Geometrical Optics: Study Notes and Problem Review

Geometrical Optics

Introduction to Geometrical Optics

Geometrical optics is the branch of physics that describes the behavior of light in terms of rays. It is used to analyze the formation of images by mirrors, lenses, and other optical devices. The main principles include reflection, refraction, and the formation of images by spherical mirrors and lenses.

Reflection and Mirrors

Types of Mirrors

• Plane Mirror: A flat mirror that forms virtual, upright images of the same size as the object.

• Concave Mirror: A spherical mirror that curves inward. It can form real or virtual images depending on the object's position.

• Convex Mirror: A spherical mirror that curves outward. It always forms virtual, diminished, and upright images.

Mirror Equation and Magnification

• Mirror Equation:

$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$ where $f$ is the focal length, $d_o$ is the object distance, and $d_i$ is the image distance.

• Magnification:

$M = -\frac{d_i}{d_o} = \frac{h_i}{h_o}$ where $M$ is magnification, $h_i$ is image height, and $h_o$ is object height.

Image Formation by Spherical Mirrors

• For concave mirrors:

  • If the object is beyond the center of curvature, the image is real, inverted, and smaller.

  • If the object is between the center and the focal point, the image is real, inverted, and larger.

  • If the object is between the focal point and the mirror, the image is virtual, upright, and larger.

• For convex mirrors:

  • The image is always virtual, upright, and smaller than the object.

Examples

• Example: An object 2.3 cm tall is placed 15 cm in front of a concave mirror with a radius of curvature of 18 cm. The image is virtual, upright, and 5.8 cm tall.

• Example: An object is placed 3.8 m in front of a plane mirror. The image appears to be 3.8 m behind the mirror's surface.

Refraction and Lenses

Refraction and Snell's Law

• Refraction: The bending of light as it passes from one medium to another with a different refractive index.

• Snell's Law:

$n_1 \sin \theta_1 = n_2 \sin \theta_2$ where $n_1$ and $n_2$ are the refractive indices of the two media, and $\theta_1$ and $\theta_2$ are the angles of incidence and refraction, respectively.

Lenses and Image Formation

• Types of Lenses:

  • Converging (Convex) Lens: Thicker at the center; converges parallel rays to a focal point.

  • Diverging (Concave) Lens: Thinner at the center; diverges parallel rays as if they originated from a focal point.

• Lens Equation:

$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$ where $f$ is the focal length, $d_o$ is the object distance, and $d_i$ is the image distance.

• Magnification for Lenses:

$M = -\frac{d_i}{d_o}$

Examples

• Example: A 3.0 cm tall statue is 41 cm in front of a biconvex lens of focal length 20 cm. The image is real, inverted, and 2.7 cm tall.

• Example: To obtain a magnification of -2 from a convex lens of focal length $f$, place a real object at a distance $3f/2$ from the lens.

Critical Angle and Total Internal Reflection

Critical Angle

• The critical angle is the angle of incidence above which total internal reflection occurs when light passes from a medium with higher refractive index to one with lower refractive index.

• Formula:

$\sin \theta_c = \frac{n_2}{n_1}$ where $n_1 > n_2$.

• Example: For Lucite ($n = 1.50$) to air ($n = 1.00$), the critical angle is $41.8^\circ$.

Dispersion and Wavelength Dependence

Dispersion

• When light of different wavelengths passes through a medium, each wavelength is refracted by a different amount due to variation in refractive index with wavelength.

• Example: A parallel light beam containing 480 nm and 700 nm wavelengths strikes glass at $60^\circ$. The angle between the two beams in the glass is $0.19^\circ$.

Summary Table: Image Characteristics for Mirrors and Lenses

Device	Object Position	Image Type	Image Orientation	Image Size
Concave Mirror	Beyond C	Real	Inverted	Smaller
Concave Mirror	Between C and F	Real	Inverted	Larger
Concave Mirror	Between F and mirror	Virtual	Upright	Larger
Convex Mirror	Any	Virtual	Upright	Smaller
Convex Lens	Beyond 2F	Real	Inverted	Smaller
Convex Lens	Between F and 2F	Real	Inverted	Larger
Convex Lens	Between F and lens	Virtual	Upright	Larger
Concave Lens	Any	Virtual	Upright	Smaller

Key Formulas and Definitions

• Mirror/Lens Equation: $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$

• Magnification: $M = -\frac{d_i}{d_o}$

• Snell's Law: $n_1 \sin \theta_1 = n_2 \sin \theta_2$

• Critical Angle: $\sin \theta_c = \frac{n_2}{n_1}$

Applications and Problem-Solving Tips

• Always identify the type of mirror or lens and the object's position relative to the focal point and center of curvature.

• Use sign conventions: For mirrors, distances measured toward the mirror are negative; for lenses, real images are on the opposite side from the object.

• Draw ray diagrams to visualize image formation.

• Apply Snell's Law for problems involving refraction and critical angles.

• For magnification, a negative value indicates an inverted image, while a positive value indicates an upright image.

Additional info: These notes are based on a set of exam-style questions and answers covering the main concepts and calculations in geometrical optics, including mirrors, lenses, refraction, and critical angles.