Science: Light – Reflection and Refraction

Introduction to Light and Visibility

• Visibility of Objects:     - Objects in a dark room are invisible. They become visible upon lighting up the room.     - During the day, sunlight facilitates vision.     - An object reflects light that falls onto it. When this reflected light is received by the human eye, it enables vision.     - For transparent media, light is transmitted through it, allowing us to see through the medium.
• Common Optical Phenomena:     - Image formation by mirrors.     - The twinkling of stars.     - The beautiful colors of a rainbow.     - Bending of light by a medium (refraction).
• Nature of Light Propagation:     - Based on common observations (like sharp shadows of opaque objects), light is traditionally thought to travel in straight lines, known as a ray of light.     - Diffraction of Light: 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 suggests light has wave-like properties.     - Quantum Theory: In the early 20th century, it was discovered that light also acts like a stream of particles when interacting with matter. Modern quantum theory reconciles these by stating light is neither purely a 'wave' nor a 'particle' but possesses characteristics of both.

Reflection of Light

• Definition: A highly polished surface, such as a mirror, reflects the majority of the light falling on it.
• Laws of Reflection:     - (i) The angle of incidence is equal to the angle of reflection ($i = r$).     - (ii) The incident ray, the reflected ray, and the normal to the mirror at the point of incidence all lie in the same plane.     - These laws apply to all reflecting surfaces, including plane and spherical surfaces.
• Image Properties in a Plane Mirror:     - The image is always virtual and erect.     - The size of the image is equal to the size of the object.     - The distance of the image behind the mirror is equal to the distance of the object in front of it.     - The image is laterally inverted.

Spherical Mirrors: Definitions and Terms

• Spherical Mirror: A mirror whose reflecting surface is a part of a hollow sphere.     - Concave Mirror: The reflecting surface is curved inwards (faces toward the center of the sphere).     - Convex Mirror: The reflecting surface is curved outwards.
• Crucial 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, $C$ lies in front of the reflecting surface.         - In a convex mirror, $C$ lies behind the reflecting surface.     - Radius of Curvature (R): The radius of the sphere of which the reflecting surface is a part. The distance $PC$ equals $R$.     - Principal Axis: A straight line passing through the pole ($P$) and the center of curvature ($C$). It is normal to the mirror at its pole.     - Principal Focus (F):         - For a concave mirror: The point on the principal axis where rays parallel to the axis converge after reflection.         - For a convex mirror: 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 ($P$) and the principal focus ($F$).     - Aperture: The diameter of the circular outline of the reflecting surface 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

• Summary of Image Characteristics (Table 9.1):     - Object at Infinity: Image at Focus $F$. Nature: Real and inverted. Size: Highly diminished, point-sized.     - Object Beyond C: Image between $F$ and $C$. Nature: Real and inverted. Size: Diminished.     - Object At C: Image at $C$. Nature: Real and inverted. Size: Same size.     - Object Between C and F: Image beyond $C$. Nature: Real and inverted. Size: Enlarged.     - Object At F: Image at Infinity. Nature: Real and inverted. Size: Highly magnified.     - Object Between P and F: Image behind the mirror. Nature: Virtual and erect. Size: Enlarged.
• Rules for Ray Diagrams:     - (i) A ray parallel to the principal axis passes through the focus ($F$) after reflection (concave) or appears to diverge from it (convex).     - (ii) A ray passing through the focus ($F$) emerges parallel to the principal axis after reflection.     - (iii) A ray passing through the center of curvature ($C$) is reflected back along the same path because it strikes the surface normally.     - (iv) A ray incident obliquely at the pole ($P$) is reflected obliquely, following the laws of reflection ($i = r$).
• Uses of 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.

Image Formation by Convex Mirrors

• Summary of Image Characteristics (Table 9.2):     - Object at Infinity: Image at focus $F$ behind the mirror. Nature: Virtual and erect. Size: Highly diminished, point-sized.     - Object Between Infinity and Pole (P): Image between $P$ and $F$ behind the mirror. Nature: Virtual and erect. Size: Diminished.
• Uses of Convex Mirrors:     - Primarily used as rear-view (wing) mirrors in vehicles.     - Advantages: They always provide an erect image and have a much wider field of view because they are curved outwards, allowing drivers to see more traffic than a plane mirror would.

New Cartesian Sign Convention

• When dealing with reflection by spherical mirrors, the following conventions are used:     - (i) The object is always placed to the left of the mirror (light travels from left to right).     - (ii) All distances parallel to the principal axis are measured from the pole ($P$), which is the origin.     - (iii) Distances measured in the direction of incident light (along $+x$ axis) are positive; distances measured against incident light (along $-x$ axis) are negative.     - (iv) Distances measured perpendicular to and above the principal axis ($+y$ axis) are positive.     - (v) Distances measured perpendicular to and below the principal axis ($-y$ axis) are negative.

Mirror Formula and Magnification

• Mirror Formula:     - Represents the relationship between object distance ($u$), image distance ($v$), and focal length ($f$).     - $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$
• Magnification (m):     - Defined as the ratio of the height of the image ($h'$) to the height of the object ($h$).     - $m = \frac{h'}{h}$     - Also related to distances: $m = -\frac{v}{u}$
• Interpretation of Magnification:     - A negative sign for $m$ indicates a real image.     - A positive sign for $m$ indicates a virtual image.

Numerical Examples (Mirrors)

• Example 9.1 (Convex Mirror):     - Given: $R = +3.00\,m$, $u = -5.00\,m$.     - Focal length: $f = \frac{R}{2} = +1.50\,m$.     - Using $\frac{1}{v} = \frac{1}{f} - \frac{1}{u} = \frac{1}{1.50} - \frac{1}{-5.00} = \frac{1}{1.50} + \frac{1}{5.00} = \frac{6.50}{7.50}$.     - $v = +1.15\,m$ (image is behind the mirror).     - $m = -\frac{v}{u} = -\frac{1.15}{-5.00} = +0.23$.     - Result: Image is virtual, erect, and smaller by a factor of 0.23.
• Example 9.2 (Concave Mirror):     - Given: $h = +4.0\,cm$, $u = -25.0\,cm$, $f = -15.0\,cm$.     - Using $\frac{1}{v} = \frac{1}{f} - \frac{1}{u} = \frac{1}{-15.0} - \frac{1}{-25.0} = -\frac{2}{75}$.     - $v = -37.5\,cm$ (screen should be placed in front).     - $h' = -\frac{v}{u} \times h = -\frac{-37.5}{-25.0} \times 4.0 = -6.0\,cm$.     - Result: Image is real, inverted, and enlarged.

Refraction of Light

• Definition: When light travels obliquely from one transparent medium to another, its direction of propagation in the second medium changes. This is due to the change in the speed of light.
• Common Observations:     - Bottom of a water tank appears raised.     - Letters under a glass slab appear raised.     - A pencil partially immersed in water appears bent at the air-water interface.     - A lemon in water appears larger when viewed from sides.
• Refraction through a Rectangular Glass Slab:     - Light bends towards the normal when entering from a rarer medium (air) to a denser medium (glass).     - Light bends away from the normal when entering from a denser medium (glass) to a rarer medium (air).     - The emergent ray is parallel to the incident ray but shifted laterally.
• Laws of Refraction:     - (i) The incident ray, the refracted ray, and the normal to the interface 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 to the sine of the angle of refraction is a constant for a given pair of media and color of light.     - $\frac{\sin(i)}{\sin(r)} = \text{constant}$

The Refractive Index

• Definition: The constant in Snell's Law is the refractive index of the second medium relative to the first ($n_{21}$).
• Basis in Light Speed:     - $n_{21} = \frac{\text{Speed of light in medium 1}}{\text{Speed of light in medium 2}} = \frac{v_{1}}{v_{2}}$
• Absolute Refractive Index (n):     - Refractive index relative to vacuum/air.     - $n_{m} = \frac{c}{v}$ (where $c = 3 \times 10^{8}\,m/s$).
• Optical Density:     - A medium with a higher refractive index is 'optically denser'.     - Light travels slower in an optically denser medium and faster in an optically rarer medium.
• Refractive Index Table (Table 9.3 highlights):     - Air: $1.0003$     - Water: $1.33$     - Crown Glass: $1.52$     - Diamond: $2.42$ (highest optical density listed).

Spherical Lenses

• Lens: A transparent material bound by two surfaces where at least one is spherical.
• Types of Lenses:     - Convex (Converging) Lens: Thicker at the middle than at the edges. Converges parallel rays of light.     - Concave (Diverging) Lens: Thicker at the edges than at the middle. Diverges parallel rays of light.
• Key Terms:     - Centres of Curvature ($C_{1}, C_{2}$): The centers of the spheres forming the lens surfaces.     - Optical Centre (O): The central point of a lens. Ray passing through $O$ suffers no deviation.     - Principal Foci ($F_{1}, F_{2}$): Foci on both sides of the lens where parallel rays converge (convex) or appear to diverge from (concave).     - Focal Length (f): Distance from the optical center to the principal focus.     - Aperture: The effective diameter of the circular outline of the lens.

Image Formation by Lenses

• Convex Lens Image Summary (Table 9.4):     - At Infinity: At focus $F_{2}$. Nature: Real and inverted. Size: Highly diminished.     - Beyond 2F1: Between $F_{2}$ and $2F_{2}$. Nature: Real and inverted. Size: Diminished.     - At 2F1: At $2F_{2}$. Nature: Real and inverted. Size: Same size.     - Between F1 and 2F1: Beyond $2F_{2}$. Nature: Real and inverted. Size: Enlarged.     - At Focus F1: At Infinity. Nature: Real and inverted. Size: Highly magnified.     - Between F1 and O: On the same side as object. Nature: Virtual and erect. Size: Enlarged.
• Concave Lens Image Summary (Table 9.5):     - Always produces a virtual, erect, and diminished image regardless of object position.

Lens Formula, Magnification, and Power

• Lens Formula:     - $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$
• Magnification (m):     - $m = \frac{h'}{h} = \frac{v}{u}$
• Power of a Lens (P):     - The degree of convergence or divergence of light rays. It is the reciprocal of focal length in meters.     - $P = \frac{1}{f}$ (where $f$ is in meters).     - Unit: Dioptre ($D$). $1\,D = 1\,m^{-1}$.     - Signs: Convex lens power is Positive (+); Concave lens power is Negative (-).
• Combination of Lenses:     - Net power $P = P_{1} + P_{2} + P_{3} + \dots$     - Used in cameras, microscopes, and telescopes to increase magnification and sharpness.

Numerical Examples (Lenses)

• Example 9.3 (Concave Lens):     - Given: $f = -15\,cm$, $v = -10\,cm$.     - Using $\frac{1}{u} = \frac{1}{v} - \frac{1}{f} = \frac{1}{-10} - \frac{1}{-15} = -\frac{1}{30}$.     - $u = -30\,cm$.     - $m = \frac{v}{u} = \frac{-10}{-30} = +0.33$ (virtual, erect, 1/3 size).
• Example 9.4 (Convex Lens):     - Given: $h = +2.0\,cm$, $f = +10\,cm$, $u = -15\,cm$.     - Using $\frac{1}{v} = \frac{1}{f} + \frac{1}{u} = \frac{1}{10} + \frac{1}{-15} = \frac{1}{30}$.     - $v = +30\,cm$.     - $m = \frac{v}{u} = \frac{30}{-15} = -2$.     - $h' = m \times h = -2 \times 2.0 = -4.0\,cm$.     - Result: Real, inverted, enlarged (double size).

Questions & Discussion

• Focus of Concave Mirror: The point on the principal axis where rays parallel to the axis meet after reflection.
• Focal Length Calculation: If $R = 20\,cm$, $f = 10\,cm$. If $R = 32\,cm$, $f = 16\,cm$.
• Concave Mirror Image: To get an erect and enlarged image, the object must be between the pole and the focus.
• Vehicle Rear-View Mirrors: Convex mirrors are used because they provide erect images and a wider field of view.
• Refractive Index Meaning: "Refractive index of diamond is 2.42" means the speed of light in diamond is $\frac{1}{2.42}$ times the speed of light in air/vacuum.
• Lens Materials: Materials like clay cannot be used for lenses as they are not transparent.
• Plane Mirror Magnification: $m = +1$ means the image is the same size as the object and is virtual and erect.
• Refraction Logic: When entering water from air obliquely, light bends towards the normal because water is optically denser than air.