Science Notes - Light: Reflection and Refraction

Introduction to Light and Its Properties

• Visibility of Objects: Objects in the world become visible due to light. In a dark room, nothing is visible until it is lit.

  • Daylight: Sunlight enables us to see objects during the day.

  • Mechanism of Sight: An object reflects light falling upon it. When this reflected light is received by the eyes, it enables vision.

  • Transparent Media: We can see through transparent media because light is transmitted through them.

• Optical Phenomena: Common wonders associated with light include:

  • Image formation by mirrors.

  • The twinkling of stars.

  • The beautiful colors of a rainbow.

  • Bending of light by a medium (refraction).

• Propagation of Light: Light generally seems to travel in straight lines. A small source of light casting a sharp shadow of an opaque object illustrates this straight-line path, referred to as a "ray of light."

• The Evolving Theory of Light:

  • Diffraction: If an opaque object on the path of light is very small, light tends to bend around it instead of traveling in a straight line. This is known as the diffraction of light. This effect renders the straight-line ray treatment of optics inadequate.

  • Wave Nature: To explain diffraction, light is treated as a wave.

  • Particle Nature: In the early 20th century, it was discovered that wave theory fails to explain light's interaction with matter. In these cases, light behaves like a stream of particles.

  • Modern Quantum Theory of Light: This theory reconciles the particle and wave properties, stating light is neither purely a wave nor purely a particle.

Reflection of Light and Laws of Reflection

• Definition: Reflection occurs when a highly polished surface, like a mirror, reflects most of the light falling on it.

• Laws of Reflection:

  1. The angle of incidence is equal to the angle of reflection ($\angle i = \angle r$).

  2. The incident ray, the normal to the mirror at the point of incidence, and the reflected ray all lie in the same plane.

• Applicability: These laws apply to all types of reflecting surfaces, including spherical surfaces.

• Characteristics of Plane Mirror Images:

  • The image is always virtual and erect.

  • The size of the image is equal to the size of the object.

  • The image is as far behind the mirror as the object is in front of it.

  • The image is laterally inverted.

Nature and Characteristics of Spherical Mirrors

• Definition: A spherical mirror is a curved mirror whose reflecting surface forms part of a sphere.

• Types of Spherical Mirrors:

  • Concave Mirror: The reflecting surface is curved inwards (faces the center of the sphere). It is approximated by the inner surface of a shining spoon.

  • Convex Mirror: The reflecting surface is curved outwards. It is approximated by the outer (bulged) surface of a shining spoon.

• Key Terminology:

  • Pole (P): The center of the reflecting surface of a spherical mirror. It lies on the surface of the mirror.

  • Centre of Curvature (C): The center of the sphere of which the mirror's reflecting surface is a part.

    • In a concave mirror, it lies in front of the reflecting surface.

    • In a convex mirror, it lies behind the reflecting surface.

  • Radius of Curvature (R): The radius of the sphere of which the mirror surface forms a part. The distance between the Pole and the Centre of Curvature ($PC = R$).

  • Principal Axis: A straight line passing through the pole and the center of curvature. It is normal to the mirror at its pole.

  • Principal Focus (F):

    • Concave: The point on the principal axis where rays parallel to the axis meet after reflection.

    • Convex: The point on the principal axis from which rays parallel to the axis appear to diverge after reflection.

  • Focal Length (f): The distance between the pole and the principal focus ($PF = f$).

  • Aperture: The diameter of the reflecting surface (circular outline) of the spherical mirror.

• Relationship between R and f: For mirrors with small apertures, the radius of curvature is twice the focal length:

  • $R = 2f$

Image Formation by Concave Mirrors

• Nature and Size: The image depends on the object's position relative to P, F, and C.

• Summary Table (Table 9.1):

  • At infinity: Image at Focus (F); Highly diminished, point-sized; Real and inverted.

  • Beyond C: Image between F and C; Diminished; Real and inverted.

  • At C: Image at C; Same size as object; Real and inverted.

  • Between C and F: Image beyond C; Enlarged; Real and inverted.

  • At F: Image at infinity; Highly enlarged; Real and inverted.

  • Between P and F: Image behind the mirror; Enlarged; Virtual and erect.

• Experimental Observation: Holding a concave mirror toward the Sun focus light into a sharp bright spot on paper. This spot is a real image of the Sun. The heat from concentrated sunlight can ignite the paper.

Image Formation by Convex Mirrors

• General Rule: Convex mirrors always produce virtual, erect, and diminished images, regardless of the object's distance.

• Summary Table (Table 9.2):

  • At infinity: Image at focus (F) behind the mirror; Highly diminished, point-sized; Virtual and erect.

  • Between infinity and Pole (P): Image between P and F behind the mirror; Diminished; Virtual and erect.

• Key Property: As an object moves away from a convex mirror, the image moves closer to the focus and becomes smaller.

Utility and Applications of Spherical Mirrors

• Concave Mirrors:

  • Used in torches, search-lights, and vehicle headlights to produce powerful parallel beams.

  • Used as shaving mirrors to see a larger image of the face.

  • Used by dentists to see large images of teeth.

  • Large concave mirrors are used in solar furnaces to concentrate sunlight for heat.

• Convex Mirrors:

  • Commonly used as rear-view (wing) mirrors in vehicles.

  • Advantages: They always give an erect image and have a wider field of view because they are curved outwards, allowing drivers to see more traffic than a plane mirror would.

New Cartesian Sign Convention and Mirror Formula

• Sign Convention Rules:

  1. The object is always placed to the left of the mirror (light travels left to right).

  2. The pole (P) is the origin. Distances are measured from the pole.

  3. Distances to the right ($+x$) are positive; distances to the left ($-x$) are negative.

  4. Distances above the principal axis ($+y$) are positive; distances below ($-y$) are negative.

• Mirror Formula: The relationship between object distance ($u$), image distance ($v$), and focal length ($f$):

  • $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

• Magnification (m): The ratio of the height of the image ($h'$) to the height of the object ($h$):

  • $m = \frac{h'}{h} = -\frac{v}{u}$

  • Sign Interpretation:

    • Negative magnification indicates a real image.

    • Positive magnification indicates a virtual image.

Quantitative Examples: Reflection

• Example 9.1 (Convex Mirror):

  • Given: $R = +3.00\,m$, $u = -5.00\,m$.

  • Step 1: $f = \frac{R}{2} = +1.50\,m$.

  • Step 2: $\frac{1}{v} = \frac{1}{f} - \frac{1}{u} = \frac{1}{1.50} - \frac{1}{-5.00} = \frac{1}{1.50} + \frac{1}{5.00} = \frac{5.00 + 1.50}{7.50} = \frac{6.50}{7.50}$.

  • Result: $v = +1.15\,m$ (image is behind the mirror).

  • Step 3: $m = -\frac{v}{u} = -\frac{1.15}{-5.00} = +0.23$.

  • Conclusion: Virtual, erect, and smaller (factor of 0.23).

• Example 9.2 (Concave Mirror):

  • Given: $h = 4.0\,cm$, $u = -25.0\,cm$, $f = -15.0\,cm$.

  • Step 1: $\frac{1}{v} = \frac{1}{f} - \frac{1}{u} = \frac{1}{-15.0} - \frac{1}{-25.0} = -\frac{2.0}{75.0}$.

  • Result: $v = -37.5\,cm$ (screen distance in front of mirror).

  • Step 2: $h' = h \times \left(-\frac{v}{u}\right) = 4.0 \times \left(-\frac{-37.5}{-25.0}\right) = -6.0\,cm$.

  • Conclusion: Real, inverted, and enlarged.

The Phenomenon of Refraction of Light

• Definition: When light travels obliquely from one transparent medium to another, the direction of propagation in the second medium changes. This is known as refraction.

• Common Observations:

  • Bottom of a water tank or pond appears raised.

  • Letters appear raised when viewed through a thick glass slab.

  • A pencil partially immersed in water appears displaced at the air-water interface.

  • A lemon in water appears larger when viewed from the sides.

• Cause: Refraction occurs because the speed of light changes as it moves from one medium to another.

Refraction through a Rectangular Glass Slab

• Mechanism: Light undergoes refraction twice—once when entering the glass (air-to-glass) and once when exiting (glass-to-air).

• Observations:

  • Air to Glass (Rarer to Denser): The light ray bends towards the normal.

  • Glass to Air (Denser to Rarer): The light ray bends away from the normal.

  • Emergent Ray: The ray exiting the slab is parallel to the incident ray but is shifted sideward (lateral displacement). This is because the extent of bending at both parallel faces is equal and opposite.

Laws of Refraction and the Refractive Index

• Laws of Refraction:

  1. The incident ray, the refracted ray, and the normal at the point of incidence all lie in the same plane.

  2. Snell's Law: The ratio of the sine of the angle of incidence ($i$) to the sine of the angle of refraction ($r$) is a constant for a given color and pair of media: $\frac{\sin(i)}{\sin(r)} = \text{constant}$.

• Refractive Index (n): This constant value ($n$) represents the second medium's refractive index relative to the first.

  • Relative Refractive Index: $n_{21} = \frac{\text{Speed of light in medium 1 } (v_1)}{\text{Speed of light in medium 2 } (v_2)}$.

  • Absolute Refractive Index ($n_m$): If medium 1 is vacuum or air, the index is called the absolute refractive index: $n_m = \frac{\text{Speed of light in air } (c)}{\text{Speed of light in medium } (v)}$.

  • Speed of Light in Vacuum ($c$): Approximately $3 \times 10^8\,m/s$.

Optical Density and Material Indices

• Optical Density: A medium's ability to refract light.

  • Optically Denser: Higher refractive index; light travels slower; rays bend towards the normal.

  • Optically Rarer: Lower refractive index; light travels faster; rays bend away from the normal.

  • Distinction from Mass Density: Optical density is not equal to mass density. For example, kerosene is optically denser than water (refractive index $1.44$ vs $1.33$) but is less dense in mass.

• Table 9.3 Examples (Absolute Refractive Indices):

  • Air: $1.0003$

  • Ice: $1.31$

  • Water: $1.33$

  • Kerosene: $1.44$

  • Crown glass: $1.52$

  • Ruby: $1.71$

  • Diamond: $2.42$

Refraction by Spherical Lenses

• Definition: A lens is a transparent material bound by at least one spherical surface.

• Types of Lenses:

  • Convex (Double Convex) Lens: Thicker at the middle than at the edges. It converges light rays and is called a converging lens.

  • Concave (Double Concave) Lens: Thicker at the edges than at the middle. It diverges light rays and is called a diverging lens.

• Key Terminology:

  • Centres of Curvature ($C_1, C_2$): The centers of the spheres forming the lens surfaces.

  • Principal Axis: Imaginary line passing through both centers of curvature.

  • Optical Centre (O): The central point of the lens. Rays passing through O do not deviate.

  • Aperture: The effective diameter of the circular outline of the lens.

  • Principal Focus (F): Point where parallel rays converge (convex) or appear to diverge from (concave).

  • Focal Length (f): Distance from the optical centre to the principal focus.

Image Formation by Spherical Lenses

• Convex Lens (Table 9.4):

  • At infinity: Image at $F_2$; highly diminished; real and inverted.

  • Beyond $2F_1$: Image between $F_2$ and $2F_2$; diminished; real and inverted.

  • At $2F_1$: Image at $2F_2$; same size; real and inverted.

  • Between $F_1$ and $2F_1$: Image beyond $2F_2$; enlarged; real and inverted.

  • At focus $F_1$: Image at infinity; infinitely large; real and inverted.

  • Between $F_1$ and O: Image on same side as object; enlarged; virtual and erect.

• Concave Lens (Table 9.5):

  • At infinity: Image at focus $F_1$; highly diminished; virtual and erect.

  • Between infinity and O: Image between $F_1$ and O; diminished; virtual and erect.

Lens Formula, Magnification, and Sign Convention

• Sign Convention: Similar to mirrors, but distances are measured from the optical centre (O). Focal length of convex lens is positive ($+$), and concave lens is negative ($-$).

• Lens Formula:

  • $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$

• Magnification (m):

  • $m = \frac{h'}{h} = \frac{v}{u}$

Quantitative Examples: Refraction

• Example 9.3 (Concave Lens):

  • Given: $f = -15\,cm$, $v = -10\,cm$.

  • Calculation: $\frac{1}{u} = \frac{1}{v} - \frac{1}{f} = \frac{1}{-10} - \frac{1}{-15} = \frac{-3 + 2}{30} = -\frac{1}{30}$.

  • Result: $u = -30\,cm$. Magnification $m = \frac{-10}{-30} = +0.33$ (virtual, erect, 1/3 size).

• Example 9.4 (Convex Lens):

  • Given: $h = +2.0\,cm$, $f = +10\,cm$, $u = -15\,cm$.

  • Calculation: $\frac{1}{v} = \frac{1}{f} + \frac{1}{u} = \frac{1}{10} + \frac{1}{-15} = \frac{3 - 2}{30} = \frac{1}{30}$.

  • Result: $v = +30\,cm$. Magnification $m = \frac{30}{-15} = -2$.

  • Image Height: $h' = h \times m = 2.0 \times (-2) = -4.0\,cm$ (real, inverted, 2x enlarged).

Power of a Lens

• Definition: The degree of convergence or divergence of light rays. It is the reciprocal of the focal length ($f$).

• Formula:

  • $P = \frac{1}{f} \text{ (where } f \text{ is in meters)}$

• Units: The SI unit is the dioptre (D). $1\,D = 1\,m^{-1}$.

• Values:

  • Power of convex lens: Positive ($+$).

  • Power of concave lens: Negative ($-$).

• Combinations: The net power ($P$) of multiple lenses in contact is the algebraic sum:

  • $P = P_1 + P_2 + P_3 + \dots$

• Application: Opticians use power additive properties to design complex lens systems for cameras, microscopes, and telescopes to minimize image defects.

Questions & Discussion

• Mirror Focus: Definition of principal focus for a concave mirror as the point where parallel incoming rays meet after reflection.

• Reflection Math: If radius is $20\,cm$, focal length is $10\,cm$.

• Mirror Utility Question: Why use convex mirrors for rear-view? Response: They provide an erect image and a wider field of view.

• Power Calculation:

  • Find focal length of lens with power $-2.0\,D$: $f = \frac{1}{-2.0} = -0.50\,m$ (Concave).

  • Prescribed lens power $+1.5\,D$: Focal length is $+0.67\,m$ (Converging/Convex).