Geometric Optics: Mirrors, Refraction, and Lenses

Geometric Optics

Introduction to Geometric Optics

Geometric optics is the branch of physics that studies the propagation of light in terms of rays. It explains how light interacts with mirrors, lenses, and other optical devices, using the principles of reflection and refraction.

• Reflection: The bouncing of light rays off a surface, such as a mirror.

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

• Mirrors: Surfaces that reflect light to form images.

• Lenses: Transparent objects that refract light to converge or diverge rays, forming images.

Reflection at Spherical Mirrors

Convex and Concave Mirrors

Spherical mirrors are sections of a sphere and can be either concave (inward-curving) or convex (outward-curving). The behavior of light and image formation depends on the mirror type.

• Concave Mirror: Reflective surface curves inward. Can form real or virtual images depending on object position.

• Convex Mirror: Reflective surface curves outward. Always forms virtual, erect, and diminished images.

Principal Rays for Spherical Mirrors

Principal rays are standard rays used to determine the location and nature of images formed by mirrors.

• Ray 1: Parallel to the principal axis, reflects through (concave) or appears to come from (convex) the focal point.

• Ray 2: Passes through (concave) or directed toward (convex) the focal point, reflects parallel to the axis.

• Ray 3: Passes through the center of curvature, reflects back on itself.

• Ray 4: Strikes the vertex and reflects symmetrically with respect to the principal axis.

Mirror Equation and Magnification

The relationship between object distance, image distance, and focal length for spherical mirrors is given by the mirror equation:

• Mirror Equation:

$\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$

• Magnification Equation:

$m = \frac{y'}{y} = -\frac{s'}{s}$

• Sign Conventions: For convex mirrors, both focal length (f) and radius of curvature (R) are negative.

• Virtual Focal Point: For convex mirrors, reflected rays appear to diverge from a point (F) behind the mirror.

Image Formation for Different Object Distances (Concave Mirror)

The nature and position of the image depend on the object's distance from the mirror:

• Object beyond center of curvature (s > 2f): Image is real, inverted, and reduced.

• Object at center of curvature (s = 2f): Image is real, inverted, and same size as object.

• Object between center and focal point (f < s < 2f): Image is real, inverted, and magnified.

• Object at focal point (s = f): Image at infinity.

• Object inside focal point (s < f): Image is virtual, erect, and magnified.

Example: Image Formation in a Convex Mirror

Example 34.3: Santa looks at his reflection in a shiny silvered Christmas tree ornament (convex mirror). Given:

• Object distance: 0.75 m

• Ornament diameter: 7.2 cm (so R = -3.6 cm, f = -1.8 cm)

• Santa's height: 1.6 m

Using the mirror equation and magnification:

$\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$

$m = -\frac{s'}{s}$

Image is virtual, erect, and much smaller than the object. For this example, the image height is approximately 3.8 cm.

Refraction at Spherical Surfaces

Refraction and Snell's Law

When light passes from one medium to another, it bends according to Snell's Law:

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

• n1: Refractive index of the first medium

• n2: Refractive index of the second medium

• θ1: Angle of incidence

• θ2: Angle of refraction

Refraction at a Spherical Surface

For a spherical refracting surface (e.g., glass-air interface), the relationship between object distance, image distance, and radius of curvature is:

$\frac{n_1}{s} + \frac{n_2}{s'} = \frac{n_2 - n_1}{R}$

• s: Object distance from the vertex

• s': Image distance from the vertex

• R: Radius of curvature of the surface

• n1: Refractive index of the medium where the object is located

• n2: Refractive index of the medium where the image is formed

Lateral Magnification for Refraction

The lateral magnification for a refracting spherical surface is:

$m = \frac{n_1 s'}{n_2 s}$

Example: Glass Rod with Spherical End

A cylindrical glass rod (n = 1.52) with a hemispherical end (R = 2.00 cm) is used to find the image distance and magnification for an object on the axis at a given distance from the vertex.

• Apply the spherical refraction equation to solve for s'.

• Calculate magnification using the formula above.

• For this example, the image is inverted and smaller than the object (m ≈ 0.93).

Lenses: Types and Image Formation

Types of Lenses

Lenses are classified based on their shape and the way they refract light:

• Converging Lenses (Positive Focal Length):

  • Double convex

  • Planoconvex

  • Meniscus (convex side stronger)

• Diverging Lenses (Negative Focal Length):

  • Double concave

  • Planoconcave

  • Meniscus (concave side stronger)

Principal Rays for Lenses

• Ray parallel to axis refracts through (converging) or appears to diverge from (diverging) the focal point.

• Ray through center of lens passes straight without deviation.

• Ray through (or toward) focal point emerges parallel to axis.

Focal Points and Image Formation

• First Focal Point (F1): Point where rays parallel to the axis before entering the lens converge (or appear to diverge from) after passing through the lens.

• Second Focal Point (F2): Point where rays parallel to the axis after passing through the lens converge (or appear to diverge from).

• Converging Lens: Forms real, inverted images when object is outside focal length; virtual, erect images when object is inside focal length.

• Diverging Lens: Always forms virtual, erect, and diminished images.

Lens Equation

The thin lens equation relates object distance, image distance, and focal length:

$\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$

• f: Focal length (positive for converging, negative for diverging lenses)

• s: Object distance

• s': Image distance

Summary Table: Mirror and Lens Sign Conventions

Quantity	Concave Mirror	Convex Mirror	Converging Lens	Diverging Lens
Focal Length (f)	Positive	Negative	Positive	Negative
Image Type	Real/Virtual	Virtual	Real/Virtual	Virtual
Image Orientation	Inverted/Erect	Erect	Inverted/Erect	Erect

Additional info: Some context and equations have been expanded for clarity and completeness, including the summary table and detailed explanations of principal rays and sign conventions.