Light: Reflection and Refraction - Chapter 9 Flashcards

The Nature and Propagation of Light

• Visibility and Light Interaction: Objects are visible because they reflect light that falls on them. When this reflected light is received by our eyes, we can see. In a dark room, nothing is visible until light is introduced. Sunlight enables visibility during the day.
• Transmission: Light is transmitted through transparent media, allowing us to see through them.
• Phenomena of Light: Common optical phenomena include the formation of images by mirrors, the twinkling of stars, the formation of rainbows, and the bending of light by different media.
• Straight-line Propagation: Observations of shadows cast by opaque objects suggest that light travels in straight lines. A small source of light casting a sharp shadow of an opaque object confirms this path, often referred to as a ray of light.
• Wave Nature (Diffraction): If an opaque object in the path of light becomes extremely small, light tends to bend around it instead of traveling in a straight line. This is known as the diffraction of light. This phenomenon fails the straight-line treatment of optics and required the development of wave theory.
• Particle Nature: Interaction of light with matter at the beginning of the 20th century showed that light can behave like a stream of particles, rendering the wave theory inadequate in certain contexts.
• Modern Quantum Theory: This theory reconciles the wave and particle properties; light is considered to have a dual nature, being neither solely a wave nor a particle.

Reflection of Light and Spherical Mirrors

• Mirrors: A highly polished surface, like a mirror, reflects most of the light falling on it.
• Laws of Reflection:
  • (i) The angle of incidence ($i$) is always equal to the angle of reflection ($r$): $\angle i = \angle r$.
  • (ii) The incident ray, the normal to the mirror at the point of incidence, and the reflected ray all lie in the same plane.
• Spherical Mirrors: Mirrors whose reflecting surfaces are part of a sphere.
  • Concave Mirror: The reflecting surface is curved inwards (faces the centre of the sphere). These are converging mirrors.
  • Convex Mirror: The reflecting surface is curved outwards. These are diverging mirrors.
• Key Geometric Terms:
  • Pole (P): The centre of the reflecting surface of a spherical mirror.
  • Centre of Curvature (C): The centre of the sphere of which the mirror is a part. For a concave mirror, it is in front; for a convex mirror, it is behind.
  • Radius of Curvature (R): The radius of the sphere of which the mirror forms a part. The distance $PC = R$.
  • Principal Axis: A straight line passing through the pole and the centre of curvature. It is normal to the mirror at the pole.
  • Principal Focus (F): For a concave mirror, the point where rays parallel to the principal axis converge after reflection. For a convex mirror, the point from which such rays appear to diverge.
  • Focal Length (f): The distance between the pole and the principal focus.
  • Aperture: The diameter of the circular outline of the reflecting surface.
• Mathematical Relationship: For mirrors with small apertures, the radius of curvature is twice the focal length: $R = 2f$.

Image Formation by Spherical Mirrors

• Ray Tracing Rules:
  • 1. A ray parallel to the principal axis passes through the focus (concave) or appears to diverge from it (convex) after reflection.
  • 2. A ray passing through or directed toward the focus emerges parallel to the principal axis.
  • 3. A ray passing through or directed toward the centre of curvature reflects back along the same path as it hits the mirror normally.
  • 4. A ray incident obliquely at the pole reflects obliquely, following the laws of reflection.
• Concave Mirror Images (Table 9.1 Summary):
  • Object at Infinity: Image at focus $F$, highly diminished (point-sized), real and inverted.
  • Object Beyond C: Image between $F$ and $C$, diminished, real and inverted.
  • Object at C: Image at $C$, same size, real and inverted.
  • Object Between C and F: Image beyond $C$, enlarged, real and inverted.
  • Object at F: Image at infinity, highly enlarged.
  • Object Between P and F: Image behind the mirror, enlarged, virtual and erect.
• Convex Mirror Images (Table 9.2 Summary):
  • Object at Infinity: Image at focus $F$ behind the mirror, highly diminished, virtual and erect.
  • Object Between Infinity and Pole: Image between $P$ and $F$ behind the mirror, diminished, virtual and erect.
• Applications:
  • Concave Mirrors: Used in torches, searchlights, vehicle headlights (parallel beams), shaving mirrors (to enlarge faces), and by dentists. Solar furnaces use them to concentrate heat.
  • Convex Mirrors: Used as rear-view (wing) mirrors in vehicles because they provide an erect image and a wider field of view due to their outward curvature.

Mirror Formula and Sign Convention

• New Cartesian Sign Convention:
  • The pole is the origin $(0,0)$. The principal axis is the x-axis.
  • Objects are placed to the left (light travels from left to right).
  • Distances to the right ($+x$) are positive; distances to the left ($-x$) are negative.
  • Heights above the axis ($+y$) are positive; heights below the axis ($-y$) 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 image height ($h'$) to object height ($h$):     $m = \frac{\text{Height of the image (h')}}{\text{Height of the object (h)}} = -\frac{v}{u}$
  • A negative $m$ value signifies a real image; a positive $m$ value signifies a virtual image.

Refraction of Light

• Phenomenon: Refraction is the change in the direction of light when it travels obliquely from one transparent medium to another due to a change in the speed of light.
• Observations:
  • Bottoms of water tanks appear raised.
  • Pencils partially immersed in water appear bent at the interface.
  • Lemons in water appear larger when viewed from sides.
• Laws of Refraction:
  • (i) The incident ray, the refracted ray, and the normal at the point of incidence all lie in the same plane.
  • (ii) Snell’s Law: The ratio of the sine of the angle of incidence ($i$) to the sine of the angle of refraction ($r$) is constant for a given pair of media and color of light:     $\frac{\sin(i)}{\sin(r)} = \text{constant}$
• Refractive Index (n): This constant represents the ratio of light speeds in different media.
  • $n_{21}$ (Refractive index of medium 2 with respect to medium 1) = $\frac{\text{Speed of light in medium 1 (v}_1)}{\text{Speed of light in medium 2 (v}_2)}$
  • Absolute Refractive Index (n_m): Refractive index with respect to vacuum ($c \approx 3 \times 10^8\,\text{m/s}$):     $n_m = \frac{c}{v}$
• Selected Refractive Indices: Air (1.0003), Water (1.33), Kerosene (1.44), Crown glass (1.52), Diamond (2.42).
• Optical Density: An optically denser medium has a higher refractive index and slower light speed. Light bending:
  • Rarer to Denser: Bends toward the normal.
  • Denser to Rarer: Bends away from the normal.

Refraction by Spherical Lenses

• Lenses: Transparent materials bound by two surfaces, at least one being spherical.
• Convex Lens (Converging): Thicker in the middle, converges parallel rays to a focus ($F$).
• Concave Lens (Diverging): Thinner in the middle, diverges parallel rays so they appear to come from a focus ($F$).
• Terms: Optical centre ($O$), Centres of curvature ($C_1, C_2$), Principal axis, Aperture, Focal length ($f$).
• Lens Formula: Gives the relation between $u$, $v$, and $f$:     $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$
• Magnification by Lens: $m = \frac{h'}{h} = \frac{v}{u}$.
• Image Formation (Convex Lens Table 9.4 Summary):
  • At Infinity: Image at $F_2$, point-sized, real, inverted.
  • Beyond 2F_1: Image between $F_2$ and $2F_2$, diminished, real, inverted.
  • At 2F_1: Image at $2F_2$, same size, real, inverted.
  • Between F_1 and 2F_1: Image beyond $2F_2$, enlarged, real, inverted.
  • Between Focus and Optical Centre: On same side as object, enlarged, virtual, erect.
• Image Formation (Concave Lens): Always produces a virtual, erect, and diminished image.

Power of a Lens

• Definition: The degree of convergence or divergence of light rays, calculated as the reciprocal of focal length in metres.     $P = \frac{1}{f}$
• Unit: Dioptre ($D$), where $1\,D = 1\,\text{m}^{-1}$.
• Polarity: Convex lenses have positive power; concave lenses have negative power.
• Combinations: For lenses in contact, net power is the algebraic sum: $P = P_1 + P_2 + P_3 + \dots$.

Questions & Discussion

• Question 1: Define the principal focus of a concave mirror.
• Response: The principal focus of a concave mirror is a point on the principal axis where all rays parallel to the principal axis meet/intersect after reflection.
• Question 2: If the radius of curvature is $20\,\text{cm}$, what is the focal length?
• Response: Using $f = \frac{R}{2}$, $f = \frac{20}{2} = 10\,\text{cm}$.
• Question 3: Name a mirror that gives an erect and enlarged image.
• Response: A concave mirror (when the object is placed between the pole and the focus).
• Question 4: Why are convex mirrors used as rear-view mirrors?
• Response: They provide an erect (though diminished) image and have a wider field of view compared to plane mirrors.
• Question 5: What is the speed of light in glass if the refractive index is 1.50 and the speed in vacuum is $3 \times 10^8\,\text{m/s}$?
• Response: Using $v = \frac{c}{n_m}$, $v = \frac{3 \times 10^8}{1.5} = 2 \times 10^8\,\text{m/s}$.
• Question 6: What is the focal length of a lens with power $-2.0\,D$?
• Response: $f = \frac{1}{P} = \frac{1}{-2.0} = -0.5\,\text{m}$. It is a concave (diverging) lens.