending on which power dominates.

• If both lenses converge (f1 > 0, f2 > 0): the combined f is positive and generally smaller than either f1 or f2. In other words, two converging lenses in contact make a stronger converger.

• If both lenses diverge (f1 < 0, f2 < 0): the combined f is negative and typically larger in magnitude than either f1 or f2 in absolute value, so the system behaves like a stronger diverging lens.

• If one converges and the other diverges, the sign of the final focal length depends on which power wins. It can still converge, diverge, or even land exactly at a certain f, depending on the numbers.

A small, concrete example to ground the idea

Let’s do a straightforward calculation with two positive (converging) lenses first:

• f1 = +0.10 m (10 cm)

• f2 = +0.20 m (20 cm)

Compute their powers: P1 = 1/f1 = 1/0.10 = 10 diopters, P2 = 1/f2 = 1/0.20 = 5 diopters.

Add them: P = P1 + P2 = 10 + 5 = 15 diopters.

Convert back to focal length: f = 1/P = 1/15 ≈ 0.0667 m, or about 6.7 cm.

So, two modest magnifying lenses pressed together behave like a single, stronger magnifier with a focal length around 6.7 cm.

Now, a quick twist to see the power of signs in action:

• f1 = +0.10 m (10 cm)

• f2 = −0.05 m (−5 cm)

Here P1 = 10 diopters, P2 = −20 diopters. Add them: P = 10 − 20 = −10 diopters. The combined focal length is f = 1/−10 = −0.10 m, i.e., −10 cm. The system now acts as a diverging lens even though one lens was converging. Signs really matter, and this is where careful bookkeeping pays off.

Where this shows up in the real world

You’ll see this relationship all over optics. In camera lenses, several elements are stacked to achieve a desired focal length and optical power. In magnifiers, you can tune how close you want to bring an image to your eye by adjusting these combinations. In the lab, double-lens setups help students observe focal properties without getting lost in a mess of distances. The takeaway is simple: if you’re using two thin lenses that sit right next to each other, treat their powers as additive.

A practical note about distance and the “in contact” assumption

The formula 1/f = 1/f1 + 1/f2 holds when the lenses are in contact or separated by a distance that is negligible compared to their focal lengths. If you slide the lenses apart, the distance matters. In the simple case where the separation is small, you can still use an approximate approach, but once the distance grows, you’ll need to use a more detailed ray-tracing method or the general thick-lens formula. In textbooks and on many problem sets, the “thin lenses in contact” assumption is the clean, teachable starting point that makes the arithmetic transparent.

A quick-fix checklist for solving these questions without getting tangled

• Check the signs first. Are you dealing with converging lenses (positive f) or diverging lenses (negative f)? Record f1 and f2 with their signs.

• Use consistent units. Either keep everything in meters or in centimeters, but don’t switch mid-calculation.

• Compute the reciprocals separately, then add. It’s easy to slip a sign here, so slow down at this step.

• Invert the final sum to get f. If the sum is negative, your f will be negative too, which means the system behaves like a diverging lens.

• Keep the physical meaning in mind. A smaller positive f means a stronger converging effect; a larger magnitude negative f means a stronger diverger.

How this connects to broader physics intuition

If you squint at the math a bit, you’ll notice a pattern that echoes other places where “effects” add. In circuits, resistances in parallel give a reciprocal sum. In magnetism, certain vectors add in a cumulative way. The lens case is a crisp, geometric illustration of the broader idea: combine the individual contributions into a single, effective quantity that governs the system’s behavior.

A playful touch of science storytelling

If you’ve ever held a magnifying glass up to sunlight and watched a spark of light form on paper, you’ve felt the power of focusing. Now imagine stacking another lens behind it. The light’s fate—the point where it concentrates—shifts. That shift is exactly what 1/f = 1/f1 + 1/f2 describes. It’s like two musicians sharing a stage; their combined sound is the sum of their individual notes, producing a new harmony that neither one could achieve alone.

Bringing it back to your toolkit

When you’re solving problems or sketching setups for NEET-level topics, this relationship is a reliable go-to. It’s quick, it’s elegant, and it’s grounded in the physics of light bending. Remember: two thin lenses in contact behave like a single lens whose focal length is determined by the reciprocal sum of the two focal lengths. That’s the neat trick you can lean on when you’re faced with a lens system in a problem, a lab demo, or a real-world optical device.

Some reflective warm-ups you can chew on later

• If you know one lens has f1 = +15 cm and the other f2 = +10 cm, what’s the effective f? Do the math and compare to each individual focal length. You’ll see the combined focal length is shorter, which means a stronger converging effect.

• Try a mixed pair: f1 = +12 cm and f2 = −8 cm. What do you get for f? This is a small exercise in signs that makes the rule feel more concrete.

• Consider the limit: what happens if f1 tends to a very large number (a very weak lens) while f2 stays finite? The result should align with your intuition: the weak lens barely changes the overall focus.

Wrapping up with a human takeaway

Two thin lenses in a tight embrace don’t complicate the world of light; they simplify it. Their combined focus is just the sum of their individual “bends,” wrapped into a single focal length that tells you where light will converge (or diverge) after passing through the pair. That little equation—1/f = 1/f1 + 1/f2—packs a surprising amount of intuition, a touch of algebra, and a lot of real-world relevance. It isn’t just a line in a problem set; it’s a lens into how engineers and physicists predict and shape what we see.

So next time you’re staring at a two-lens setup, remember the tune these two notes play together. Add the reciprocals, invert the sum, and you’ve got the system’s focal length in hand. It’s one of those simple truths in physics that feels almost poetic—two simple ideas, joined to reveal a clearer picture of the world. And that clarity, honestly, is what makes learning feel a little brighter than a bare diagram on a page.