Exhaustive Notes on Light: Reflection and Refraction

Fundamentals of Light

• Definition: Light is a form of electromagnetic radiation and energy that enables the perception of objects.
• Mechanism of Vision: When light falls on an object, the surface reflects a portion of that light. When this reflected light enters human eyes, vision occurs.

• Rectilinear Propagation: Light travels in straight lines under uniform atmospheric conditions.
• Common Optical Phenomena:
  • Formation of shadows when opaque objects obstruct light paths.
  • Formation of images by plane, spherical mirrors, and spherical lenses.
  • Bending of light rays as they transition across different medium boundaries (refraction).
  • Twinkling of stars in the night sky due to atmospheric refraction.
  • Formation of rainbows due to dispersion and internal reflection within water droplets.

Reflection of Light

• Definition: Reflection is the phenomenon in which light striking a highly polished, reflective surface (such as a mirror) is sent back into the same medium.

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

$i = r$

*   **Second Law**: The incident ray, the reflected ray, and the normal to the mirror surface at the point of incidence all lie in the same plane.

Image Formation by a Plane Mirror

• Characteristics of Images Formed by Plane Mirrors:
  • Orientation: The image formed is erect (upright).
  • Size: The image size is equal to the object size ($h_i = h_o$).
  • Position: The image distance behind the mirror is identical to the object distance in front of the mirror ($v = u$).
  • Nature: The image is virtual, meaning light rays do not actually intersect at the image point and it cannot be projected or captured onto a screen.
  • Lateral Inversion: The image exhibits lateral inversion, where the left side of the object appears as the right side of the image and vice versa.

Spherical Mirrors

• Definition: A spherical mirror is a curved mirror whose reflective surface forms part of a hollow sphere of glass.
• Types of Spherical Mirrors:
  • Concave Mirror: A spherical mirror whose reflective surface is curved inward toward the center of the sphere. It acts as a converging mirror because rays parallel to its principal axis converge to a point after reflection.

*   **Convex Mirror**: A spherical mirror whose reflective surface is curved outward away from the center of the sphere. It acts as a diverging mirror because rays parallel to its principal axis diverge upon reflection and appear to emanate from a virtual focal point behind the mirror.

Terms Used in the Study of Spherical Mirrors

• Center of Curvature ($C$): The center of the hollow sphere of glass of which the mirror surface forms a part.
• Radius of Curvature ($R$ or $CP$): The radius of the hollow sphere of glass of which the mirror forms a part; it represents the linear distance between the center of curvature ($C$) and the pole ($P$).
• Pole ($P$): The geometric midpoint or center point of the reflecting surface of a spherical mirror.
• Principal Axis ($X\text{--}Y$): The straight reference line passing through the center of curvature ($C$) and the pole ($P$) of the mirror.
• Principal Focus ($F$):
  • For a Concave Mirror: The point on the principal axis where light rays traveling parallel to the principal axis actually converge/meet after reflecting off the mirror surface.
  • For a Convex Mirror: The point on the principal axis behind the mirror from which light rays traveling parallel to the principal axis appear to diverge after reflection.
• Focal Length ($f$): The linear distance measured between the pole ($P$) and the principal focus ($F$) of the spherical mirror.
• Relationship Between Radius of Curvature and Focal Length: The radius of curvature ($R$) of a spherical mirror is twice its focal length ($f$):

$R = 2f$

$f = \frac{R}{2}$

Rules for Ray Tracing in Spherical Mirrors

• Rule 1 (Rays Parallel to Principal Axis):
  • Concave Mirror: A ray parallel to the principal axis passes directly through the principal focus ($F$) after reflection.
  • Convex Mirror: A ray parallel to the principal axis reflects such that it appears to diverge from the principal focus ($F$) located behind the mirror.

• Rule 2 (Rays Passing Through or Directed Towards Focus):
  • Concave Mirror: A light ray passing through the principal focus ($F$) emerges parallel to the principal axis after reflection.
  • Convex Mirror: A light ray directed toward the principal focus ($F$) reflects and emerges parallel to the principal axis.
• Rule 3 (Rays Passing Through or Directed Towards Center of Curvature):
  • Concave Mirror: A ray passing through the center of curvature ($C$) strikes the mirror surface normally (i = 0^\n\circ) and reflects back along the exact same line.
  • Convex Mirror: A ray directed toward the center of curvature ($C$) reflects back along its original path.
• Rule 4 (Rays Directed Obliquely to the Pole):
  • Concave & Convex Mirrors: A ray directed obliquely toward the pole ($P$) reflects obliquely, obeying the law of reflection ($i = r$) relative to the principal axis.

Image Formation by Concave Mirror

• Object Position: At Infinity
  • Image Position: At the principal focus ($F$).
  • Image Size: Highly diminished, point-sized.
  • Image Nature: Real and inverted.
• Object Position: Beyond Center of Curvature ($C$)
  • Image Position: Between the focus ($F$) and center of curvature ($C$).
  • Image Size: Diminished.
  • Image Nature: Real and inverted.

• Object Position: At Center of Curvature ($C$)
  • Image Position: At the center of curvature ($C$).
  • Image Size: Same size as the object.
  • Image Nature: Real and inverted.
• Object Position: Between Center of Curvature ($C$) and Focus ($F$)
  • Image Position: Beyond the center of curvature ($C$).
  • Image Size: Enlarged.
  • Image Nature: Real and inverted.
• Object Position: At Focus ($F$)
  • Image Position: At infinity.
  • Image Size: Highly enlarged/magnified.
  • Image Nature: Real and inverted.
• Object Position: Between Focus ($F$) and Pole ($P$)
  • Image Position: Behind the mirror.
  • Image Size: Enlarged.
  • Image Nature: Virtual and erect.

Image Formation by Convex Mirror

• Object Position: At Infinity
  • Image Position: Behind the mirror at principal focus ($F$).
  • Image Size: Highly diminished, point-sized.
  • Image Nature: Virtual and erect.
• Object Position: Between Infinity and Pole ($P$)
  • Image Position: Behind the mirror between pole ($P$) and focus ($F$).
  • Image Size: Diminished.
  • Image Nature: Virtual and erect.

Practical Applications of Spherical Mirrors

• Applications of Concave Mirrors:
  • Torches, Searchlights, and Vehicle Headlights: The light source is placed at the principal focus of a concave reflector to generate powerful, parallel beams of light.
  • Shaving Mirrors and Makeup Mirrors: When held close to the face (within the focal length), they produce an enlarged, virtual, and erect image of the face.
  • Dentists' Mirrors: Dentists use concave mirrors to see enlarged virtual images of patients' teeth.

*   *Solar Furnaces*: Large concave mirrors concentrate parallel solar radiation at their focus to achieve high temperatures for thermal power generation.

• Applications of Convex Mirrors:
  • Vehicle Rear-View Mirrors: Used as side/rear-view mirrors on automobiles because they always yield an erect, diminished image and afford a significantly wider field of view compared to plane mirrors.

New Cartesian Sign Convention for Spherical Mirrors

• Origin Reference: The pole ($P$) of the mirror is treated as the origin ($0,0$) on a Cartesian coordinate plane.
• Object Placement: The object is always placed to the left of the mirror, meaning incident light travels from left to right.
• Horizontal Measurement: All distances parallel to the principal axis are measured from the pole ($P$).
  • Distances measured in the direction of incident light (to the right of the pole along the $+x$-axis) are positive ($+$).
  • Distances measured opposite to the direction of incident light (to the left of the pole along the $-x$-axis) are negative ($-$).
• Vertical Measurement:
  • Heights measured perpendicular to and above the principal axis (along the $+y$-axis) are positive ($+$).
  • Heights measured perpendicular to and below the principal axis (along the $-y$-axis) are negative ($-$).
• Convention Summary for Mirrors:
  • Object distance ($u$) is always negative ($-$).
  • Focal length ($f$) of a concave mirror is always negative ($-$).
  • Focal length ($f$) of a convex mirror is always positive ($+$).

Mirror Formula and Magnification

• Mirror Formula: Mathematical relationship connecting 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 ($h_i$) to object height ($h_o$):

$m = \frac{h_i}{h_o}$

• Distance Relation to Magnification: Magnification can also be expressed in terms of image distance ($v$) and object distance ($u$):

$m = -\frac{v}{u}$

• Combined Magnification Formula:

$m = \frac{h_i}{h_o} = -\frac{v}{u}$

• Sign Interpretation for Magnification:
  • A negative ($-$-) magnification value signifies a real and inverted image.
  • A positive ($+$-) magnification value signifies a virtual and erect image.

Refraction of Light

• Definition: Refraction is the bending of light at the boundary when it travels obliquely from one transparent medium into another due to changes in optical density and light speed.
• Behavior Across Media:
  • Optically Rarer to Optically Denser Medium: Light slows down and bends towards the normal.
  • Optically Denser to Optically Rarer Medium: Light speeds up and bends away from the normal.

Refraction Through a Rectangular Glass Slab

• Interface Refractions: When light passes through a glass slab, it undergoes refraction twice: first at the air-glass interface (bending toward the normal) and second at the glass-air interface (bending away from the normal).

• Parallel Emergence: The emergent ray is parallel to the initial incident ray because the extent of bending at the two parallel opposite faces of the slab is equal and opposite.
• Lateral Displacement: The perpendicular distance separating the path of the original straight-line incident ray and the emergent ray is known as lateral displacement or lateral shift.
• Angles Involved: Angle of incidence ($i$), angle of refraction ($r$), and angle of emergence ($e$). For parallel slab boundaries, $i = e$.

Laws of Refraction of Light

• First Law: 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.
• Second Law (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 a constant value for light of a given color and for a given pair of media:

$\frac{\sin(i)}{\sin(r)} = \text{constant}$

Refractive Index

• Absolute Refractive Index ($n$): The ratio of the speed of light in vacuum or air ($c$) to the speed of light in a specific medium ($v$):

$n = \frac{c}{v}$

• Relative Refractive Index ($n_{21}$): The refractive index of medium 2 with respect to medium 1 is defined as the ratio of light speed in medium 1 ($v_1$) to light speed in medium 2 ($v_2$):

$n_{21} = \frac{\text{Speed of light in medium 1}}{\text{Speed of light in medium 2}} = \frac{v_1}{v_2}$

Spherical Lenses

• Definition: A spherical lens is a transparent optical material bounded by two surfaces, where at least one or both of the surfaces are spherical.
• Types of Lenses:
  • Convex Lens (Converging Lens): Thicker in the middle and thinner at the edges. Light rays traveling parallel to the principal axis converge to a real point on the principal axis after passing through the lens.
  • Concave Lens (Diverging Lens): Thinner in the middle and thicker at the edges. Light rays traveling parallel to the principal axis diverge after refraction, appearing to originate from a focal point located on the same side of the lens.

Rules for Ray Tracing in Spherical Lenses

• Rule 1 (Parallel Rays):
  • Convex Lens: A light ray parallel to the principal axis refracts through the lens and passes through the principal focus ($F_2$) on the opposite side.
  • Concave Lens: A light ray parallel to the principal axis diverges upon refraction, appearing to originate from the focus ($F_1$) located on the same side as the incident ray.
• Rule 2 (Focal Rays):
  • Convex Lens: A light ray passing through the focus ($F_1$) emerges parallel to the principal axis after refraction.
  • Concave Lens: A light ray directed toward the focus ($F_2$) on the opposite side emerges parallel to the principal axis after refraction.
• Rule 3 (Optical Center Rays):
  • Convex & Concave Lenses: A light ray passing through the optical center ($O$) of the lens travels straight through without undergoing any observable angular deviation.

Image Formation by Convex Lens

• Object Position: At Infinity
  • Image Position: At Focus $F_2$.
  • Image Size: Highly diminished, point-sized.
  • Image Nature: Real and inverted.
• Object Position: Beyond $2F_1$
  • Image Position: Between $F_2$ and $2F_2$
  • Image Size: Diminished.
  • Image Nature: Real and inverted.
• Object Position: At $2F_1$
  • Image Position: At $2F_2$
  • Image Size: Same size as the object.
  • Image Nature: Real and inverted.
• Object Position: Between $2F_1$ and $F_1$
  • Image Position: Beyond $2F_2$
  • Image Size: Enlarged.
  • Image Nature: Real and inverted.
• Object Position: At Focus $F_1$
  • Image Position: At infinity.
  • Image Size: Highly enlarged/magnified.
  • Image Nature: Real and inverted.
• Object Position: Between Focus $F_1$ and Optical Center ($O$)
  • Image Position: On the same side of the lens as the object.
  • Image Size: Enlarged.
  • Image Nature: Virtual and erect.

Image Formation by Concave Lens

• Object Position: At Infinity
  • Image Position: At Focus $F_1$ on the same side of the lens.
  • Image Size: Highly diminished, point-sized.
  • Image Nature: Virtual and erect.
• Object Position: Between Infinity and Optical Center ($O$)
  • Image Position: Between Focus $F_1$ and Optical Center ($O$) on the same side.
  • Image Size: Diminished.
  • Image Nature: Virtual and erect.

Sign Convention for Spherical Lenses

• Reference Point: Distances are measured from the Optical Center ($O$) acting as the Cartesian origin.
• Directions: Follows identical rules to spherical mirrors regarding left ($-$), right ($+$), above principal axis ($+$), and below principal axis ($-$).
• Focal Length Conventions:
  • Focal length ($f$) of a Convex lens is positive ($+$).
  • Focal length ($f$) of a Concave lens is negative ($-$).

Lens Formula and Magnification

• Lens Formula: Mathematical relationship connecting object distance ($u$), image distance ($v$), and focal length ($f$):

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

• Magnification Produced by Lenses ($m$): Ratio of image height ($h_i$) to object height ($h_o$):

$m = \frac{h_i}{h_o}$

• Distance Relation to Lens Magnification:

$m = \frac{v}{u}$

• Combined Lens Magnification Formula:

$m = \frac{h_i}{h_o} = \frac{v}{u}$

Power of a Lens

• Definition: Power of a lens measures its ability to converge or diverge light rays; it is quantitatively defined as the reciprocal of the focal length expressed in meters.
• Formula:

$P = \frac{1}{f(\text{m})}$

$f(\text{m}) = \frac{1}{P}$

• SI Unit: The SI unit of lens power is Dioptre ($\text{D}$).
• Definition of 1 Dioptre: 1 Dioptre is the optical power of a lens having a focal length of 1 meter ($1\,\text{D} = 1\,\text{m}^{-1}$).
• Sign Conventions for Power:
  • A convex lens has a positive focal length, so its power is positive ($+$).
  • A concave lens has a negative focal length, so its power is negative ($-$).