Move an Object Inside the Focus and the Same Lens Starts Magnifying Instead
Tell a converging lens from a diverging one, name every part from the centre of curvature to the focal plane, see a lens as a stack of prisms, locate images by ray diagram for each object position, apply the sign convention and the lens formula, and calculate the power in dioptre.
How can one lens both shrink an image and magnify it?
Hold a convex lens well above a page of print and you see the letters upside down and smaller. Bring the lens close to the page and the same letters appear the right way up and larger. Nothing about the lens has changed.
What changed is where the object sits relative to the focus.
- Beyond the focus, the lens brings the rays together on the far side and forms a real, inverted image that could be caught on a screen
- Inside the focus, the rays leaving the lens are still spreading apart, so they never meet — and the image is virtual, erect and magnified, on the same side as the object
The focus is the boundary between the two behaviours, and a convex lens is a projector on one side of it and a magnifying glass on the other.
That single observation is why this part of the chapter spends so much effort on positions. A lens question is almost never about the lens — it is about where the object is. And there are only a handful of positions that matter: beyond twice the focal length, at twice the focal length, between there and the focus, at the focus, and inside it.
Two tools handle all of them.
- Ray diagrams, which show you the nature, position and size of the image at a glance
- The lens formula, which gives the same information as numbers — provided the signs are put in correctly
And the signs are where the marks go. The lens formula is one short equation, but it only works inside a strict convention about which distances count as positive. Get the convention right and every lens numerical becomes two lines; get it wrong and every answer is wrong by a sign.
The part closes with the power of a lens, which is simply the reciprocal of the focal length — the quantity an optician actually writes on a prescription.
This page covers the third part of the ICSE Class 10 Physics chapter on light: converging and diverging lenses, ray diagrams, the lens formula and the sign convention, and the power of a lens.
What changed is where the object sits relative to the focus.
- Beyond the focus, the lens brings the rays together on the far side and forms a real, inverted image that could be caught on a screen
- Inside the focus, the rays leaving the lens are still spreading apart, so they never meet — and the image is virtual, erect and magnified, on the same side as the object
The focus is the boundary between the two behaviours, and a convex lens is a projector on one side of it and a magnifying glass on the other.
That single observation is why this part of the chapter spends so much effort on positions. A lens question is almost never about the lens — it is about where the object is. And there are only a handful of positions that matter: beyond twice the focal length, at twice the focal length, between there and the focus, at the focus, and inside it.
Two tools handle all of them.
- Ray diagrams, which show you the nature, position and size of the image at a glance
- The lens formula, which gives the same information as numbers — provided the signs are put in correctly
And the signs are where the marks go. The lens formula is one short equation, but it only works inside a strict convention about which distances count as positive. Get the convention right and every lens numerical becomes two lines; get it wrong and every answer is wrong by a sign.
The part closes with the power of a lens, which is simply the reciprocal of the focal length — the quantity an optician actually writes on a prescription.
This page covers the third part of the ICSE Class 10 Physics chapter on light: converging and diverging lenses, ray diagrams, the lens formula and the sign convention, and the power of a lens.
What are the parts of a lens, and how do you measure its focal length?
A lens is a piece of transparent material bounded by two surfaces, at least one of which is curved. A converging lens is thicker at the middle; a diverging lens is thinner at the middle.
The two families.
- A convex or converging lens is thicker at the centre than at the edges. It brings a parallel beam to a real point of convergence, so a beam of sunlight passing through it is concentrated to a bright spot
- A concave or diverging lens is thinner at the centre than at the edges. It spreads a parallel beam apart, so the rays appear to diverge from a point behind the lens
The terms to define, each of which a question may ask for.
- Centre of curvature — the centre of the sphere of which a lens surface forms a part. A lens has two, one for each surface, written and
- Radius of curvature — the radius of that sphere. Again there are two, and
- Principal axis — the straight line joining the two centres of curvature, passing through the optical centre
- Optical centre — the point at the middle of the lens on the principal axis. A ray passing through it goes straight on without any deviation
- First and second principal focus — a lens has two foci, one on each side, at equal distances from the optical centre for a thin lens in air. **The second focus is the point at which a beam parallel to the principal axis converges after passing through a convex lens, or from which it appears to diverge after a concave lens
- Focal length — the distance between the optical centre and a principal focus
- Focal plane — the plane through a focus, perpendicular to the principal axis. A parallel beam arriving at an angle to the axis is brought to a point on the focal plane rather than at the focus itself, and that is exactly what the focal plane is for
The focus of a convex lens is real; the focus of a concave lens is virtual. For the concave lens the rays never actually pass through the focus — they only appear to come from it — which is why its focal length is taken as negative.
Three methods of finding the focal length of a convex lens.
The distant-object method. Point the lens at a distant object — the sun, or a building across the road — and move a screen behind it until a sharp, small, inverted image forms.
- Rays from a very distant object arrive practically parallel, so they converge at the focus
- The distance from the lens to the screen is therefore the focal length
- It is quick but only approximate, because the object is not infinitely far away
The plane-mirror method, which is more accurate. Fix a plane mirror behind the lens, facing it, and place an illuminated pin in front.
- Move the pin until its image, formed after refraction through the lens, reflection at the mirror and refraction back, coincides with the pin itself with no parallax
- That happens when the pin is at the focus, because the rays then strike the mirror normally as a parallel beam and return along their own path
- The distance from the pin to the lens is the focal length
The lens-formula method. Form a real image of an object on a screen, measure the object distance and the image distance, and substitute in the lens formula. Repeating this for several positions and averaging gives a good value.
One boundary case that explains why the first method is only approximate. The focal length is defined for a beam that is exactly parallel, which needs an object at infinity. A building a hundred metres away is not at infinity, so its image forms slightly beyond the focus and the measured value is slightly too large. The plane-mirror method avoids this entirely**, which is why it is the preferred laboratory method.
The two families.
- A convex or converging lens is thicker at the centre than at the edges. It brings a parallel beam to a real point of convergence, so a beam of sunlight passing through it is concentrated to a bright spot
- A concave or diverging lens is thinner at the centre than at the edges. It spreads a parallel beam apart, so the rays appear to diverge from a point behind the lens
The terms to define, each of which a question may ask for.
- Centre of curvature — the centre of the sphere of which a lens surface forms a part. A lens has two, one for each surface, written and
- Radius of curvature — the radius of that sphere. Again there are two, and
- Principal axis — the straight line joining the two centres of curvature, passing through the optical centre
- Optical centre — the point at the middle of the lens on the principal axis. A ray passing through it goes straight on without any deviation
- First and second principal focus — a lens has two foci, one on each side, at equal distances from the optical centre for a thin lens in air. **The second focus is the point at which a beam parallel to the principal axis converges after passing through a convex lens, or from which it appears to diverge after a concave lens
- Focal length — the distance between the optical centre and a principal focus
- Focal plane — the plane through a focus, perpendicular to the principal axis. A parallel beam arriving at an angle to the axis is brought to a point on the focal plane rather than at the focus itself, and that is exactly what the focal plane is for
The focus of a convex lens is real; the focus of a concave lens is virtual. For the concave lens the rays never actually pass through the focus — they only appear to come from it — which is why its focal length is taken as negative.
Three methods of finding the focal length of a convex lens.
The distant-object method. Point the lens at a distant object — the sun, or a building across the road — and move a screen behind it until a sharp, small, inverted image forms.
- Rays from a very distant object arrive practically parallel, so they converge at the focus
- The distance from the lens to the screen is therefore the focal length
- It is quick but only approximate, because the object is not infinitely far away
The plane-mirror method, which is more accurate. Fix a plane mirror behind the lens, facing it, and place an illuminated pin in front.
- Move the pin until its image, formed after refraction through the lens, reflection at the mirror and refraction back, coincides with the pin itself with no parallax
- That happens when the pin is at the focus, because the rays then strike the mirror normally as a parallel beam and return along their own path
- The distance from the pin to the lens is the focal length
The lens-formula method. Form a real image of an object on a screen, measure the object distance and the image distance, and substitute in the lens formula. Repeating this for several positions and averaging gives a good value.
One boundary case that explains why the first method is only approximate. The focal length is defined for a beam that is exactly parallel, which needs an object at infinity. A building a hundred metres away is not at infinity, so its image forms slightly beyond the focus and the measured value is slightly too large. The plane-mirror method avoids this entirely**, which is why it is the preferred laboratory method.
Why does a lens behave like a set of prisms, and what images does it form?
Because each part of a lens is a small prism, and the prisms are tilted progressively more as you move away from the centre.
Think of a convex lens as two prisms placed base to base, with a parallel-sided slab between them.
- The upper prism has its base below, so it bends a ray downward toward the axis
- The lower prism has its base above, so it bends a ray upward toward the axis
- The middle behaves like a parallel-sided slab, which does not deviate the ray at all — which is exactly why a ray through the optical centre goes straight on
And a concave lens is two prisms placed apex to apex, which bend rays away from the axis for the same reason.
But a stack of a few prisms would not focus a beam to a point, because each prism deviates every ray passing through it by the same amount. A lens works because its curvature makes the deviation increase smoothly with distance from the axis — a ray far from the centre meets a more steeply inclined surface and is bent more, by just enough to reach the same point. That is what the spherical shape achieves and a stack of prisms cannot.
The three rays used in every ray diagram.
- A ray parallel to the principal axis passes, after refraction, through the second focus of a convex lens, or appears to come from for a concave lens
- A ray through the optical centre continues undeviated
- **A ray through the first focus emerges parallel to the principal axis
Any two of them locate the image. The third is a check.
The six object positions for a convex lens, with the image in each case:
- Object at infinity — image at the focus , real, inverted, highly diminished to a point
- Object beyond — image between and , real, inverted, diminished
- Object at — image at , real, inverted, the same size as the object
- Object between and — image beyond , real, inverted, magnified
- Object at — image at infinity, real, inverted, highly magnified
- Object between and the lens — image on the same side as the object, virtual, erect, magnified
Notice the pattern rather than memorising the list. As the object moves in from infinity toward the focus, the image moves out from the focus toward infinity and grows all the while.** The two cross over at , which is the only position where the image is the same size as the object — and that is the easiest entry in the table to remember, because it anchors the rest.
For a concave lens there is only one case. Wherever the object is placed, the image is between the focus and the optical centre, and it is always virtual, erect and diminished.
Which is why a concave lens can never be used as a magnifier or a projector. It cannot form a real image of a real object at all, and a question asking for a concave lens to throw an image on a screen has no answer.
**Worked observation — the crossover at .** Place an object exactly from a convex lens and the image forms on the other side, the same size and inverted. Move the object a little closer and the image runs outward and grows; move it a little further and the image comes inward and shrinks. The object and image distances are locked together, and the lens formula of the next section is the equation that locks them.
Think of a convex lens as two prisms placed base to base, with a parallel-sided slab between them.
- The upper prism has its base below, so it bends a ray downward toward the axis
- The lower prism has its base above, so it bends a ray upward toward the axis
- The middle behaves like a parallel-sided slab, which does not deviate the ray at all — which is exactly why a ray through the optical centre goes straight on
And a concave lens is two prisms placed apex to apex, which bend rays away from the axis for the same reason.
But a stack of a few prisms would not focus a beam to a point, because each prism deviates every ray passing through it by the same amount. A lens works because its curvature makes the deviation increase smoothly with distance from the axis — a ray far from the centre meets a more steeply inclined surface and is bent more, by just enough to reach the same point. That is what the spherical shape achieves and a stack of prisms cannot.
The three rays used in every ray diagram.
- A ray parallel to the principal axis passes, after refraction, through the second focus of a convex lens, or appears to come from for a concave lens
- A ray through the optical centre continues undeviated
- **A ray through the first focus emerges parallel to the principal axis
Any two of them locate the image. The third is a check.
The six object positions for a convex lens, with the image in each case:
- Object at infinity — image at the focus , real, inverted, highly diminished to a point
- Object beyond — image between and , real, inverted, diminished
- Object at — image at , real, inverted, the same size as the object
- Object between and — image beyond , real, inverted, magnified
- Object at — image at infinity, real, inverted, highly magnified
- Object between and the lens — image on the same side as the object, virtual, erect, magnified
Notice the pattern rather than memorising the list. As the object moves in from infinity toward the focus, the image moves out from the focus toward infinity and grows all the while.** The two cross over at , which is the only position where the image is the same size as the object — and that is the easiest entry in the table to remember, because it anchors the rest.
For a concave lens there is only one case. Wherever the object is placed, the image is between the focus and the optical centre, and it is always virtual, erect and diminished.
Which is why a concave lens can never be used as a magnifier or a projector. It cannot form a real image of a real object at all, and a question asking for a concave lens to throw an image on a screen has no answer.
**Worked observation — the crossover at .** Place an object exactly from a convex lens and the image forms on the other side, the same size and inverted. Move the object a little closer and the image runs outward and grows; move it a little further and the image comes inward and shrinks. The object and image distances are locked together, and the lens formula of the next section is the equation that locks them.
Formula
What is the lens formula and how do the signs work?
One equation relates the object distance, the image distance and the focal length — and a strict sign convention makes it work for both kinds of lens and both kinds of image.
where is the object distance, the image distance, the focal length, the object height and the image height.
The Cartesian sign convention, which the formula depends on entirely.
- All distances are measured from the optical centre of the lens
- Distances measured in the direction of the incident light are positive; those measured against it are negative
- Heights measured upward from the principal axis are positive; those measured downward are negative
What that means in practice, which is the working list to use:
- ** is always negative for a real object, because the object is on the side the light comes from
- is positive for a real image on the far side, and negative for a virtual image on the same side as the object
- is positive for a convex lens and negative for a concave lens
- is negative for an inverted image and positive for an erect one
And the sign of tells you the nature for free. A negative magnification means the image is inverted, which for a single lens means it is real; a positive magnification means erect, which means virtual.
Worked example 1 — object beyond .** An object is placed cm from a convex lens of focal length cm. Find the image position, the magnification and the nature of the image.
**The image is cm from the lens on the other side, real, inverted and half the size of the object.
Check against the ray-diagram table.** The object is beyond , which is cm, so the image should lie between and — that is between cm and cm — and be real, inverted and diminished. **The answer of cm with fits exactly. That cross-check between the formula and the table is worth doing every time.
Worked example 2 — object between and .** The same lens, with the object now cm away.
**Real, inverted and twice the size, at cm** — beyond , as the table requires.
**Worked example 3 — the object at .** The same lens, object at cm.
Real, inverted and the same size, at on the other side — the crossover case.
Worked example 4 — object inside the focus, the magnifying glass. The same lens, object at cm.
**The negative means the image is on the same side as the object, so it is virtual**, and the positive magnification of means erect and twice the size. This is a magnifying glass in use.
Worked example 5 — a concave lens. An object is placed cm from a concave lens of focal length cm. Find the image.
**Virtual, erect and diminished, cm from the lens on the same side as the object** — and cm is between the focus at cm and the lens, exactly as the concave-lens rule requires. A concave lens always gives this, whatever the object distance, and if your answer ever comes out real for a concave lens, a sign has been dropped.
The rearrangement worth memorising. Since , the working form is
and with negative that second term is a subtraction in practice. **Substituting without its minus sign is the commonest single error in this chapter**, and it turns a real image into a virtual one.
where is the object distance, the image distance, the focal length, the object height and the image height.
The Cartesian sign convention, which the formula depends on entirely.
- All distances are measured from the optical centre of the lens
- Distances measured in the direction of the incident light are positive; those measured against it are negative
- Heights measured upward from the principal axis are positive; those measured downward are negative
What that means in practice, which is the working list to use:
- ** is always negative for a real object, because the object is on the side the light comes from
- is positive for a real image on the far side, and negative for a virtual image on the same side as the object
- is positive for a convex lens and negative for a concave lens
- is negative for an inverted image and positive for an erect one
And the sign of tells you the nature for free. A negative magnification means the image is inverted, which for a single lens means it is real; a positive magnification means erect, which means virtual.
Worked example 1 — object beyond .** An object is placed cm from a convex lens of focal length cm. Find the image position, the magnification and the nature of the image.
**The image is cm from the lens on the other side, real, inverted and half the size of the object.
Check against the ray-diagram table.** The object is beyond , which is cm, so the image should lie between and — that is between cm and cm — and be real, inverted and diminished. **The answer of cm with fits exactly. That cross-check between the formula and the table is worth doing every time.
Worked example 2 — object between and .** The same lens, with the object now cm away.
**Real, inverted and twice the size, at cm** — beyond , as the table requires.
**Worked example 3 — the object at .** The same lens, object at cm.
Real, inverted and the same size, at on the other side — the crossover case.
Worked example 4 — object inside the focus, the magnifying glass. The same lens, object at cm.
**The negative means the image is on the same side as the object, so it is virtual**, and the positive magnification of means erect and twice the size. This is a magnifying glass in use.
Worked example 5 — a concave lens. An object is placed cm from a concave lens of focal length cm. Find the image.
**Virtual, erect and diminished, cm from the lens on the same side as the object** — and cm is between the focus at cm and the lens, exactly as the concave-lens rule requires. A concave lens always gives this, whatever the object distance, and if your answer ever comes out real for a concave lens, a sign has been dropped.
The rearrangement worth memorising. Since , the working form is
and with negative that second term is a subtraction in practice. **Substituting without its minus sign is the commonest single error in this chapter**, and it turns a real image into a virtual one.
How do you calculate the power of a lens, and what is a magnifying glass for?
The power of a lens is the reciprocal of its focal length in metres, and it measures how strongly the lens bends light.
with in metres and in dioptre (D). One dioptre is the power of a lens of focal length one metre.
The sign of the power follows the sign of the focal length.
- A convex lens has a positive focal length and therefore a positive power
- A concave lens has a negative focal length and therefore a negative power
And a short focal length means a high power. A lens of focal length cm bends light much more sharply than one of focal length cm, and its power is five times as large — which is why "power" is the natural quantity for an optician to quote.
Worked example 1 — a convex lens. Find the power of a convex lens of focal length cm.
Worked example 2 — a concave lens. Find the power of a concave lens of focal length cm.
Worked example 3 — working backwards. The power of a lens is D. Find its focal length and state its type.
The power is positive, so it is a convex lens of focal length cm.
Worked example 4 — a prescription. An optician prescribes a lens of power D. What is it, and what is its focal length?
**A concave lens of focal length cm, which is the kind used to correct short-sightedness.
The conversion that costs marks. The focal length must be in metres before taking the reciprocal. A focal length of cm gives D, not D, and substituting centimetres directly is the error this section exists to prevent.
The magnifying glass, which is a convex lens used inside its focus. It is also called a simple microscope.
- The object is placed between the lens and its focus**, so is smaller than
- The image formed is virtual, erect and magnified, on the same side as the object, exactly as worked example 4 of the previous section showed
- A shorter focal length gives greater magnification, so a magnifying glass with a strongly curved lens magnifies more
And a magnifying glass has a second use that surprises people. Held far from a page, with the page beyond its focus, the same lens acts as a projector and forms a small inverted real image — which is what you see when you hold it at arm's length. One lens, two behaviours, decided entirely by the object distance.
Other applications of lenses, each worth naming:
- A convex lens as a burning glass, concentrating sunlight from a distant source at its focus
- A camera, which uses a convex lens to form a real, inverted, diminished image on the film or sensor
- A projector, which uses a convex lens with the slide between and to throw a real, inverted, magnified image on a screen
- The eye, whose lens forms a real, inverted, diminished image on the retina
- Spectacles, using convex lenses to correct long sight and concave lenses to correct short sight
- A compound microscope and a telescope, each using two convex lenses
One connection back to the ray-diagram table. The camera, the projector and the burning glass are the three object positions of a convex lens that give real images — **beyond , between and , and at infinity. So the table is not an abstract list: each row of it is a device.**
with in metres and in dioptre (D). One dioptre is the power of a lens of focal length one metre.
The sign of the power follows the sign of the focal length.
- A convex lens has a positive focal length and therefore a positive power
- A concave lens has a negative focal length and therefore a negative power
And a short focal length means a high power. A lens of focal length cm bends light much more sharply than one of focal length cm, and its power is five times as large — which is why "power" is the natural quantity for an optician to quote.
Worked example 1 — a convex lens. Find the power of a convex lens of focal length cm.
Worked example 2 — a concave lens. Find the power of a concave lens of focal length cm.
Worked example 3 — working backwards. The power of a lens is D. Find its focal length and state its type.
The power is positive, so it is a convex lens of focal length cm.
Worked example 4 — a prescription. An optician prescribes a lens of power D. What is it, and what is its focal length?
**A concave lens of focal length cm, which is the kind used to correct short-sightedness.
The conversion that costs marks. The focal length must be in metres before taking the reciprocal. A focal length of cm gives D, not D, and substituting centimetres directly is the error this section exists to prevent.
The magnifying glass, which is a convex lens used inside its focus. It is also called a simple microscope.
- The object is placed between the lens and its focus**, so is smaller than
- The image formed is virtual, erect and magnified, on the same side as the object, exactly as worked example 4 of the previous section showed
- A shorter focal length gives greater magnification, so a magnifying glass with a strongly curved lens magnifies more
And a magnifying glass has a second use that surprises people. Held far from a page, with the page beyond its focus, the same lens acts as a projector and forms a small inverted real image — which is what you see when you hold it at arm's length. One lens, two behaviours, decided entirely by the object distance.
Other applications of lenses, each worth naming:
- A convex lens as a burning glass, concentrating sunlight from a distant source at its focus
- A camera, which uses a convex lens to form a real, inverted, diminished image on the film or sensor
- A projector, which uses a convex lens with the slide between and to throw a real, inverted, magnified image on a screen
- The eye, whose lens forms a real, inverted, diminished image on the retina
- Spectacles, using convex lenses to correct long sight and concave lenses to correct short sight
- A compound microscope and a telescope, each using two convex lenses
One connection back to the ray-diagram table. The camera, the projector and the burning glass are the three object positions of a convex lens that give real images — **beyond , between and , and at infinity. So the table is not an abstract list: each row of it is a device.**
Exam tip
Which steps protect the marks in a lens numerical?
**Write down , and with their signs on separate lines before substituting, and check the answer against the ray-diagram table.
- Make negative for every real object — this is the single most important habit in the chapter
- Make positive for a convex lens and negative for a concave one
- Use ** as the working form, and remember the second term subtracts because is negative
- **Read the sign of : positive means a real image on the far side, negative means a virtual image on the near side
- Read the sign of : negative means inverted and real, positive means erect and virtual
- Convert the focal length to metres before computing the power in dioptre
- Check against the table** — an object beyond must give an image between and , and so on
- Check a concave lens answer: it must always be virtual, erect, diminished, and between the focus and the lens
- Draw the two easy rays — the one parallel to the axis and the one through the optical centre — and mark the focus on both sides
- State the nature, position and size in words at the end, not just a number
The misconception to name. A concave lens cannot form a real image of a real object. Its image is always virtual, erect and diminished, no matter where the object is placed, so it can never be used to throw a picture on a screen or to burn paper with sunlight. If a calculation gives a positive for a concave lens, **a sign has been dropped — almost always the minus on .
A second trap. Taking the power in dioptre from a focal length in centimetres. A cm convex lens has a power of D, because cm is m** — and using directly gives D, a hundred times too small. The unit of power is defined with the focal length in metres, and the conversion has to come first.
- Make negative for every real object — this is the single most important habit in the chapter
- Make positive for a convex lens and negative for a concave one
- Use ** as the working form, and remember the second term subtracts because is negative
- **Read the sign of : positive means a real image on the far side, negative means a virtual image on the near side
- Read the sign of : negative means inverted and real, positive means erect and virtual
- Convert the focal length to metres before computing the power in dioptre
- Check against the table** — an object beyond must give an image between and , and so on
- Check a concave lens answer: it must always be virtual, erect, diminished, and between the focus and the lens
- Draw the two easy rays — the one parallel to the axis and the one through the optical centre — and mark the focus on both sides
- State the nature, position and size in words at the end, not just a number
The misconception to name. A concave lens cannot form a real image of a real object. Its image is always virtual, erect and diminished, no matter where the object is placed, so it can never be used to throw a picture on a screen or to burn paper with sunlight. If a calculation gives a positive for a concave lens, **a sign has been dropped — almost always the minus on .
A second trap. Taking the power in dioptre from a focal length in centimetres. A cm convex lens has a power of D, because cm is m** — and using directly gives D, a hundred times too small. The unit of power is defined with the focal length in metres, and the conversion has to come first.
Did you know
Why can a magnifying glass also set paper alight?
The same lens that enlarges small print will, held in sunlight, concentrate enough energy at a point to scorch paper. Those two uses look unrelated, and they are the same property seen from two directions.
A convex lens takes rays that are spread out and makes them converge. So:
- Rays from a nearby object, spreading widely, are bent inward but not enough to meet — they emerge still diverging, and the eye traces them back to a large virtual image. That is magnification
- Rays from the sun, which arrives so far away that its rays are effectively parallel, are bent inward enough to meet at the focus. All the light falling on the whole area of the lens is squeezed into a tiny spot, and that is what burns
The burning works because of area, not because of any extra energy. The lens does not create light. It collects the light falling on its full aperture and delivers it to a spot a few millimetres across, so the energy arriving per unit area at that spot is enormously larger than it was at the lens. A bigger lens burns faster, and a lens with a shorter focal length concentrates the light into a smaller spot.
Which is why the burning spot is at the focus, and why that gives you the focal length. The distant-object method of the second section is nothing but this: find where the sun burns and you have found the focus.
The same reasoning explains why a water bottle left on a table can be a hazard. A clear bottle full of water is, in cross-section, a crude convex lens — thicker at the middle than at the edges — so sunlight through it can be concentrated onto whatever lies behind. It is also why droplets of water left on leaves after watering at midday can scorch them.
And it explains why the image on a cinema screen is enormous while the one in a camera is tiny. Both use a convex lens; the difference is the object distance.
- **In a camera the object is far beyond , so the image is small, inverted and close to the focus
- In a projector the slide is just beyond , so the image is far away, inverted and hugely magnified
Same lens family, opposite ends of the same table — and this is why the projector image must be put into the projector upside down to come out the right way up on the screen.
One last thing the lens-as-prisms picture explains. A prism deviates every ray by the same amount, so a stack of prisms would send rays to many different points rather than one. A lens focuses because the deviation grows smoothly with distance from the axis. And when that growth is not quite right — as it never is for a perfectly spherical surface — the outer rays meet at a slightly different point from the inner ones, so the image is not perfectly sharp. That is why good camera lenses are built from several elements rather than one**, and it is a genuine limitation of the single thin lens this chapter treats.
A convex lens takes rays that are spread out and makes them converge. So:
- Rays from a nearby object, spreading widely, are bent inward but not enough to meet — they emerge still diverging, and the eye traces them back to a large virtual image. That is magnification
- Rays from the sun, which arrives so far away that its rays are effectively parallel, are bent inward enough to meet at the focus. All the light falling on the whole area of the lens is squeezed into a tiny spot, and that is what burns
The burning works because of area, not because of any extra energy. The lens does not create light. It collects the light falling on its full aperture and delivers it to a spot a few millimetres across, so the energy arriving per unit area at that spot is enormously larger than it was at the lens. A bigger lens burns faster, and a lens with a shorter focal length concentrates the light into a smaller spot.
Which is why the burning spot is at the focus, and why that gives you the focal length. The distant-object method of the second section is nothing but this: find where the sun burns and you have found the focus.
The same reasoning explains why a water bottle left on a table can be a hazard. A clear bottle full of water is, in cross-section, a crude convex lens — thicker at the middle than at the edges — so sunlight through it can be concentrated onto whatever lies behind. It is also why droplets of water left on leaves after watering at midday can scorch them.
And it explains why the image on a cinema screen is enormous while the one in a camera is tiny. Both use a convex lens; the difference is the object distance.
- **In a camera the object is far beyond , so the image is small, inverted and close to the focus
- In a projector the slide is just beyond , so the image is far away, inverted and hugely magnified
Same lens family, opposite ends of the same table — and this is why the projector image must be put into the projector upside down to come out the right way up on the screen.
One last thing the lens-as-prisms picture explains. A prism deviates every ray by the same amount, so a stack of prisms would send rays to many different points rather than one. A lens focuses because the deviation grows smoothly with distance from the axis. And when that growth is not quite right — as it never is for a perfectly spherical surface — the outer rays meet at a slightly different point from the inner ones, so the image is not perfectly sharp. That is why good camera lenses are built from several elements rather than one**, and it is a genuine limitation of the single thin lens this chapter treats.
Exam relevance
How are lenses tested in JEE and NEET?
This is foundation work for Class 12 Ray Optics and Optical Instruments, one of the most reliably examined chapters in both JEE Main and NEET Physics.
Where the lens formula leads. Class 12 keeps unchanged and adds the lens maker's formula, which expresses in terms of the refractive index and the two radii of curvature. The two centres of curvature and the two radii you define here are precisely the quantities that formula uses, and JEE Main sets numericals in which a lens is reground or immersed in a liquid and the new focal length must be found.
Where the sign convention leads. It is used unchanged for mirrors and lenses throughout Class 12, and it is the single most common source of lost marks there as well. **The habit of writing , and with signs on separate lines before substituting transfers exactly, and candidates who never built it lose marks on problems they understood perfectly.
Where the power leads.** Class 12 adds the rule for lenses in contact, , which makes the dioptre far more than a convenience — powers add while focal lengths do not. That is why an optician works in dioptre, and it is a standard JEE Main one-liner. NEET uses it in the questions on correcting defects of vision.
Where the ray-diagram table leads. Class 12 builds the compound microscope and the astronomical telescope from two convex lenses, and the position of the intermediate image in each is read straight off this table. The magnifying glass of this chapter becomes the simple microscope, with its magnification quantified in terms of the least distance of distinct vision.
Where the lens-as-prisms picture leads. The observation that a spherical surface does not focus all rays to exactly one point becomes spherical aberration, and the wavelength dependence of the refractive index becomes chromatic aberration — both named and explained in Class 12.
Question types to expect. At this level: image position and magnification by the lens formula, ray diagrams for named object positions, power in dioptre, and the magnifying glass. In competitive papers: the lens maker's formula, combinations of lenses in contact and separated, magnification of microscopes and telescopes, and the effect of immersing a lens in a liquid.
The single trap that costs marks. Substituting as a positive number. **For a real object is always negative, and omitting the sign turns the subtraction into an addition, which converts a real image into a virtual one and reverses the whole answer. At JEE level the same error propagates through a two-lens system and destroys every subsequent step.
A second trap. Reporting a real image for a concave lens. A concave lens acting alone on a real object always gives a virtual, erect, diminished image between its focus and its optical centre** — so a positive is a signal that the minus on was dropped. In competitive problems a concave lens can contribute to a real final image only as part of a combination, never alone.
Board versus competitive emphasis. The ICSE paper marks the labelled ray diagram, the sign-marked substitution, the nature of the image in words and the unit of the power; a competitive paper marks a single distance, magnification or power, often for a two-lens arrangement. The transferable habit is checking every formula answer against the ray-diagram table — it costs five seconds and it catches a dropped sign before it reaches the next step.
Where the lens formula leads. Class 12 keeps unchanged and adds the lens maker's formula, which expresses in terms of the refractive index and the two radii of curvature. The two centres of curvature and the two radii you define here are precisely the quantities that formula uses, and JEE Main sets numericals in which a lens is reground or immersed in a liquid and the new focal length must be found.
Where the sign convention leads. It is used unchanged for mirrors and lenses throughout Class 12, and it is the single most common source of lost marks there as well. **The habit of writing , and with signs on separate lines before substituting transfers exactly, and candidates who never built it lose marks on problems they understood perfectly.
Where the power leads.** Class 12 adds the rule for lenses in contact, , which makes the dioptre far more than a convenience — powers add while focal lengths do not. That is why an optician works in dioptre, and it is a standard JEE Main one-liner. NEET uses it in the questions on correcting defects of vision.
Where the ray-diagram table leads. Class 12 builds the compound microscope and the astronomical telescope from two convex lenses, and the position of the intermediate image in each is read straight off this table. The magnifying glass of this chapter becomes the simple microscope, with its magnification quantified in terms of the least distance of distinct vision.
Where the lens-as-prisms picture leads. The observation that a spherical surface does not focus all rays to exactly one point becomes spherical aberration, and the wavelength dependence of the refractive index becomes chromatic aberration — both named and explained in Class 12.
Question types to expect. At this level: image position and magnification by the lens formula, ray diagrams for named object positions, power in dioptre, and the magnifying glass. In competitive papers: the lens maker's formula, combinations of lenses in contact and separated, magnification of microscopes and telescopes, and the effect of immersing a lens in a liquid.
The single trap that costs marks. Substituting as a positive number. **For a real object is always negative, and omitting the sign turns the subtraction into an addition, which converts a real image into a virtual one and reverses the whole answer. At JEE level the same error propagates through a two-lens system and destroys every subsequent step.
A second trap. Reporting a real image for a concave lens. A concave lens acting alone on a real object always gives a virtual, erect, diminished image between its focus and its optical centre** — so a positive is a signal that the minus on was dropped. In competitive problems a concave lens can contribute to a real final image only as part of a combination, never alone.
Board versus competitive emphasis. The ICSE paper marks the labelled ray diagram, the sign-marked substitution, the nature of the image in words and the unit of the power; a competitive paper marks a single distance, magnification or power, often for a two-lens arrangement. The transferable habit is checking every formula answer against the ray-diagram table — it costs five seconds and it catches a dropped sign before it reaches the next step.
Key takeaways
What must you be able to do from this part?
One formula, one sign convention and one table of positions.
- A convex or converging lens is thicker at the centre and brings a parallel beam to a real focus; a concave or diverging lens is thinner at the centre and has a virtual focus
- Terms to define: two centres of curvature, two radii, the principal axis, the optical centre, the two principal foci, the focal length and the focal plane, which catches a parallel beam arriving at an angle
- A ray through the optical centre is undeviated, because the lens behaves there like a parallel-sided slab
- A convex lens is two prisms base to base; a concave lens is two prisms apex to apex — but a lens focuses because the deviation increases smoothly with distance from the axis
- **Methods of finding **: the distant-object method (quick but approximate), the plane-mirror no-parallax method (accurate), and the lens formula with a measured and
- The three construction rays: parallel to the axis, through the optical centre, and through the first focus
- Convex lens image table: object at infinity gives a point image at ; beyond gives real, inverted, diminished between and ; at gives real, inverted, same size at ; between and gives real, inverted, magnified beyond ; at gives an image at infinity; inside gives virtual, erect, magnified on the same side
- A concave lens always gives a virtual, erect, diminished image between its focus and the optical centre
- Lens formula: , with
- Sign convention: is negative for a real object, positive for a real image and negative for a virtual one, positive for convex and negative for concave, and negative for an inverted image
- ** cm with cm** gives cm and ; ** cm** gives cm and ; ** cm** gives cm and
- ** cm with cm** gives cm and — a magnifying glass
- ** cm with cm** gives cm and , virtual and diminished
- Power with in metres, measured in dioptre — cm gives D, cm gives D, and D means a convex lens of cm
- A magnifying glass is a convex lens used inside its focus, and a shorter focal length magnifies more
- Applications: burning glass, camera, projector, the eye, spectacles, microscope and telescope
The sharpest self-test needs a magnifying glass and a window. Hold it near a page and then at arm's length, describe each image in three words, and then predict from the lens formula which of the two should be catchable on a piece of paper — and check.
- A convex or converging lens is thicker at the centre and brings a parallel beam to a real focus; a concave or diverging lens is thinner at the centre and has a virtual focus
- Terms to define: two centres of curvature, two radii, the principal axis, the optical centre, the two principal foci, the focal length and the focal plane, which catches a parallel beam arriving at an angle
- A ray through the optical centre is undeviated, because the lens behaves there like a parallel-sided slab
- A convex lens is two prisms base to base; a concave lens is two prisms apex to apex — but a lens focuses because the deviation increases smoothly with distance from the axis
- **Methods of finding **: the distant-object method (quick but approximate), the plane-mirror no-parallax method (accurate), and the lens formula with a measured and
- The three construction rays: parallel to the axis, through the optical centre, and through the first focus
- Convex lens image table: object at infinity gives a point image at ; beyond gives real, inverted, diminished between and ; at gives real, inverted, same size at ; between and gives real, inverted, magnified beyond ; at gives an image at infinity; inside gives virtual, erect, magnified on the same side
- A concave lens always gives a virtual, erect, diminished image between its focus and the optical centre
- Lens formula: , with
- Sign convention: is negative for a real object, positive for a real image and negative for a virtual one, positive for convex and negative for concave, and negative for an inverted image
- ** cm with cm** gives cm and ; ** cm** gives cm and ; ** cm** gives cm and
- ** cm with cm** gives cm and — a magnifying glass
- ** cm with cm** gives cm and , virtual and diminished
- Power with in metres, measured in dioptre — cm gives D, cm gives D, and D means a convex lens of cm
- A magnifying glass is a convex lens used inside its focus, and a shorter focal length magnifies more
- Applications: burning glass, camera, projector, the eye, spectacles, microscope and telescope
The sharpest self-test needs a magnifying glass and a window. Hold it near a page and then at arm's length, describe each image in three words, and then predict from the lens formula which of the two should be catchable on a piece of paper — and check.