Geometrical Optics - Class 12 Odisha Board Physics

Geometrical Optics: Reflection and Refraction of Light

Fundamental Concepts of Geometrical Optics

Geometrical optics deals with the study of light propagation in terms of rays, focusing on the phenomena of reflection and refraction. The main principles include the behavior of light at plane and curved surfaces, the formation of images by mirrors and lenses, and the mathematical relationships governing these processes.

• Reflection: The bouncing back of light in the same medium from a surface.
• Refraction: The bending of light as it passes from one medium to another with a different refractive index.
• Spherical Surfaces: Mirrors…

Geometrical Optics: Reflection and Refraction of Light

Fundamental Concepts of Geometrical Optics

Geometrical optics deals with the study of light propagation in terms of rays, focusing on the phenomena of reflection and refraction. The main principles include the behavior of light at plane and curved surfaces, the formation of images by mirrors and lenses, and the mathematical relationships governing these processes.

• Reflection: The bouncing back of light in the same medium from a surface.
• Refraction: The bending of light as it passes from one medium to another with a different refractive index.
• Spherical Surfaces: Mirrors and lenses derived from spheres, classified as concave or convex based on the reflecting/refracting surface.

Laws of Reflection and Types

Reflection of light follows two fundamental laws. These laws apply to both plane and curved surfaces, with specific considerations for normals at each point.

• Law 1: The angle of incidence equals the angle of reflection ($i = r$).
• Law 2: The incident ray, reflected ray, and the normal at the point of incidence all lie in the same plane.
• Types of Reflection:
  • Specular Reflection: Occurs on smooth surfaces (plane mirrors), where normals are parallel.
  • Diffuse Reflection: Occurs on rough or curved surfaces, where normals are not parallel.

Spherical Mirrors: Structure and Image Formation

Spherical mirrors are classified as concave or convex, each with specific parts and image formation rules. The principal axis, pole, center of curvature, focus, and focal length are key elements.

• Concave Mirror: Reflecting surface is the inner side of the sphere.
• Convex Mirror: Reflecting surface is the outer side of the sphere.
• Key Points:
  • Pole (P): Center of the mirror surface.
  • Center of Curvature (C): Center of the sphere from which the mirror is cut.
  • Principal Focus (F): Point where parallel rays converge (concave) or appear to diverge from (convex).
  • Focal Length (f): Distance from pole to focus.
  • Aperture: Width of the mirror.

Mirror Formula and Magnification

The mirror formula relates object distance ($u$), image distance ($v$), and focal length ($f$) for spherical mirrors. Magnification describes the ratio of image height to object height.

• Mirror Formula:$$\frac{1}{v} + \frac{1}{u} = \frac{2}{r}$$or$$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$$where $f = \frac{r}{2}$.
• Magnification ($m$):$$m = \frac{\text{height of image}}{\text{height of object}} = -\frac{v}{u}$$
• Sign Convention:
  • Distances measured in the direction of incident light are positive.
  • Distances against the direction of incident light are negative.

Image Formation by Spherical Mirrors: Special Cases

Image formation depends on the position of the object relative to the mirror. For concave mirrors, six cases are considered; for convex mirrors, two main cases.

• Concave Mirror Cases:
  • Object at infinity: Image at focus, highly diminished, real, inverted.
  • Object beyond $C$: Image between $F$ and $C$, diminished, real, inverted.
  • Object at $C$: Image at $C$, same size, real, inverted.
  • Object between $C$ and $F$: Image beyond $C$, enlarged, real, inverted.
  • Object at $F$: Image at infinity, highly enlarged, real, inverted.
  • Object between $P$ and $F$: Image behind mirror, enlarged, virtual, erect.
• Convex Mirror Cases:
  • Object at infinity: Image at focus, point-sized, virtual, erect.
  • Object at finite distance: Image between $P$ and $F$, diminished, virtual, erect.

Refraction of Light and Refractive Index

Refraction is the bending of light as it passes from one medium to another. The refractive index quantifies the bending power of a medium.

• Definition: The bending of light when it changes medium is called refraction.
• Absolute Refractive Index ($\mu$):$$\mu = \frac{c}{v}$$where $c$ is the speed of light in vacuum, $v$ is the speed in the medium.
• Relative Refractive Index:$$\mu_{21} = \frac{\mu_2}{\mu_1}$$where $\mu_2$ and $\mu_1$ are the refractive indices of medium 2 and 1, respectively.
• Relation with Wavelength: $\mu$ is inversely proportional to wavelength; higher $\mu$ means more bending.

Laws of Refraction (Snell’s Law) and Critical Angle

Snell’s Law governs the relationship between the angles of incidence and refraction and the refractive indices of the two media.

• Snell’s Law:$$\mu_1 \sin i = \mu_2 \sin r$$or$$\frac{\sin i}{\sin r} = \frac{\mu_2}{\mu_1}$$
• Critical Angle ($\theta_c$): The angle of incidence in the denser medium for which the angle of refraction is $90^\circ$.$$\sin \theta_c = \frac{\mu_2}{\mu_1}$$(for light going from medium 1 to medium 2, with $\mu_1 > \mu_2$).

Total Internal Reflection and Its Applications

Total internal reflection (TIR) occurs when light attempts to move from a denser to a rarer medium at an angle greater than the critical angle, resulting in all light being reflected internally.

• Conditions for TIR:
  • Light must travel from a denser to a rarer medium.
  • Angle of incidence must exceed the critical angle.
• Applications:
  • Mirage: Optical illusion due to TIR in hot conditions.
  • Diamond Sparkle: High refractive index leads to multiple TIRs, causing brilliance.
  • Optical Fibers: Light undergoes repeated TIR, enabling long-distance data transmission.
  • Endoscopy: Medical imaging using optical fibers.

Refraction at Spherical Surfaces and Lens Formula

Refraction at spherical surfaces is governed by a formula relating object distance, image distance, refractive indices, and radius of curvature.

• Formula for Refraction at Spherical Surface:$$\frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{r}$$where $u$ is object distance, $v$ is image distance, $r$ is radius of curvature, $\mu_1$ and $\mu_2$ are refractive indices.
• Magnification for Spherical Surface:$$m = \frac{\mu_1}{\mu_2} \cdot \frac{v}{u}$$

Lenses: Types, Image Formation, and Lens Formula

Lenses are transparent objects with two refracting surfaces. Types include biconvex, plano-convex, biconcave, and plano-concave.

• Convex Lens: Converging lens, thicker at the center.
• Concave Lens: Diverging lens, thinner at the center.
• Lens Formula:$$\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$$
• Magnification for Lenses:$$m = \frac{v}{u}$$
• Sign Convention: Same as mirrors; distances measured from the optical center.

Lens Maker’s Formula and Power of a Lens

The lens maker’s formula relates the focal length of a lens to the refractive index and the radii of curvature of its surfaces. The power of a lens quantifies its converging or diverging ability.

• Lens Maker’s Formula:$$\frac{1}{f} = \left(\frac{\mu_2}{\mu_1} - 1\right) \left(\frac{1}{r_1} - \frac{1}{r_2}\right)$$For a lens in air ($\mu_1 = 1$):$$\frac{1}{f} = (\mu - 1) \left(\frac{1}{r_1} - \frac{1}{r_2}\right)$$
• Power of a Lens ($P$):$$P = \frac{1}{f}$$(with $f$ in meters; SI unit is diopter, $D$)
• Sign of Power:
  • Convex lens: $f$ positive, $P$ positive.
  • Concave lens: $f$ negative, $P$ negative.

Combination of Lenses

When multiple lenses are placed coaxially in contact, their combined focal length and power can be calculated.

• Combined Focal Length:$$\frac{1}{f_{\text{eq}}} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} + \ldots$$
• Combined Power:$$P_{\text{eq}} = P_1 + P_2 + P_3 + \ldots$$
• If the net focal length is positive, the combination behaves as a convex lens; if negative, as a concave lens.

Exam-Focused Problem Types and Calculation Strategies

Several problem types are emphasized, including image position, nature, and magnification for mirrors and lenses, as well as calculation of focal length, power, and refractive index.

• Mirror and Lens Formula Application: Substitute given values with correct sign convention.
• Critical Angle and Refractive Index: Use $\sin \theta_c = 1/\mu$ for calculation.
• Combination Problems: For lenses in contact, sum reciprocals of focal lengths or powers.
• Nature of Image: Use sign of $v$ and $m$ to determine if image is real/virtual, erect/inverted, enlarged/diminished.
• Dependence on Wavelength: Focal length increases with increasing wavelength, as refractive index decreases.

These principles and formulas form the core of geometrical optics, essential for solving both conceptual and numerical problems in exams.