Geometric Optics: Mirrors, Lenses, and Refraction (5)

Geometric Optics

Using Rays to Locate the Image of a Spherical Mirror

Ray diagrams are essential tools for determining the position and nature of images formed by spherical mirrors. By tracing specific rays from the top of an object, one can locate the image produced by the mirror.

• P ray: Parallel to the mirror's axis; reflects through the focal point.

• F ray: Passes through (or toward) the focal point; reflects parallel to the axis.

• C ray: Passes through (or toward) the center of curvature; reflects back on itself.

• Image type: Rays intersect at the image location (real image) or appear to diverge from a point (virtual image).

• Application: Used for both concave and convex mirrors.

Ray Diagrams and the Mirror Equation

Ray diagrams can be used to derive the mirror equation, which relates the object distance, image distance, and focal length of a spherical mirror.

• Mirror Equation:

$\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}$

• Variables: $d_o$ = object distance, $d_i$ = image distance, $f$ = focal length.

• Magnification: $\frac{h_i}{h_o} = -\frac{d_i}{d_o}$, where $h_i$ and $h_o$ are image and object heights.

Mirror Image Characteristics

The characteristics of images formed by convex and concave mirrors depend on the object's position relative to the mirror.

Mirror Type	Object Location	Image Orientation	Image Size	Image Type
Convex	Arbitrary	Upright	Reduced	Virtual
Concave	Beyond C	Inverted	Reduced	Real
	Between F and C	Inverted	Enlarged	Real
	Just beyond F	Inverted	Approaching infinity	Real
	Just inside F	Upright	Approaching infinity	Virtual
	Between mirror and F	Upright	Enlarged	Virtual

Isaac Newton's Prism Experiment

Newton's experiment with prisms demonstrated that white light is composed of a spectrum of colors. By passing sunlight through a prism, he observed the separation of light into its constituent colors, proving that prisms do not create colors but separate them.

• Application: Foundation for understanding the nature of light and color.

Light Propagation Through a Glass Slab

When light passes through a glass slab, it undergoes refraction twice, resulting in a sideways displacement of the ray while maintaining its original direction.

• Refraction: Bending of light as it passes from one medium to another.

• Sideways displacement: The ray emerges parallel to its original path but shifted laterally.

Snell's Law and Indices of Refraction

Snell's Law governs the relationship between the angles of incidence and refraction when light passes between two media with different refractive indices.

• Snell's Law:

$n_1 \sin \theta_1 = n_2 \sin \theta_2$

• Index of refraction ($n$): Ratio of the speed of light in vacuum to that in the medium.

• Table of indices: Common substances have characteristic refractive indices (e.g., air ≈ 1.00, water ≈ 1.33, glass ≈ 1.5).

Total Internal Reflection

Total internal reflection occurs when light attempts to move from a medium with a higher refractive index to one with a lower index at an angle greater than the critical angle, resulting in all light being reflected internally.

• Critical angle ($\theta_c$): $\sin \theta_c = \frac{n_2}{n_1}$, where $n_1 > n_2$.

• Applications: Used in optical fibers for efficient light transmission.

Optical Fibers and Total Internal Reflection

Optical fibers utilize total internal reflection to guide light along their length, allowing for high-speed data transmission over long distances.

• Core and cladding: Light is confined within the core by repeated internal reflections at the core-cladding boundary.

Refraction in a Prism

When light enters a prism, it is refracted toward the normal, and upon exiting, it is refracted away from the normal, resulting in an overall deviation of the light path.

• Application: Prisms are used to disperse light into its component colors.

Convex and Concave Lenses Compared with Prisms

Convex and concave lenses can be conceptually compared to pairs of prisms. Convex lenses converge light, similar to two prisms placed base-to-base, while concave lenses diverge light, similar to two prisms placed tip-to-tip.

• Converging lens: Focuses parallel rays to a point.

• Diverging lens: Spreads parallel rays outward.

Types of Lenses

Lenses are classified as converging or diverging based on their shape and effect on light rays.

• Converging lenses: Convex, plano-convex, double convex.

• Diverging lenses: Concave, plano-concave, double concave.

• Shape: Converging lenses are thicker in the middle; diverging lenses are thinner in the middle.

Principal Rays for Ray Tracing with Lenses

Ray tracing for lenses involves three principal rays to determine image location and characteristics.

• P ray: Parallel to axis; refracted through (convex) or away from (concave) the focal point.

• F ray: Passes through the focal point; refracted parallel to axis.

• M ray: Passes through the center of the lens; continues straight.

Image Formation on Lenses

The position and nature of images formed by lenses depend on the object's location relative to the lens and its focal points.

• Convex lens: Can produce real or virtual images, inverted or upright, depending on object position.

• Concave lens: Always produces virtual, upright, and reduced images.

Thin Lens Image Types

Image characteristics for thin lenses are summarized in the following table:

Lens Type	Object Location	Image Orientation	Image Size	Image Type
Concave	Arbitrary	Upright	Reduced	Virtual
Convex	Beyond F	Inverted	Reduced or enlarged	Real
	Just beyond F	Inverted	Approaching infinity	Real
	Just inside F	Upright	Approaching infinity	Virtual
	Between lens and F	Upright	Enlarged	Virtual

Summary of Image Formation by Lenses

• Convex lens:

  • Distant object: Real, inverted, smaller

  • Object between 2F and F: Real, inverted, larger

  • Object at F: Image at infinity

  • Object between F and lens: Virtual, upright, larger

• Concave lens:

  • Always virtual, upright, and reduced

Additional info: These notes expand on the original slides by providing definitions, equations, and context for each concept, ensuring a self-contained study guide for geometric optics.