Light – Reflection and Refraction

Nature and Propagation of Light

Light exhibits straight-line propagation under everyday macroscopic conditions, casting sharp shadows of opaque objects along defined linear trajectories known as rays of light. When an opaque obstacle encountered by a beam becomes exceedingly small, light bends around its edges instead of propagating along a purely straight path, a phenomenon termed the diffraction of light. This diffraction demonstrates the limitation of ray optics and requires modeling light as a wave. In phenomena involving the interaction of light with matter explored in the early twentieth century, the wave model proves inadequate because light displays particle behavior resembling a stream of discrete packets. Modern quantum theory resolves this conflict by reconciling particle properties with wave nature, establishing that light possesses a dual character rather than acting strictly as a classical wave or particle.

Reflection of Light and Spherical Mirrors

Reflection at any smooth interface conforms to two universal laws: the angle of incidence equals the angle of reflection, and the incident ray, the reflected ray, and the normal to the reflecting surface at the point of incidence all reside within the same plane. An ideal plane mirror generates an image that is virtual, erect, laterally inverted, identical in size to the object, and positioned at an equal distance behind the reflecting surface as the object is in front of it. When reflecting surfaces are curved, they are classified as spherical mirrors if their surface constitutes a segment of a sphere. A spherical mirror whose reflecting surface curves inward toward the centre of the sphere is a concave mirror, whereas a spherical mirror whose reflecting surface curves outward is a convex mirror.

Specific geometric landmarks define spherical mirrors. The pole ($P$) represents the geometric centre of the reflecting surface and lies directly on the mirror. The centre of curvature ($C$) is the centre of the original sphere of which the mirror forms a part; it lies in front of a concave mirror and behind a convex mirror. The radius of curvature ($R$) corresponds to the radius of this sphere, matching the linear distance between $P$ and $C$. The principal axis is the straight line traversing both the pole and the centre of curvature, oriented perpendicular to the mirror surface at $P$. Rays propagating parallel to the principal axis reflect to converge at a single point on the principal axis in a concave mirror, or appear to diverge from such a point in a convex mirror. This point is the principal focus ($F$), and the distance from $P$ to $F$ is the focal length ($f$). For mirrors with an aperture much smaller than their radius of curvature, the focus lies precisely midway between the pole and centre of curvature, establishing the relation: $R = 2f$

Image Formation by Spherical Mirrors

Tracing the path of at least two known rays from any point on an extended object determines the position and characteristics of an image. A ray traveling parallel to the principal axis reflects directly through the principal focus of a concave mirror or diverges along a line passing through the focus of a convex mirror. A ray passing through or directed toward the principal focus emerges parallel to the principal axis upon reflection. A ray directed through the centre of curvature strikes the mirror along the surface normal and reflects back along its original trajectory. A ray incident obliquely at the pole reflects obliquely at an equal angle with respect to the principal axis.

In a concave mirror, an object located at infinity forms a real, inverted, point-sized image at $F$. An object placed beyond $C$ produces a diminished, real, inverted image located between $F$ and $C$. When the object is situated at $C$, the image is real, inverted, identical in size, and formed at $C$. Positioning the object between $C$ and $F$ results in an enlarged, real, inverted image situated beyond $C$. An object placed exactly at $F$ yields a highly magnified real image at infinity. When the object is placed between the pole $P$ and the focus $F$, the reflected rays diverge, producing an enlarged, virtual, and erect image behind the mirror. Concave mirrors are used in headlights, torches, and searchlights to produce parallel beams, in shaving and dental mirrors to provide magnified views of faces and teeth, and in solar furnaces to concentrate sunlight.

Convex mirrors always produce virtual, erect, and diminished images behind the mirror regardless of object position. When an object is at infinity, the image is formed at the focus $F$ as a highly diminished point; when placed at a finite distance from the mirror, the image forms between $P$ and $F$. Due to their outward curvature, convex mirrors provide an exceptionally wide field of view, making them standard rear-view mirrors in vehicles to view large regions of trailing traffic.

Sign Convention, Mirror Formula, and Magnification

Calculations for spherical mirrors employ the New Cartesian Sign Convention, wherein the pole ($P$) acts as the origin and the principal axis serves as the horizontal x-axis. The object is always oriented to the left of the mirror, ensuring incident light travels from left to right. All linear distances along the principal axis are measured from $P$. Distances measured in the direction of incident light (to the right of the origin) are assigned positive values, while distances measured against the direction of incident light (to the left of the origin) are negative. Heights measured perpendicularly above the principal axis are positive, and heights measured perpendicularly downwards are negative. Consequently, the focal length ($f$) of a concave mirror is negative, whereas the focal length of a convex mirror is positive.

The mirror formula defines the mathematical dependency linking object distance ($u$), image distance ($v$), and focal length ($f$): $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

Magnification ($m$) describes the ratio between the height of the image ($h'$) and the height of the object ($h$). It also relates directly to image distance and object distance: $m = \frac{h'}{h} = -\frac{v}{u}$

A negative sign in magnification signifies an inverted, real image, whereas a positive sign indicates an erect, virtual image.

Refraction of Light and Snell's Law

Refraction is the change in the direction of light propagation when traversing obliquely from one transparent medium into another, resulting from variations in the speed of light across different substances. Light reaches its maximum speed in a vacuum at $3 \times 10^8\,\text{m}\,\text{s}^{-1}$, traveling slightly slower in air and substantially slower in dense media like water and glass. When light traverses a rectangular glass slab obliquely, it refracts toward the normal at the entering air-glass boundary and away from the normal by an equal and opposite magnitude at the exiting glass-air boundary. This renders the emergent ray strictly parallel to the incident ray, displaced sideways by a distance termed lateral shift.

Refraction obeys two fundamental laws. The incident ray, the refracted ray, and the normal at the boundary point of incidence all lie within the identical plane. Furthermore, for light of a given color and a specified pair of media, the ratio of the sine of the angle of incidence ($i$) to the sine of the angle of refraction ($r$) remains constant, known as Snell's law: $\frac{\sin(i)}{\sin(r)} = \text{constant} = n_{21}$

The relative refractive index ($n_{21}$) represents the ratio of the speed of light in medium 1 ($v_1$) to that in medium 2 ($v_2$): $n_{21} = \frac{v_1}{v_2}$

The absolute refractive index ($n$) describes the ratio of light speed in air or vacuum ($c$) to the speed in the material medium ($v$): $n = \frac{c}{v}$

Optical density is distinct from mass density and reflects the degree to which a medium retards light. A substance with a higher refractive index is optically denser, causing transmitted light to slow and bend toward the normal. Conversely, an optically rarer medium exhibits a lower refractive index, allowing light to accelerate and bend away from the normal.

Refraction by Spherical Lenses

A lens comprises a transparent optical material bounded by two surfaces, of which at least one is spherical. A double convex lens bulges outward on both faces and is thicker at the centre than at the margins, functioning as a converging lens that directs incident parallel rays toward a real focus. A double concave lens curves inward on both sides and is thicker at the edges than at the centre, functioning as a diverging lens that causes parallel incident rays to diverge as if emanating from a virtual focus. Each lens features an optical centre ($O$) through which light passes without angular deviation, two centres of curvature ($C_1$ and $C_2$), and two principal foci ($F_1$ and $F_2$) located on opposite sides at a distance equal to the focal length ($f$).

Convex lenses produce real, inverted images for object positions ranging from infinity down to the principal focus ($F_1$). When an object is at infinity, a point-sized real image forms at $F_2$; between infinity and $2F_1$, a diminished real image forms between $F_2$ and $2F_2$; at $2F_1$, a real image of identical size forms at $2F_2$; between $F_1$ and $2F_1$, an enlarged real image forms beyond $2F_2$; and at $F_1$, an image forms at infinity. Placing the object between $F_1$ and the optical centre ($O$) produces an enlarged, virtual, and erect image on the same side of the lens as the object. Concave lenses consistently produce virtual, erect, and diminished images situated between the optical centre ($O$) and the focus ($F_1$) on the object's side of the lens.

Lens Formula, Magnification, and Power of a Lens

Adhering to the Cartesian sign convention with reference to the optical centre ($O$), the focal length of a convex lens is assigned a positive value, while that of a concave lens is assigned a negative value. The lens formula connects object distance ($u$), image distance ($v$), and focal length ($f$): $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$

The magnification ($m$) generated by a lens is given by the ratio of the image height ($h'$) to the object height ($h$), which relates to image and object distances as: $m = \frac{h'}{h} = \frac{v}{u}$

The power ($P$) of a lens represents its capacity to converge or diverge incident light rays and is defined quantitatively as the reciprocal of its focal length expressed in metres: $P = \frac{1}{f}$

The SI unit of lens power is the dioptre ($\text{D}$), where $1\,\text{D} = 1\,\text{m}^{-1}$. Converging convex lenses possess positive power, whereas diverging concave lenses possess negative power. When multiple thin lenses are placed in direct contact, the net optical power equals the algebraic sum of their individual powers: $P = P_1 + P_2 + P_3 + \dots$

Questions & Discussion

The principal focus of a concave mirror is defined as the fixed point on its principal axis where rays of light propagating parallel to the axis converge after undergoing reflection. For a spherical mirror possessing a radius of curvature of $20\,\text{cm}$, applying the relation $f = \frac{R}{2}$ yields a focal length of $10\,\text{cm}$. An erect and enlarged image of an object is obtained using a concave mirror by placing the object between its pole and principal focus. Vehicles utilize convex mirrors as rear-view mirrors because they invariably produce an erect, though diminished, image and provide a wider field of view by virtue of curving outwards. Similarly, a convex mirror having a radius of curvature of $32\,\text{cm}$ has a focal length of $+16\,\text{cm}$. When a concave mirror creates a real image magnified three times ($m = -3$) for an object stationed at $10\,\text{cm}$ in front of it ($u = -10\,\text{cm}$), evaluating $m = -\frac{v}{u}$ determines that $v = -30\,\text{cm}$, indicating the image is located $30\,\text{cm}$ in front of the mirror.

When a light ray enters obliquely from air into water, it bends toward the normal because water is optically denser than air, causing the propagation speed of the wave to decrease. For light entering glass with a refractive index of $1.50$ from a vacuum where $c = 3 \times 10^8\,\text{m}\,\text{s}^{-1}$, the speed within the glass is $v = \frac{c}{n} = \frac{3 \times 10^8\,\text{m}\,\text{s}^{-1}}{1.50} = 2 \times 10^8\,\text{m}\,\text{s}^{-1}$. In terms of optical density across standard media, diamond possesses the highest optical density with a refractive index of $2.42$, signifying that light travels $2.42$ times faster in vacuum than in diamond, while air exhibits the lowest optical density with an index of $1.0003$. Among kerosene ($n = 1.44$), turpentine ($n = 1.47$), and water ($n = 1.33$), light travels fastest through water due to its lowest refractive index.

One dioptre ($1\,\text{D}$) denotes the optical power of a lens having a focal length of exactly $1\,\text{m}$. If a convex lens forms an inverted, real image of a needle identical in size to the object at a distance of $50\,\text{cm}$ ($v = +50\,\text{cm}$), the needle is situated at $2F_1$ where $u = -50\,\text{cm}$, yielding a focal length of $f = +25\,\text{cm} = +0.25\,\text{m}$ and a lens power of $P = +4.0\,\text{D}$. A concave lens with a focal length of $2\,\text{m}$ ($f = -2\,\text{m}$) exhibits an optical power of $P = -0.5\,\text{D}$. A lens prescribed with a power of $-2.0\,\text{D}$ corresponds to a concave diverging lens of focal length $-0.50\,\text{m}$, whereas a prescribed corrective lens of power $+1.5\,\text{D}$ represents a convex converging lens with a focal length of $+0.67\,\text{m}$.