Comprehensive Study Notes on Light: Reflection, Refraction, Mirrors, and Lenses

Fundamental Nature and Propagation of Light

• Visibility and Light Reflection

  • Objects in a dark room are invisible because no light enters the eye from them.

  • Illuminating a room makes objects visible because they reflect light falling on them, and this reflected light is received by the human eye.

  • Transparent media allow vision through them because light is transmitted through the medium.

• Optical Phenomena in Daily Life

  • Image formation by spherical and plane mirrors.

  • Twinkling of stars in the night sky.

  • Formation of the rainbow with its vivid colors.

  • Bending of light when passing from one medium to another.

• Rectilinear Propagation and Light Rays

  • Light generally travels along straight-line paths in a homogeneous medium.

  • The formation of sharp shadows when an opaque object blocks a small light source provides direct evidence of straight-line propagation.

  • A straight line path along which light travels is represented as a ray of light.

• Evolution of Theories on the Nature of Light

  • Diffraction of Light: When an opaque object blocking light becomes very small, light tends to bend around the edges of the object rather than casting a simple sharp shadow. Under these conditions, straight-line ray optics fails.

  • Wave Theory: Wave concepts were introduced to explain wave-like phenomena such as diffraction and interference.

  • Particle Behavior: Early 20th-century experiments revealed that wave theory is inadequate when describing light-matter interactions, where light behaves as a stream of localized energy packets or particles.

  • Modern Quantum Theory: Modern physics reconciles particle and wave behaviors through quantum mechanics, defining light as possessing dual wave-particle properties rather than being strictly one or the other.

Reflection of Light and Spherical Mirrors

• Laws of Reflection

  • Reflection occurs when light strikes a highly polished surface, such as a mirror, and bounces back into the same medium.

  • First Law of Reflection: The angle of incidence ($i$) is strictly equal to the angle of reflection ($r$):     $i = r$

  • Second Law of Reflection: The incident ray, the normal to the mirror at the point of incidence, and the reflected ray all lie within the exact same plane.

  • These laws apply universally to all types of reflecting surfaces, including plane and curved spherical surfaces.

• Image Characteristics of a Plane Mirror

  • The image formed is always virtual and erect.

  • The size of the image is equal to the size of the object ($h' = h$).

  • The image distance behind the mirror equals the object distance in front of the mirror ($v = u$).

  • The image is laterally inverted (left and right sides are reversed).

• Introduction to Spherical Mirrors

  • A curved surface can be modeled as a curved mirror; a common household example is a shining metallic spoon.

  • Spherical mirrors are curved mirrors whose reflecting surfaces form part of a hollow sphere of glass.

  • Concave Mirror: A spherical mirror whose reflecting surface is curved inwards (facing toward the centre of the sphere). The curved inner surface of a spoon models a concave mirror.

  • Convex Mirror: A spherical mirror whose reflecting surface is curved outwards (bulging away from the centre). The outer bulged surface of a spoon models a convex mirror.

• Key Terminology of Spherical Mirrors

  • Pole ($P$): The geometric centre of the reflecting surface of a spherical mirror, lying directly on its surface.

  • Centre of Curvature ($C$): The centre of the sphere of which the reflecting surface forms a part. It is not part of the mirror itself and lies outside the surface.

  • For a concave mirror, $C$ lies in front of its reflecting surface.

  • For a convex mirror, $C$ lies behind its reflecting surface.

  • Radius of Curvature ($R$): The radius of the sphere of which the mirror surface forms a part; it equals the linear distance $PC$:     $PC = R$

  • Principal Axis: An imaginary straight line passing through the pole ($P$) and centre of curvature ($C$). The principal axis is normal to the mirror surface at its pole.

  • Aperture ($MN$): The effective diameter of the circular outline of the spherical mirror's reflecting surface. Discussions here focus on thin spherical mirrors whose aperture is significantly smaller than their radius of curvature ($MN \ll R$).

  • Principal Focus ($F$):

  • Concave Mirror: Incident rays parallel to the principal axis reflect and converge at a distinct point on the principal axis in front of the mirror.

  • Convex Mirror: Incident rays parallel to the principal axis reflect and appear to diverge from a point on the principal axis located behind the mirror.

  • Focal Length ($f$): The distance along the principal axis between the pole ($P$) and the principal focus ($F$):     $PF = f$

  • Relationship Between Radius of Curvature and Focal Length:

  • For spherical mirrors with small apertures, the radius of curvature is twice the focal length:       $R = 2f$

  • This relationship places the principal focus $F$ directly midway between the pole $P$ and the centre of curvature $C$.

Image Formation and Ray Diagrams for Spherical Mirrors

• Image Formation Properties of Concave Mirrors

  • Depending on the relative position of the object along the principal axis with respect to points $P$, $F$, and $C$, the image characteristics vary systematically.

• Standard Rays Used in Ray Diagrams

  • To trace the exact location of an image point, at least two intersecting reflected rays must be drawn:

  1. Parallel Ray: A ray parallel to the principal axis passes through the principal focus $F$ after reflection (concave mirror) or appears to diverge from $F$ (convex mirror).

  2. Focal Ray: A ray passing through or directed toward focus $F$ reflects parallel to the principal axis.

  3. Centre of Curvature Ray: A ray passing through or directed toward centre of curvature $C$ strikes normally and reflects back along its own path.

  4. Oblique Ray: A ray incident obliquely at pole $P$ reflects obliquely at an equal angle with respect to the principal axis ($i = r$).

• Practical Applications of Concave Mirrors

  • Torches, Search-lights, and Vehicle Headlights: Light sources are placed at focus $F$ to obtain powerful parallel beams.

  • Shaving and Makeup Mirrors: Used to obtain a magnified, erect image when the face is positioned between $P$ and F$.\n - **Dental Mirrors**: Used by dentists to view magnified images of teeth.\n - **Solar Furnaces**: Large concave mirrors concentrate parallel sunlight at focus F to generate high temperatures.\n\n- **Image Formation and Applications of Convex Mirrors**\n\n| Position of Object | Position of Image | Relative Size of Image | Nature of Image |\n| :--- | :--- | :--- | :--- |\n| At infinity | At focus F, behind mirror | Highly diminished, point-sized | Virtual and erect |\n| Between infinity and pole P$| Between$P$and$F, behind mirror | Diminished | Virtual and erect |\n\n - **Rear-View (Wing) Mirrors in Vehicles**:\n - Preferred because they produce an erect, albeit diminished, image of trailing traffic.\n - Curved outward geometry provides a wider field of view than plane mirrors, enabling drivers to observe a significantly broader region.\n - **Security Mirrors**: Used in large walls or spaces (e.g., Agra Fort viewing wall for Taj Mahal) to provide full-length diminished images of wide distant structures.\n\n# New Cartesian Sign Convention, Mirror Formula, and Magnification\n\n- **New Cartesian Sign Convention**\n - Pole P$of the mirror is taken as the origin$(0,0).\n - Principal axis is taken as the x-axis (X'X).\n - Light is always assumed to fall on the mirror from the left-hand side.\n - All distances parallel to the principal axis are measured from the pole P\n - Distances measured in the direction of incident light (to the right of origin along the +x-axis) are taken as **positive**.\n - Distances measured opposite to incident light (to the left of origin along the -x-axis) are taken as **negative**.\n - Distances measured perpendicular to and above the principal axis (along the +y-axis) are taken as **positive**.\n - Distances measured perpendicular to and below the principal axis (along the -y-axis) are taken as **negative**.\n\n\n\n- **Mirror Formula**\n - Expresses the mathematical relation between object distance (u$), image distance ($v$), and focal length ($f):\n    \frac{1}{v} + \frac{1}{u} = \frac{1}{f}\n - Applies universally to all spherical mirrors for all object positions when proper sign conventions are applied.\n\n- **Linear Magnification (m)**\n - Defined as the ratio of image height (h'$) to object height ($h):\n    m = \frac{\text{Height of image }(h')}{\text{Height of object }(h)} = \frac{h'}{h}\n - Expressed in terms of object distance (u$) and image distance ($v):\n    m = \frac{h'}{h} = -\frac{v}{u}\n - **Sign Rules for Magnification**:\n - Object height h is positive for upright objects placed above the principal axis.\n - Real and inverted images have negative image height (h' < 0$), leading to negative magnification ($m < 0).\n - Virtual and erect images have positive image height (h' > 0$), leading to positive magnification ($m > 0).\n\n# Refraction of Light and Rectangular Glass Slab\n\n- **The Phenomenon of Refraction**\n - Refraction is the change in direction of propagation of light when passing obliquely from one transparent medium to another.\n - Occurs due to the variation in the speed of light across different transparent media.\n\n- **Observable Everyday Refraction Effects**\n - Bottom of a water tank or pond appears raised.\n - Printed text appears raised when viewed through a glass slab placed over it.\n - A straight pencil partially immersed in water appears bent or displaced at the air-water interface.\n - Lemons submerged in water in a glass vessel appear larger than their actual physical dimensions when viewed from the side.\n\n- **Refraction through a Rectangular Glass Slab**\n - Light passing obliquely through a glass slab undergoes two sequential refractions:\n 1. **Air-to-Glass Interface (AB)**: Light enters from a rarer medium (air) to a denser medium (glass) and bends **towards the normal**.\n 2. **Glass-to-Air Interface (CD)**: Light enters from a denser medium (glass) to a rarer medium (air) and bends **away from the normal**.\n - **Emergent Ray Characteristics**:\n - The emergent ray is strictly parallel to the direction of the original incident ray because the extent of bending at opposite parallel faces is equal and opposite.\n - The emergent ray undergoes a sideways shift known as **lateral displacement**.\n - **Normal Incidence**: Light entering perpendicular to the interface (i = 0^\circ$) suffers zero deviation ($r = 0^\circ).\n\n# Laws of Refraction and Refractive Index\n\n- **Laws of Refraction**\n 1. The incident ray, the refracted ray, and the normal to the interface of two transparent media at the point of incidence all lie in the same plane.\n 2. **Snell's Law of Refraction**: The ratio of the sine of the angle of incidence (i$) to the sine of the angle of refraction ($r$) is constant for light of a given color and a given pair of media (for$0 < i < 90^\circ):\n     \frac{\sin(i)}{\sin(r)} = \text{constant}\n - This constant value is the refractive index of the second medium relative to the first.\n\n- **Refractive Index and Speed of Light**\n - Speed of light in vacuum c = 3 \times 10^8\,m\,s^{-1}; speed in air is negligibly lower.\n - **Relative Refractive Index (n_{21})**: Refractive index of medium 2 with respect to medium 1:\n    n_{21} = \frac{\text{Speed of light in medium 1 }(v_1)}{\text{Speed of light in medium 2 }(v_2)} = \frac{v_1}{v_2}\n - Relative refractive index of medium 1 with respect to medium 2:\n    n_{12} = \frac{v_2}{v_1}\n - **Absolute Refractive Index (n_m)**: Refractive index of a medium with respect to vacuum or air:\n    n_m = \frac{\text{Speed of light in air }(c)}{\text{Speed of light in medium }(v)} = \frac{c}{v}\n\n- **Absolute Refractive Index Values of Common Media**\n - Air: 1.0003\n - Ice: 1.31\n - Water: 1.33\n - Alcohol: 1.36\n - Kerosene: 1.44\n - Fused quartz: 1.46\n - Turpentine oil: 1.47\n - Benzene: 1.50\n - Crown glass: 1.52\n - Canada Balsam: 1.53\n - Rock salt: 1.54\n - Carbon disulphide: 1.63\n - Dense flint glass: 1.65\n - Ruby: 1.71\n - Sapphire: 1.77\n - Diamond: 2.42\n\n- **Optical Density versus Mass Density**\n - **Optical Density**: Measures the refractive capability of a medium. Higher refractive index indicates an optically denser medium.\n - Speed of light is higher in an optically rarer medium than in an optically denser medium.\n - Light traveling from optically rarer to denser medium: Slows down, bends towards normal.\n - Light traveling from optically denser to rarer medium: Speeds up, bends away from normal.\n - Optical density does not equal mass density. For example, kerosene has a higher refractive index (1.44$) than water ($1.33) and is optically denser, yet kerosene's mass density is lower than water (it floats on water).\n\n# Refraction by Spherical Lenses\n\n- **Definition and Types of Lenses**\n - A lens is a transparent medium bound by two surfaces, at least one of which is spherical.\n - **Double Convex Lens (Convex Lens)**: Bounded by two spherical surfaces bulging outwards; thicker at the middle than at the edges. Converges parallel incident light rays to a point; known as a **converging lens**.\n - **Double Concave Lens (Concave Lens)**: Bounded by two spherical surfaces curved inwards; thicker at the edges than at the middle. Diverges parallel light rays; known as a **diverging lens**.\n\n- **Key Terminology of Spherical Lenses**\n - **Centres of Curvature (C_1, C_2)**: The centres of the two hollow spheres of which the lens surfaces form parts.\n - **Principal Axis**: An imaginary straight line passing through both centres of curvature C_1$and$C_2 of the lens.\n - **Optical Centre (O$)**: The central point of a lens. A ray passing through$O suffers no net angular deviation.\n - **Aperture**: The effective diameter of the circular outline of the spherical lens surface.\n - **Principal Focus (F_1, F_2)**:\n - *Convex Lens*: Incident rays parallel to the principal axis converge after refraction to a focus F_2 on the opposite side.\n - *Concave Lens*: Incident rays parallel to the principal axis appear to diverge from focus F_1 on the same side.\n - A lens has two principal foci equidistant from O when radii of curvature are equal.\n - **Focal Length (f$)**: The distance between the optical centre$O$and principal focus$F$($OF = f).\n\n# Image Formation and Ray Diagrams for Spherical Lenses\n\n- **Image Formation Properties of Convex Lenses**\n\n| Position of Object | Position of Image | Relative Size of Image | Nature of Image |\n| :--- | :--- | :--- | :--- |\n| At infinity | At focus F_2 | Highly diminished, point-sized | Real and inverted |\n| Beyond 2F_1$| Between$F_2$and$2F_2 | Diminished | Real and inverted |\n| At 2F_1$| At$2F_2 | Same size | Real and inverted |\n| Between F_1$and$2F_1$| Beyond$2F_2 | Enlarged | Real and inverted |\n| At focus F_1 | At infinity | Highly enlarged / Infinitely large | Real and inverted |\n| Between focus F_1$and optical centre$O | On same side as object | Enlarged | Virtual and erect |\n\n- **Image Formation Properties of Concave Lenses**\n\n| Position of Object | Position of Image | Relative Size of Image | Nature of Image |\n| :--- | :--- | :--- | :--- |\n| At infinity | At focus F_1 | Highly diminished, point-sized | Virtual and erect |\n| Between infinity and optical centre O$| Between focus$F_1$and optical centre$O | Diminished | Virtual and erect |\n\n- **Standard Rays for Lens Ray Diagrams**\n 1. **Parallel Ray**: Parallel to principal axis, passes through F_2$after refraction (convex) or appears to diverge from$F_1 (concave).\n 2. **Focal Ray**: Passing through F_1$(convex) or directed toward$F_2 (concave), refracts parallel to principal axis.\n 3. **Optical Centre Ray**: Passes through optical centre O without deviation.\n\n# Lens Formula, Magnification, and Power of a Lens\n\n- **Sign Convention for Spherical Lenses**\n - Measurements are taken from optical centre O.\n - Focal length of a convex lens is **positive** (f > 0).\n - Focal length of a concave lens is **negative** (f < 0).\n\n- **Lens Formula**\n - Relates object distance (u$), image distance ($v$), and focal length ($f):\n    \frac{1}{v} - \frac{1}{u} = \frac{1}{f}\n\n- **Lens Magnification (m)**\n - Ratio of image height (h'$) to object height ($h):\n    m = \frac{h'}{h} = \frac{v}{u}\n\n- **Power of a Lens (P)**\n - Measures the degree of convergence or divergence achieved by a lens; defined as the reciprocal of its focal length in metres:\n    P = \frac{1}{f}\n - **SI Unit**: **Dioptre** (D$), where$1\,D = 1\,m^{-1}.\n - **Sign Rules for Power**:\n - Convex Lens: Positive focal length \implies$Positive power ($P > 0).\n - Concave Lens: Negative focal length \implies$Negative power ($P < 0).\n - **Combination of Lenses in Contact**:\n - When multiple thin lenses of powers P_1, P_2, P_3, \dots$are placed in direct contact, the net power$P is:\n      P = P_1 + P_2 + P_3 + \dots\n - Lens combinations increase image sharpness, enlarge magnification, and reduce optical aberrations in cameras, microscopes, and telescopes.\n\n# Solved Examples and Practical Numerical Problems\n\n- **Example 9.1**:\n - *Problem*: A convex mirror used for rear-view on an automobile has a radius of curvature R = +3.00\,m$. A bus is located at$u = -5.00\,m. Find the position, nature, and size of the image.\n - *Calculation*:\n - Focal length f = \frac{R}{2} = \frac{+3.00\,m}{2} = +1.50\,m\n - Mirror Formula: \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \implies \frac{1}{v} = \frac{1}{f} - \frac{1}{u}\n - \frac{1}{v} = \frac{1}{+1.50\,m} - \frac{1}{-5.00\,m} = \frac{1}{1.50\,m} + \frac{1}{5.00\,m} = \frac{5.00 + 1.50}{7.50\,m} = \frac{6.50}{7.50\,m}\n - v = +\frac{7.50}{6.50}\,m = +1.15\,m\n - Magnification m = -\frac{v}{u} = -\frac{+1.15\,m}{-5.00\,m} = +0.23\n - *Conclusion*: Image is formed 1.15\,m$behind the mirror; it is virtual, erect, and diminished by a scale factor of$0.23$.

• Example 9.2:

  • Problem: An object $h = +4.0\,cm$ tall is placed at $u = -25.0\,cm$ in front of a concave mirror of focal length $f = -15.0\,cm$. Find screen location $v$, image size $h'$, and nature.

  • Calculation:

  • $\frac{1}{v} = \frac{1}{f} - \frac{1}{u} = \frac{1}{-15.0\,cm} - \frac{1}{-25.0\,cm} = \frac{-5.0 + 3.0}{75.0\,cm} = \frac{-2.0}{75.0\,cm}$

  • $v = -37.5\,cm$

  • $m = \frac{h'}{h} = -\frac{v}{u} \implies h' = -\frac{v \times h}{u} = -\frac{(-37.5\,cm) \times (+4.0\,cm)}{(-25.0\,cm)} = -6.0\,cm$

  • Conclusion: Screen must be positioned $37.5\,cm$ in front of mirror. The image is real, inverted, and enlarged ($6.0\,cm$ tall).

• Example 9.3:

  • Problem: Concave lens with $f = -15\,cm$ forms image at $v = -10\,cm$. Find object distance $u$ and magnification m$.\n - *Calculation*:\n - Lens Formula: \frac{1}{v} - \frac{1}{u} = \frac{1}{f} \implies \frac{1}{u} = \frac{1}{v} - \frac{1}{f}\n - \frac{1}{u} = \frac{1}{-10\,cm} - \frac{1}{-15\,cm} = -\frac{1}{10\,cm} + \frac{1}{15\,cm} = \frac{-3 + 2}{30\,cm} = -\frac{1}{30\,cm}\n - u = -30\,cm\n - Magnification m = \frac{v}{u} = \frac{-10\,cm}{-30\,cm} = +\frac{1}{3} \approx +0.33\n - *Conclusion*: Object distance is 30\,cm in front of lens. Image is virtual, erect, and one-third of object size.\n\n- **Example 9.4**:\n - *Problem*: Object h = +2.0\,cm$is placed at$u = -15\,cm$from convex lens of focal length$f = +10\,cm$. Find position$v$, magnification$m$, and size$h'.\n - *Calculation*:\n - \frac{1}{v} = \frac{1}{u} + \frac{1}{f} = \frac{1}{-15\,cm} + \frac{1}{10\,cm} = \frac{-2 + 3}{30\,cm} = +\frac{1}{30\,cm}\n - v = +30\,cm\n - m = \frac{v}{u} = \frac{+30\,cm}{-15\,cm} = -2.0\n - h' = m \times h = (-2.0) \times (+2.0\,cm) = -4.0\,cm\n - *Conclusion*: Real, inverted image formed 30\,cm$on opposite side of lens, magnified two times ($4.0\,cm tall below axis).\n\n# Questions and Discussion\n\n- **Question 1**: Define the principal focus of a concave mirror.\n - *Answer*: The principal focus of a concave mirror is a point on its principal axis where rays of light coming parallel to the principal axis converge after reflection from the mirror.\n\n- **Question 2**: The radius of curvature of a spherical mirror is 20\,cm. What is its focal length?\n - *Answer*: Using f = \frac{R}{2}$, focal length$f = \frac{20\,cm}{2} = 10\,cm.\n\n- **Question 3**: Name a mirror that can give an erect and enlarged image of an object.\n - *Answer*: A concave mirror yields an erect and enlarged image when the object is placed between its pole P$and focus$F$.

• Question 4: Why do we prefer a convex mirror as a rear-view mirror in vehicles?

  • Answer: Convex mirrors always yield erect, diminished images and offer a wider field of view because they bulge outward.

• Question 5: A ray of light traveling in air enters obliquely into water. Does the light ray bend towards the normal or away from the normal? Why?

  • Answer: Bends towards the normal because water is optically denser than air ($n_w = 1.33 > n_{air} = 1.0003$), causing light speed to decrease.

• Question 6: Light enters from air to glass having refractive index $1.50$. What is the speed of light in the glass? (Speed of light in vacuum = $3 \times 10^8\,m\,s^{-1}$).

  • Answer: Using $v = \frac{c}{n} = \frac{3 \times 10^8\,m\,s^{-1}}{1.50} = 2 \times 10^8\,m\,s^{-1}$.

• Question 7: Identify media with highest and lowest optical density from absolute refractive indices.

  • Answer: Highest optical density is Diamond ($n = 2.42$); lowest optical density is Air ($n = 1.0003$).

• Question 8: Given kerosene ($n = 1.44$), turpentine ($n = 1.47$), and water ($n = 1.33$), in which does light travel fastest?

  • Answer: Light travels fastest in water because it has the lowest refractive index among the three.

• Question 9: The refractive index of diamond is $2.42$. What is the meaning of this statement?

  • Answer: The ratio of the speed of light in air/vacuum to the speed of light in diamond is $2.42$.

• Question 10: Define $1\,\text{dioptre}$ of power of a lens.

  • Answer: One dioptre ($1\,D$) is the optical power of a lens whose focal length is equal to $1\,\text{metre}$ ($1\,D = 1\,m^{-1}$).

• Question 11: A convex lens forms a real and inverted image of a needle at a distance of $50\,cm$ from it. Where is the needle placed if image size equals object size? Find power.

  • Answer: For equal size real image, object is at $2F_1$ and image at $2F_2$. Given $v = +50\,cm$, object distance $u = -50\,cm$. Focal length $f = +25\,cm = +0.25\,m$. Power P = \frac{1}{+0.25\,m} = +4.0\,D$.\n\n- **Question 12**: Find the power of a concave lens of focal length 2\,m$.

  • Answer: Concave lens has negative focal length ($f = -2\,m$). Power P = \frac{1}{-2\,m} = -0.5\,D$.\n\n- **Question 13**: Which material cannot be used to make a lens: Water, Glass, Plastic, or Clay?\n - *Answer*: Clay, because it is opaque and cannot transmit or refract light.\n\n- **Question 14**: A spherical mirror and a thin spherical lens have each a focal length of -15\,cm$$. What are they?

  • Answer: Both are concave, as negative focal length is assigned to concave mirrors and concave lenses.