Light: Reflection and Refraction Study Guide

Physics: Reflection of Light

• Definition of Reflection: The bouncing back of an incident ray of light when it hits a surface.

• Laws of Reflection:

  • (1) The incident ray, the reflected ray, and the normal to the point of incidence all lie on the same plane.

  • (2) The angle of incidence $\angle i$ is always equal to the angle of reflection $\angle r$.

Types of Images

• Real Image:

  • It is an image that can be taken or captured on a screen.

  • Real images are always inverted (upside down).

  • Rays of light actually meet at a point after reflection.

  • Example: An image formed on a cinema hall screen.

• Virtual Image:

  • It is an image that cannot be taken on a screen.

  • Virtual images are always erect (upright).

  • Rays of light appear to meet at a point after reflection (they do not actually meet).

  • Example: An image formed in a plane mirror.

Characteristics of Image Formation by a Plane Mirror

• The image formed is always virtual and erect.

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

• The distance between the object and the mirror is equal to the distance between the image and the mirror.

• The image undergoes Lateral Inversion: the left side of the object appears to be the right side of the image, and vice versa.

• Magnification ($m$): Always equals $1$ because the object and image sizes are identical.

• Focal Length: The focal length of a plane mirror is infinite ($\infty$).

Spherical Mirrors

• Concave Mirror: A mirror that is curved inward; the reflecting surface is on the inner side.

• Convex Mirror: A mirror that is curved outward; the reflecting surface is on the outer side.

• Important Terms:

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

  • Center of Curvature ($C$): The center of the sphere of which the mirror is a part.

  • Radius of Curvature ($R$): The radius of the sphere of which the mirror is a part, or the line joining the pole and the center of curvature.

  • Principal Axis: An imaginary line extended through the center of curvature ($C$) and the pole ($P$).

  • Principal Focus ($F$):

    • In a concave mirror, it is the point on the principal axis where light rays parallel to the principal axis converge after reflection.

    • In a convex mirror, it is the point from which parallel rays appear to diverge.

  • Focal Length ($f$): The distance between the pole and the principal focus. It is denoted by $f$.

  • Aperture: The diameter of the reflecting surface area of the spherical mirror.

Rules for Drawing Ray Diagrams in Spherical Mirrors

• (1) A ray parallel to the principal axis passes through the focus ($F$) after reflection.

• (2) A ray passing through the focus ($F$) becomes parallel to the principal axis after reflection.

• (3) A ray passing through the center of curvature ($C$) strikes the mirror normally and retraces its own path.

• (4) A ray of light falling obliquely on the pole ($P$) is reflected such that the principal axis acts as the normal, maintaining equal angles of incidence and reflection.

Image Formation by a Concave Mirror

• Object at Infinity: Image is formed at the focus ($F$). The nature is real and inverted, and it is highly diminished (point-sized).

• Object Beyond $C$: Image is formed between $C$ and $F$. The nature is real and inverted, and the size is small (diminished).

• Object at $C$: Image is formed at $C$. The nature is real and inverted, and the size is the same as the object.

• Object Between $C$ and $F$: Image is formed beyond $C$. The nature is real and inverted, and the size is large (enlarged).

• Object at $F$: Image is formed at infinity. The nature is real and inverted, and it is highly magnified.

• Object Between $P$ and $F$: Image is formed beyond the mirror. The nature is virtual and erect, and the image is enlarged.

Uses of Concave Mirrors

• Used in torches, searchlights, and vehicle headlights to produce powerful parallel beams of light.

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

• Used by dentists to see large and erect images of teeth.

• Used by ENT (Ear, Nose, Throat) doctors.

• Used in solar furnaces to concentrate solar energy and rays to produce high heat.

Image Formation by a Convex Mirror

• Object at Infinity: Image is formed at focus ($F$) behind the mirror. The nature is virtual and erect, and it is highly diminished (point-sized).

• Object Between Infinity and Pole: Image is formed between the pole ($P$) and focus ($F$) behind the mirror. The nature is virtual and erect, and it is small in size (diminished).

• Uses: Used as rearview mirrors in vehicles because they provide a wide field of view and always form an erect (though diminished) image.

Mirror Formula and Magnification

• Mirror Formula:     $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$     Where:

  • $u$ = Object Distance

  • $v$ = Image Distance

  • $f$ = Focal Length

• Magnification ($m$): The ratio of the height of the image to the height of the object.     $m = \frac{h_i}{h_o} = -\frac{v}{u}$

  • If $m$ is positive ($+ve$): A virtual and erect image is formed.

  • If $m$ is negative ($-ve$): A real and inverted image is formed.

  • If $m = 1$: Height of image = height of object (same size).

  • If $m < 1$: Image is diminished (height of image < height of object).

  • If $m > 1$: Image is enlarged (height of image > height of object).

Refraction of Light

• Refraction: The bending of light when a light ray enters from one transparent medium to another transparent medium.

• Optical Media:

  • Denser Medium: Light travels slower (e.g., Glass).

  • Rarer Medium: Light travels faster (e.g., Air, Water).

• Speed of Light ($c$):

  • Speed of light in air/vacuum: $3 \times 10^8\,m/s$.

  • Speed of light in water: $2.25 \times 10^8\,m/s$.

  • Speed of light in glass: $2 \times 10^8\,m/s$.

• Rules of Refraction:

  • Case I: When light enters from a rarer medium to a denser medium, it bends towards the normal.

  • Case II: When light enters from an optically denser medium to an optically rarer medium, it bends away from the normal.

Laws of Refraction and Refractive Index

• Laws of Refraction:

  • (1) The incident ray, the refracted ray, and the normal to the interface at the point of incidence all lie in the same plane.

  • (2) The ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant for a given pair of media. This is known as Snell's Law.     $\frac{\sin(i)}{\sin(r)} = \text{Constant} = \mu \text{ (Refractive Index)}$

• Refractive Index ($\mu$ or $n$): The ratio of the speed of light in air/vacuum ($c$) to the speed of light in another medium ($v$).     $\mu = \frac{c}{v}$

  • Example Calculation for Water:         $\mu_{water} = \frac{3 \times 10^{8}\,m/s}{2.25 \times 10^{8}\,m/s} = 1.33$

• Key Relationships:

  • Optical Density $\propto$ Refractive Index $\propto$ Bending of light.

  • Refractive Index $\propto \frac{1}{\text{Speed of light}}$.

Refraction through Lenses

• Lens: A transparent material bounded by two spherical surfaces.

• Types of Lenses:

  1. Convex Lens: Thick at the center and thin at the edges. It is a converging lens.

  2. Concave Lens: Thin at the center and thick at the edges. It is a diverging lens.

• Important Terms:

  • Optical Center ($O$): The center of the lens through which light passes undeviated.

  • Center of Curvature ($C_1, C_2$): The centers of the spheres of which the lens surfaces are parts.

  • Principal Axis: The line joining the two centers of curvature.

  • Principal Focus ($F$): The point where rays parallel to the principal axis meet (convex) or appear to diverge from (concave) after refraction.

  • Focal Length ($f$): The distance between the optical center and the focus.

• Rules for Image Formation:

  1. A ray parallel to the principal axis passes through the focus ($F$).

  2. A ray passing through the focus ($F$) becomes parallel to the principal axis.

  3. A ray passing through the optical center ($O$) passes through without any deviation.

Image Formation by a Convex Lens

1. Object at Infinity: Image at focus ($F_2$). Nature: Real and inverted. Size: Highly diminished (point-sized).

2. Object Beyond $2F_1$: Image between $F_2$ and $2F_2$. Nature: Real and inverted. Size: Diminished.

3. Object at $2F_1$: Image at $2F_2$. Nature: Real and inverted. Size: Same size as object.

4. Object Between $F_1$ and $2F_1$: Image beyond $2F_2$. Nature: Real and inverted. Size: Enlarged.

5. Object at $F_1$: Image at infinity. Nature: Real and inverted. Size: Infinitely large/Highly enlarged.

6. Object Between $F_1$ and $O$: Image formed on the same side of the lens as the object. Nature: Virtual and erect. Size: Enlarged.

Lens Formula, Magnification, and Power

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

• Magnification ($m$):     $m = \frac{h_i}{h_o} = \frac{v}{u}$

  • $m$ is positive ($+ve$) for virtual and erect images.

  • $m$ is negative ($-ve$) for real and inverted images.

• Power of a Lens ($P$): The ability of a lens to converge or diverge light. It is the reciprocal of the focal length in meters.     $P = \frac{1}{f(m)} \text{ or } P = \frac{100}{f(cm)}$

  • SI Unit: Dioptre ($D$).

  • Sign: Convex lenses have positive ($+$) power; concave lenses have negative ($-$) power.

  • Combination of Power: $P = P_1 + P_2 + P_3 + \dots$

  • Example: If $P_1 = +2.5\,D, P_2 = -3\,D, P_3 = +3\,D$, then $P = 2.5 + (-3) + 3 = +2.5\,D$.

Refraction through a Glass Slab

• When light enters from air to glass, it bends towards the normal.

• When light exits from glass to air, it bends away from the normal.

• Emergent Ray: The ray that comes out of the glass slab is parallel to the incident ray but slightly shifted.

• Lateral Displacement: The perpendicular distance between the path of the original incident ray and the emergent ray.

• Factors affecting Lateral Displacement:

  1. Angle of incidence ($i$).

  2. Thickness of the glass slab.

  3. Refractive index of the material.

Sign Convention for Spherical Lenses and Mirrors

1. The object is always placed on the left-hand side.

2. All distances parallel to the principal axis are measured from the pole (mirror) or optical center (lens).

3. Distances measured in the direction of incident light (to the right of origin) are positive ($+ve$).

4. Distances measured opposite to the direction of incident light (to the left of origin) are negative ($-ve$).

5. Heights perpendicular to and above the principal axis are positive ($+ve$).

6. Heights perpendicular to and below the principal axis are negative ($-ve$).

Chemistry Appendix: Gas Tests

• Test for Carbon Dioxide ($CO_2$):

  • $CO_2$ reacts with Calcium Hydroxide solution (lime water) to produce a white precipitate of Calcium Carbonate ($CaCO_3$).

  • Limewater turns milky: $Ca(OH)_2 + CO_2 \rightarrow CaCO_3 + H_2O$.

• Test for Chlorine ($Cl_2$):

  • Chlorine is an acidic gas that acts as a bleach.

  • Damp blue litmus paper will first turn red and then be bleached white when placed in chlorine gas.

Questions & Discussion

• Q: What is the relation between focal length and radius of curvature?

  • A: $f = \frac{R}{2}$. For a sphere with $R = 20\,cm$, $f = 10\,cm$.

• Q: Name a mirror that can give an erect and enlarged image.

  • A: Concave Mirror (when the object is between P and F).

• Q: Why do we prefer convex mirrors as rearview mirrors?

  • A: (1) They always produce an erect image. (2) They provide a wider field of view. (3) The images formed are diminished, allowing much more traffic to be seen in a small mirror.

• Q: What is the meaning of the statement "Refractive index of diamond is 2.42"?

  • A: It means the ratio of the speed of light in vacuum to the speed of light in diamond is $2.42$. Alternatively, speed of light in vacuum is $2.42$ times the speed of light in diamond.

• Q: Describe the appearance of a pencil partly submerged in water.

  • A: The pencil appears to be bent at the water surface because light coming from the submerged portion refracts (bends) as it enters the air, appearing to come from a position slightly above the actual position.

• Q: Define 1 Dioptre.

  • A: One dioptre is the power of a lens whose focal length is exactly $1\,meter$.

• Q: Why does a ray of light bend when traveling from one medium to another?

  • A: Due to the change in velocity of light in different media and to minimize the time taken to travel the distance.

• Q: What happens when a ray passes through the center of curvature ($C$)?

  • A: It retraces its path because it strikes the mirror surface normally ($\angle i = 0^\circ, \angle r = 0^\circ$).

• Q: In which direction should a beam of light pass through a convex lens to continue in the same direction?

  • A: Through the optical center.

• Q: Name a liquid with mass density less than water but optical density greater than water.

  • A: Kerosene.

• Q: Power of Accommodation of the human eye.

  • A: The ability of the eye lens to adjust its focal length to form sharp images of objects at varying distances on the retina.

• Q: Intensity comparison for a lens half-covered with black paper.

  • A: The lens will still produce a complete image, but the intensity of the image will be reduced because the amount of light reaching the image position is cut in half.

• Q: Numerical Problem - Mirror Magnification: If $u = -16\,cm$ and $m = -3$:

  • $m = -\frac{v}{u} \Rightarrow -3 = -\frac{v}{-16} \Rightarrow v = -48\,cm$.

  • Formula: $\frac{1}{f} = \frac{1}{v} + \frac{1}{u} = \frac{1}{-48} + \frac{1}{-16} = \frac{-1-3}{48} = -\frac{4}{48} \Rightarrow f = -12\,cm$.

• Q: Comparison of light speed changes via Refractive Index: Media A (1.33), B (1.50), C (1.52), D (2.40).

  • Minimum change: Between B and C ($1.50 \leftrightarrow 1.52$).

  • Maximum change: Between A and D ($1.33 \leftrightarrow 2.40$).