Difficulty: Introductory to Intermediate | Prerequisites: Part 1 of these notes (ray model, reflection, refraction, Snell's law), basic algebra, trigonometry, comfort with small-angle approximations.
This second half of Chapter 18 takes the ray model from Part 1 and applies it to image formation. You will learn how refraction at a flat boundary shifts where objects appear to be, how converging and diverging lenses bend rays to produce real or virtual images, how concave and convex mirrors do the same, and how the thin lens equation ties everything together with a single formula. These tools are the building blocks for understanding cameras, eyeglasses, microscopes and telescopes.
Lenses and curved mirrors form images by bending light rays to converge or diverge. The thin lens equation (1/s + 1/s' = 1/f) and the magnification formula (m = -s'/s) tell you where the image lands, how big it is and whether it is upright or inverted. Converging optics can make both real and virtual images depending on object placement; diverging optics always make virtual, upright, smaller images.
Converging lens
A lens that is thicker in the middle than at the edges. Parallel rays passing through it bend inward and meet at the far focal point. Think of it as a lens that gathers light to a point.
Diverging lens
A lens that is thinner in the middle than at the edges. Parallel rays passing through it spread outward, and the diverging rays appear to come from the near focal point. Think of it as a lens that spreads light apart.
Focal point (F)
The point where parallel rays converge (converging lens/mirror) or appear to diverge from (diverging lens/mirror) after interacting with the optic.
Focal length (f)
The distance from the centre of a lens (or surface of a mirror) to its focal point. Positive for converging optics, negative for diverging optics.
Real image
An image formed where actual light rays converge and pass through. You can project a real image onto a screen. Real images are inverted.
Virtual image
An image formed where light rays appear to diverge from but do not actually pass through. You cannot project it on a screen, but your eye can see it. Virtual images are upright.
Object distance (s)
The distance from the object to the lens or mirror. By convention, s is always positive.
Image distance (s')
The distance from the lens or mirror to the image. Positive for real images (opposite side of a lens from the object, or in front of a mirror), negative for virtual images.
Magnification (m)
The ratio of image height to object height: m = -s'/s. When |m| > 1, the image is larger; when |m| < 1, it is smaller. Negative m means inverted; positive m means upright.
Optical axis
The straight line passing through the centre of a lens or mirror, perpendicular to its surface. Rays along this line pass through undeviated.
Concave mirror (converging mirror)
A mirror whose reflective surface curves inward, like the inside of a spoon. Parallel rays reflect and converge at the focal point.
Convex mirror (diverging mirror)
A mirror whose reflective surface curves outward, like the back of a spoon. Parallel rays reflect and diverge; their extensions converge at a virtual focal point behind the mirror.
Radius of curvature (R)
The radius of the sphere that the mirror surface is part of. Related to focal length by f = R / 2.
Principal rays
The three standard rays used in ray tracing to locate an image: (1) a ray parallel to the optical axis, (2) a ray through the focal point, (3) a ray through the centre of the lens or mirror.
When light crosses a flat boundary between two media, refraction shifts the apparent position of an object.
Setup
An object at point P sits in medium 1 (index n₁). Light crosses into medium 2 (index n₂). The observer is in medium 2.
Assuming n₁ > n₂ (for example, an object under water viewed from air).
Virtual image location
Rays diverging from P refract at the boundary. When extended backward, the refracted rays appear to come from a point P' that is closer to the boundary than P actually is.
The virtual image looks like it is behind the surface, in the same medium as the object.
Small-angle approximation
For rays close to the optical axis (small angles), tan θ ≈ sin θ.
Using Snell's law with this approximation: s' = s (n₂ / n₁), where s is the actual object distance and s' is the apparent (image) distance.
This is why a pool looks shallower than it is: n_water > n_air, so s' < s.
Real images by refraction
When rays from an object refract and actually converge on the other side of the boundary, a real image forms. All rays leaving one point on the object converge to one point in the image plane.
Real images are inverted relative to the object.
A thin lens is one whose thickness is negligible compared with its focal length. All the bending is treated as happening at a single plane.
Converging lens
Thicker in the middle. Parallel rays converge at the far focal point.
The near focal point is the same distance f from the lens, on the object side. Rays passing through it emerge parallel.
Focal length f is positive.
Diverging lens
Thinner in the middle. Parallel rays diverge after passing through; their backward extensions converge at the near focal point.
Focal length f is negative.
Three principal rays for ray tracing
Ray 1 (parallel ray): Travels parallel to the optical axis, then refracts through (or away from) the far focal point.
Ray 2 (focal ray): Passes through (or aims toward) the near focal point, then emerges parallel to the axis.
Ray 3 (central ray): Passes straight through the centre of the lens without bending.
Where any two of these rays intersect on the far side, a real image forms. If they diverge, extend them backward to find the virtual image.
Converging lens: object outside the focal point
Rays converge on the far side of the lens, producing a real, inverted image.
The image can be larger or smaller than the object, depending on object distance.
Converging lens: object inside the focal point
Rays diverge after passing through the lens. Extended backward, they appear to come from a virtual image on the same side as the object.
The virtual image is upright and magnified. This is how a magnifying glass works.
Diverging lens: any object position
Diverging lenses always produce virtual, upright images that are smaller than the object and located closer to the lens than the object.
Magnification
|m| = h' / h, where h' is image height and h is object height.
m = -s' / s.
If m < 0, the image is inverted. If m > 0, the image is upright.
Spherical mirrors work on the same principles as lenses, but with reflection instead of refraction.
Concave mirror (converging)
Reflective surface curves inward. Parallel rays converge at the focal point, which is in front of the mirror.
Focal length f is positive.
Can produce real or virtual images depending on object placement.
Convex mirror (diverging)
Reflective surface curves outward. Parallel rays diverge after reflection; their backward extensions meet at a virtual focal point behind the mirror.
Focal length f is negative.
Always produces virtual, upright, reduced images.
Principal rays for mirrors
Converging mirror: (1) A ray parallel to the axis reflects through the focal point. (2) A ray through the focal point reflects parallel to the axis. (3) A ray aimed at the centre of curvature reflects back on itself.
Diverging mirror: (1) A ray parallel to the axis reflects as if it came from the focal point behind the mirror. (2) A ray aimed toward the focal point behind the mirror reflects parallel to the axis. (3) A ray aimed at the centre of curvature reflects back on itself.
Concave mirror image types
Object beyond the centre of curvature: real, inverted, smaller image.
Object at the centre of curvature: real, inverted, same-size image.
Object between the centre of curvature and the focal point: real, inverted, larger image.
Object inside the focal point: virtual, upright, larger image (behind the mirror).
Radius of curvature
R = 2f. The centre of curvature is twice the focal length from the mirror surface.
Summary of mirror types
Plane mirror: virtual image, not flipped, same distance behind as object is in front.
Concave mirror: real and flipped when the object is outside f; virtual and upright when inside f. No virtual image forms at the focal point itself.
Convex mirror: always virtual, upright, smaller. No real image.
The thin lens equation applies to both thin lenses and spherical mirrors (the mirror equation has the same form).
The equation
1/s + 1/s' = 1/f
s = object distance, s' = image distance, f = focal length.
Sign conventions (critical to get right)
s is always positive (the object is always on the incoming-light side).
s' is positive for a real image (opposite side of a lens from the object, or in front of a mirror). s' is negative for a virtual image (same side as the object for a lens, or behind a mirror).
f is positive for converging lenses and converging (concave) mirrors. f is negative for diverging lenses and diverging (convex) mirrors.
m is positive when the image is upright, negative when inverted.
Converging lens or mirror, object outside f
Produces a real, inverted image on the opposite side. s' is positive, m is negative.
Converging lens or mirror, object inside f
Produces a virtual, upright image. s' is negative, m is positive.
Diverging lens or mirror, any object position
Always produces a virtual, upright image closer to the lens/mirror than the object. s' is negative, m is positive, |m| < 1.
Using the equation
To find the image distance: 1/s' = 1/f - 1/s.
To find magnification: m = -s'/s.
A negative s' means the image is virtual (on the same side as the object for a lens, behind the mirror for a mirror). A positive s' means the image is real.
Thin lens / mirror equation: 1/s + 1/s' = 1/f
Magnification: m = -s'/s = h'/h
Radius of curvature to focal length: f = R/2
Image distance by refraction at a flat boundary (small angle): s' = s (n₂ / n₁)
Sign conventions summary: s always positive; f positive for converging, negative for diverging; s' positive for real images, negative for virtual; m positive for upright, negative for inverted.
Converging lenses are the basis of cameras, projectors and the human eye, all of which form real images on a sensor, screen or retina. A magnifying glass is a converging lens used with the object inside the focal length, producing a magnified virtual image. Convex mirrors are used as car wing mirrors and shop security mirrors because they always produce upright images with a wide field of view (the trade-off is that objects appear smaller and further away than they are). Concave mirrors are used in telescopes (reflecting telescopes), torch reflectors and satellite dishes to focus incoming parallel rays to a point.
Students often assume that blocking half a lens removes half the image. It does not. The full image still forms; it just gets dimmer, because fewer rays contribute.
A frequent error is forgetting the sign conventions. If you plug in a positive f for a diverging lens, every answer comes out wrong. Diverging optics always carry a negative f.
Students sometimes think virtual images are invisible or "not real." Your eye sees virtual images perfectly well (you see one every time you look in a mirror). "Virtual" only means the light rays do not physically pass through the image location.
Confusing magnification with image distance is common. A large |m| means the image is bigger than the object, but it says nothing on its own about where the image sits. Use the thin lens equation for position, magnification for size.
⚠️ The thin lens equation (1/s + 1/s' = 1/f) and magnification (m = -s'/s) will almost certainly appear. Practise solving for each variable.
⚠️ Ray-tracing diagrams are high-value exam questions. Know the three principal rays for both lenses and mirrors.
⚠️ Sign conventions are the most common source of lost marks. Memorise the table: s always positive; f positive for converging, negative for diverging; s' positive for real, negative for virtual; m positive for upright, negative for inverted.
⚠️ Be ready to identify image type (real/virtual, upright/inverted, magnified/reduced) from a given object position and optic type.
⚠️ The relationship R = 2f for mirrors is frequently tested, sometimes as part of a multi-step problem.
True or false: A diverging lens can produce a real image. (False – diverging lenses always produce virtual images.)
Fill in the blank: The focal length of a concave mirror is ______ (positive/negative). (Positive)
True or false: If m = -2, the image is inverted and twice the height of the object. (True)
Fill in the blank: The relationship between focal length and radius of curvature for a spherical mirror is f = ___. (R/2)
True or false: A converging lens with the object placed inside the focal point produces a real, inverted image. (False – it produces a virtual, upright image.)
Q: An object is placed 30 cm from a converging lens with a focal length of 10 cm. Where does the image form, and what is the magnification?
A: 1/s' = 1/f - 1/s = 1/10 - 1/30 = 3/30 - 1/30 = 2/30. So s' = 15 cm. The image is real (s' positive) and on the far side of the lens. Magnification m = -s'/s = -15/30 = -0.5. The image is inverted and half the size of the object.
Q: An object is placed 5 cm from a converging lens with a focal length of 10 cm. Describe the image.
A: 1/s' = 1/10 - 1/5 = 1/10 - 2/10 = -1/10. So s' = -10 cm. The negative sign means the image is virtual, on the same side as the object. m = -(-10)/5 = +2. The image is upright and twice the size of the object.
Q: A concave mirror has a radius of curvature of 40 cm. An object is placed 30 cm from the mirror. Find the image distance and describe the image.
A: f = R/2 = 20 cm. 1/s' = 1/20 - 1/30 = 3/60 - 2/60 = 1/60. So s' = 60 cm. The image is real (positive s'), in front of the mirror. m = -60/30 = -2. The image is inverted and twice the size.
Q: A diverging lens has a focal length of -15 cm. An object is 20 cm away. Where is the image?
A: 1/s' = 1/(-15) - 1/20 = -4/60 - 3/60 = -7/60. So s' ≈ -8.6 cm. The image is virtual (negative s'), upright, and on the same side as the object. m = -(-8.6)/20 ≈ +0.43. The image is smaller than the object.
Q: Why does a convex mirror always produce a virtual image, regardless of where the object is placed?
A: A convex mirror has a negative focal length, and reflected rays always diverge. Because diverging rays never converge on the reflective side, the image can only be found by extending the rays backward behind the mirror. The result is always a virtual, upright, reduced image.
The thin lens equation reappears in later chapters on optical instruments: compound microscopes use two converging lenses in series, and refracting telescopes do the same at different scales. Mirrors connect to wave optics through the concept of focal points and image formation, which becomes relevant when studying diffraction-limited resolution. The sign conventions here also carry directly into the lensmaker's equation and multi-lens systems. If you continue to electromagnetism, the index of refraction ties back to the dielectric properties of the medium.
Thin lens equation, mirror equation, converging lens, diverging lens, concave mirror, convex mirror, focal length, focal point, real image, virtual image, magnification, image distance, object distance, ray tracing, principal rays, radius of curvature, sign conventions optics, upright image, inverted image, magnifying glass, optical axis, lens plane, image plane, object plane, spherical mirror, R = 2f