Light – Reflection and Refraction

The chapter has two connected halves, both built on the straight-line travel of light. The first half, reflection by spherical mirrors, introduces the vocabulary (pole, centre of curvature, principal axis, principal focus, focal length), the rule $R = 2f$, the four standard rays used for ray diagrams, and the image tables for concave and convex mirrors, then formalises this with the New Cartesian Sign Convention and the mirror formula plus magnification, worked through NCERT's Examples 9.1 and 9.2. The second half, refraction, starts from everyday observations — a raised pond bottom, a displaced pencil — states the laws of refraction including Snell's law, defines refractive index through the speed of light, and studies refraction through a glass slab. It then extends the same machinery to spherical lenses: vocabulary, ray-construction rules, image tables, sign convention, lens formula, magnification and power in dioptres, worked through Examples 9.3 and 9.4. Carry two ideas through both halves: every image is located by intersecting just two rays, and every calculation depends on getting the sign of each distance right before substituting.

Spherical mirrors: key terms and R = 2f

A spherical mirror's reflecting surface forms part of a sphere. If the surface curves inward, toward the centre of that sphere, it is a concave mirror; if it curves outward, away from the centre, it is a convex mirror. The centre of the sphere is the centre of curvature (C), lying in front of a concave mirror and behind a convex one. The midpoint of the reflecting surface is the pole (P), and the radius of the sphere, the distance PC, is the radius of curvature (R). A straight line through P and C is the principal axis, normal to the mirror at P. Rays parallel to the principal axis, after reflection, converge at (concave) or appear to diverge from (convex) a single point on the axis called the principal focus (F); the distance PF is the focal length (f). For a mirror of small aperture — reflecting surface diameter much smaller than R — experiment shows the focus lies exactly halfway between pole and centre of curvature:

where: $R$ is the radius of curvature and $f$ is the focal length, both measured from the pole along the principal axis. This single relation is why locating C on a ray diagram automatically fixes F, and it holds for both concave and convex mirrors.

Ray diagrams and image formation by mirrors

Any two of four standard rays fix an image, since where two reflected rays cross (or appear to cross) is where the image forms. A ray parallel to the principal axis reflects through F for a concave mirror, or reflects so that it appears to come from F for a convex mirror. A ray directed through F (concave) or toward F (convex) emerges parallel to the axis after reflection. A ray directed through C, striking the mirror along the normal, reflects straight back along its own path. A ray striking the pole obliquely reflects making an equal angle with the axis on the other side, exactly as ordinary reflection from a flat surface would. For a concave mirror, the object's position relative to F and C entirely decides the image: at infinity gives a point-sized real inverted image at F; beyond C gives a diminished real inverted image between F and C; at C gives a same-size real inverted image at C; between C and F gives an enlarged real inverted image beyond C; at F sends the image to infinity; and between P and F — the one case with no real image — gives an enlarged, virtual, erect image behind the mirror. A convex mirror never runs through this sequence: for every object position it forms a virtual, erect, diminished image that stays between P and F, behind the mirror, exactly what the spoon and rear-view mirror show — a convex surface cannot converge rays into a real image.

The New Cartesian Sign Convention and the mirror formula

Numerical work needs a consistent bookkeeping rule, the New Cartesian Sign Convention. The pole P is the origin and the principal axis is the x-axis. The object is always placed to the left, so light travels left to right. Distances measured in the direction of incident light (to the right of P) are positive; distances measured against it (to the left of P) are negative. Heights measured upward, above the axis, are positive; heights measured downward are negative. Under this convention the object distance $u$ is always negative, the focal length of a concave mirror is negative and that of a convex mirror is positive, and the sign of $v$ that comes out of a calculation tells you where the image actually is. With these signs, the object distance ($u$), image distance ($v$) and focal length ($f$) of any spherical mirror are related by the mirror formula:

where: all three distances are measured from the pole P along the principal axis and carry the signs fixed by the convention above; this single equation reproduces every row of the image table once the correct signs go in. Magnification, the ratio of image height to object height, is given by

where: $h$ is object height (positive, since the object stands above the axis) and $h'$ is image height, positive for a virtual image and negative for a real image; correspondingly, a negative $m$ signals a real, inverted image and a positive $m$ signals a virtual, erect one.

Refraction, Snell's law and refractive index

Light bends when it crosses obliquely from one transparent medium into another, because its speed changes — this is refraction, and it is why a pencil in a glass of water looks displaced and a pond's bottom looks raised. Two laws govern it: the incident ray, the refracted ray and the normal at the point of incidence all lie in one plane, and for a given pair of media and a given colour of light, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant — Snell's law.

where: $i$ is the angle of incidence and $r$ the angle of refraction, both from the normal, and $n_{21}$ is the refractive index of medium 2 with respect to medium 1, valid for angles of incidence between zero and ninety degrees. This constant ties directly to speed: $n_{21} = v_1/v_2$. When medium 1 is air or vacuum, $n_m = c/v$ is the absolute refractive index of the medium — water's is 1.33, crown glass's 1.52, diamond's 2.42. A medium with a larger refractive index is optically denser; light entering a denser medium slows down and bends toward the normal, and entering a rarer medium it speeds up and bends away from the normal. Optical density is not mass density — kerosene has a higher refractive index than water, so it is optically denser, even though it floats on water.

Refraction through a rectangular glass slab

Shine a ray obliquely onto a rectangular glass slab and it bends toward the normal entering the glass (rarer to denser), travels straight inside, then bends away from the normal leaving the glass (denser to rarer). Because the two faces of the slab are parallel, the bending at entry and exit is equal and opposite, so the emergent ray leaves parallel to the original incident ray — only shifted a little sideways, a lateral shift. A ray striking a boundary exactly along the normal (zero angle of incidence) passes straight through undeviated at every interface, since there is nothing to bend it toward or away from.

Spherical lenses: convex, concave and ray diagrams

A lens is a transparent solid bounded by at least one spherical surface. A convex lens, thicker at the middle than at the edges, bends parallel rays inward to a real focus and is called converging; a concave lens, thicker at the edges, spreads parallel rays outward from an apparent focus and is called diverging. The optical centre (O) is the lens's central point, through which a ray passes without any deviation; the principal axis joins the two centres of curvature; and, since a lens has two faces, it has two principal foci, $F_1$ on the incoming side and $F_2$ on the outgoing side, each at focal length $f$ from O. Ray construction uses any two of three rays: parallel to the axis, refracting through $F_2$ (convex) or appearing to diverge from $F_1$ (concave); through or aimed at a principal focus, emerging parallel to the axis; and through the optical centre, passing straight through undeviated. A convex lens runs through the same full sequence as a concave mirror as the object approaches: infinity gives a point image at $F_2$; beyond $2F_1$ gives a diminished real inverted image between $F_2$ and $2F_2$; at $2F_1$ gives a same-size real inverted image at $2F_2$; between $F_1$ and $2F_1$ gives an enlarged real inverted image beyond $2F_2$; at $F_1$ sends the image to infinity; and between $F_1$ and O gives an enlarged, virtual, erect image on the object's own side. A concave lens, like a convex mirror, never varies: every position gives a virtual, erect, diminished image between $F_1$ and O.

Sign convention, lens formula, magnification and power

Lenses use the same left-negative, right-positive, up-positive, down-negative rules as mirrors, except that every distance is now measured from the optical centre O rather than the pole. Under this convention a convex lens has positive focal length and a concave lens has negative focal length. The object distance ($u$), image distance ($v$) and focal length ($f$) of any spherical lens are related by the lens formula:

where: the only difference from the mirror formula is the minus sign before $1/u$, a direct consequence of the object and a real image now lying on opposite sides of the lens rather than the same side of a mirror (the second derivation below shows exactly why). Magnification is

where: unlike the mirror case there is no leading minus sign, because $u$ and $v$ already carry opposite signs for a real image, which automatically makes $m$ negative. A lens's ability to bend light is expressed as power,

where: $f$ must be in metres, giving $P$ in dioptres (D), the SI unit of lens power; one dioptre is the power of a lens of focal length one metre. Power is positive for a converging convex lens and negative for a diverging concave lens, and when several thin lenses are held in contact their powers simply add, $P = P_1 + P_2 + P_3 + \ldots$ — the arithmetic an optician uses while testing your eyes.

Board level - A convex rear-view mirror (NCERT Example 9.1, recomputed)

A convex mirror used for rear-view on an automobile has a radius of curvature of 3.00 m. If a bus is located 5.00 m from this mirror, find the position, nature and size of the image relative to the bus.

Step 1. List the signed quantities. Radius of curvature $R = +3.00$ m (positive, since a convex mirror's centre of curvature lies behind it). Focal length $f = R/2 = +1.50$ m. Object distance $u = -5.00$ m (object always to the left, hence negative).

Step 2. Substitute into the mirror formula, $1/v = 1/f - 1/u = 1/1.50 - 1/(-5.00) = 0.6667 + 0.2000 = 0.8667$.

Step 3. Invert: $v = 1/0.8667 = +1.15$ m. The positive sign means the image forms 1.15 m behind the mirror, on the same side as a virtual image always does for a convex mirror.

Step 4. Magnification, $m = -v/u = -(+1.15)/(-5.00) = +0.23$. Answer: the image forms 1.15 m behind the mirror, virtual, erect, and diminished to 0.23 times the bus's actual size — why a convex mirror gives a wide, safe rear view rather than a true-size one.

Board level - Screen distance for a concave mirror (NCERT Example 9.2, recomputed)

An object 4.0 cm tall is placed 25.0 cm in front of a concave mirror of focal length 15.0 cm. Find where a screen must be placed for a sharp image, and describe the image's nature and size.

Step 1. List the signed quantities. Object height $h = +4.0$ cm. Object distance $u = -25.0$ cm. Focal length $f = -15.0$ cm (negative, since a concave mirror's focus is in front of it).

Step 2. Substitute into the mirror formula, $1/v = 1/f - 1/u = 1/(-15.0) - 1/(-25.0) = -0.0667 + 0.0400 = -0.0267$.

Step 3. Invert: $v = 1/(-0.0267) = -37.5$ cm. The negative sign places the image 37.5 cm in front of the mirror — that is where the screen must be positioned to catch a sharp image.

Step 4. Magnification, $m = -v/u = -(-37.5)/(-25.0) = -1.5$. Image height $h' = m \times h = -1.5 \times 4.0 = -6.0$ cm. Answer: the screen goes 37.5 cm in front of the mirror; the image is real, inverted, and enlarged to 6.0 cm tall (1.5 times the object).

Board HOTS / foundation JEE-NEET - A ray crossing air, glass and water in sequence

A ray of light strikes a glass slab (refractive index 1.50) from air at an angle of incidence of 30 degrees. The slab's lower face is in contact with water (refractive index 1.33) instead of air. Find the angle of refraction inside the glass, and the angle at which the ray finally enters the water, using the chain rule from the derivation above.

Step 1. Apply the general Snell's law at the air-glass interface: $n_{\text{air}}\sin\theta_{\text{air}} = n_{\text{glass}}\sin\theta_{\text{glass}}$, so $1 \times \sin 30^{\circ} = 1.50 \times \sin\theta_{\text{glass}}$. Since $\sin 30^{\circ} = 0.500$, this gives $\sin\theta_{\text{glass}} = 0.500/1.50 = 0.3333$, so $\theta_{\text{glass}} = 19.5^{\circ}$ — bent toward the normal, since glass is optically denser than air, exactly as the traps below expect.

Step 2. Because the glass slab's two faces are parallel, the ray's angle of incidence at the lower glass-water face equals its angle of refraction at the upper face, $19.5^{\circ}$ (the same reasoning as the sideways-shift discussion for a glass slab in air).

Step 3. Apply the general Snell's law again at the glass-water interface: $n_{\text{glass}}\sin\theta_{\text{glass}} = n_{\text{water}}\sin\theta_{\text{water}}$, so $1.50 \times 0.3333 = 1.33 \times \sin\theta_{\text{water}}$, giving $\sin\theta_{\text{water}} = 0.500/1.33 = 0.3759$, so $\theta_{\text{water}} = 22.1^{\circ}$ — bent away from the normal compared with the glass angle, since water is optically rarer than glass.

Step 4. Check using the conserved invariant from the derivation: $n_{\text{air}}\sin\theta_{\text{air}} = 1 \times 0.500 = 0.500$, and $n_{\text{water}}\sin\theta_{\text{water}} = 1.33 \times 0.3759 = 0.500$ — the two match, confirming the chain of interfaces was handled consistently. Answer: the ray refracts to 19.5 degrees inside the glass, and enters the water at 22.1 degrees from the normal. Timing note: aim for under three minutes. The chain is one repeated move — apply $n_1\sin\theta_1 = n_2\sin\theta_2$ at each interface, carrying the previous angle forward as the next angle of incidence — so spot the parallel-face shortcut (refraction angle at one face equals incidence angle at the next) before touching a calculator; recomputing geometry from scratch at the second face is what eats up time.

The single most repeated slip is dropping or misplacing signs: forgetting that the object distance $u$ is always negative, or mixing up that a concave mirror's focal length is negative while a convex lens's is positive. A second common error is mixing up the two magnification formulas — $m = -v/u$ for mirrors carries a minus sign that $m = v/u$ for lenses does not, because a lens's real image sits opposite the object while a mirror's sits on the same side, as the two derivations above show. Students also apply a mirror's ray-construction rules to a lens or vice versa — a lens has no equivalent of "the ray through the centre of curvature reflects straight back"; instead its ray through the optical centre passes straight through undeviated. Another frequent error is assuming a convex mirror or concave lens can, for some special object position, produce a real or enlarged image — neither ever can; both always give a virtual, erect, diminished image regardless of object position. In refraction, students often invert the refractive-index ratio, writing $n_{21} = v_2/v_1$ instead of $v_1/v_2$, or forget that optical density and mass density differ (kerosene bends light more than water despite floating on it). Finally, Snell's law and the bending rules apply only to oblique incidence — a ray along the normal passes straight through without bending, even though its speed still changes.

• State the laws of reflection and identify the pole, centre of curvature, principal axis, principal focus and focal length of a spherical mirror, using $R = 2f$.
• Construct a ray diagram for image formation by a concave or convex mirror using any two of the four standard rays, for a stated object position.
• Apply the New Cartesian Sign Convention correctly to substitute $u$, $v$, $f$, $R$ and $h$ into the mirror formula and solve for an unknown quantity.
• Compute the magnification produced by a mirror and read its sign to state whether the image is real and inverted or virtual and erect.
• Explain refraction in terms of a change in the speed of light, state the two laws of refraction, and apply Snell's law, $\sin i/\sin r = n_{21}$.
• Compute absolute and relative refractive index from speed-of-light data, and explain why optical density and mass density are different properties.
• Construct a ray diagram for image formation by a convex or concave lens using any two of the three standard rays, and state the resulting image's nature, position and size.
• Apply the lens formula and its magnification formula with the correct sign convention, and calculate the power of a lens (and of lenses in contact) in dioptres.

Ask the tutor: Want to work through more ray diagrams and mirror or lens formula numericals together, or check your own worked solution against the sign convention step by step?