Lecture Notes: Spherical Mirrors and Lenses

Types of Spherical Mirrors

1. Concave Mirrors

Definition: If the reflecting surface is cave-like (curving inward), it's a concave mirror.
Construction: The back of the mirror is silvered, and the reflecting side is concave.
Characteristics: Reflecting side is inward (cave-like).

2. Convex Mirrors

Definition: If the reflecting surface bulges outwards, it's a convex mirror.
Construction: The back of the mirror is silvered, and the front surface bulges outwards.
Characteristics: Reflecting side is bulging outward.

Key Terms Related to Mirrors

Concave Mirror Example

Center of Curvature: The center of the sphere from which the mirror segment is cut.
Pole (P): The center point of the mirror.
Principal Axis: The line joining the pole (P) and the center of curvature (C).
Focus (F): The midpoint on the principal axis between the pole and center of curvature.
Radius of Curvature (PC): The distance between the pole and center of curvature.
Focal Length (PF): The distance between the pole and focus (F), often half of the radius of curvature.

Rules for Image Formation in Concave Mirrors

Rule 1

Parallel Rays: Rays parallel to the principal axis reflect and pass through the focus.

Rule 2

Focus Rays: Rays passing through the focus after reflection become parallel to the principal axis.

Rule 3

Center of Curvature Rays: Rays passing through the center of curvature reflect back along the same path.

Image Formation with Concave Mirrors

Case 1: Object Beyond the Center of Curvature (C)

Process: Use two rays—one parallel to the principal axis and one through the center of curvature.
Image Position: Between C and F.
Characteristics: Real, inverted, and diminished image.

Case 2: Object at the Center of Curvature (C)

Process: Use the same ray rules.
Image Position: At C.
Characteristics: Real, inverted, and same size.

Case 3: Object Between Center of Curvature (C) and Focus (F)

Process: Use parallel and focus rays.
Image Position: Beyond C.
Characteristics: Real, inverted, and magnified image.

Case 4: Object at Focus (F)

Process: Reflected rays become parallel.
Image Position: At infinity.
Characteristics: Real, inverted, and highly magnified.

Case 5: Object Between Focus (F) and Pole (P)

Process: Use parallel ray and optical center ray.
Image Position: Behind the mirror.
Characteristics: Virtual, erect, and magnified image.

Principle of Reversibility of Light

Definition: If a light path is reversed, it will follow the same path back.

Lenses

Convex Lenses

Construction: Middle is thicker than the edges.
Characteristics: Converging lens, bringing parallel rays to a focus.

Image Formation with Convex Lenses

Image Formation Steps

Two rays needed: one parallel to the principal axis and one through the optical center.
Use provided rules to determine the intersection and therefore the image.

Case 1: Object Beyond 2F1

Image: Between F2 and 2F2, real, inverted, and diminished.

Case 2: Object at 2F1

Image: At 2F2, real, inverted, and same size.

Case 3: Object Between F1 and 2F1

Image: Beyond 2F2, real, inverted, and magnified.

Case 4: Object at Focus (F1)

Image: At infinity, real, inverted, and highly magnified.

Case 5: Object Between Focus (F1) and Optical Center (O)

Image: On the same side as the object, virtual, erect, and magnified.

Concave Lenses

Construction: Middle is thinner than the edges.
Characteristics: Diverging lens, spreading out parallel rays.
Key Rules: Parallel rays diverge, appearing to come from the focus. Rays through the optical center pass undeviated.

Image Formation with Concave Lenses

Image Formation Steps

Two rays needed: one parallel to the principal axis and one passing through the optical center.
Image Characteristics: Virtual, erect, and diminished at all positions except at infinity.

Special Case for Concave Lenses: Object at Infinity

Image: At focus, virtual, erect, and diminished.

Summary Characteristics of Images for Lenses

Convex Lens: Can form real, inverted, diminished/same size/magnified images, or virtual, erect, and magnified images.
Concave Lens: Always forms virtual, erect, and diminished images.

Sign Conventions for Spherical Lenses

Positive Direction: Direction of the incident light.
Negative Direction: Opposite to the positive direction.
Above Principal Axis: Positive.
Below Principal Axis: Negative.

Convex Lenses

Focal Length (F): Positive.
Image Distance (V): Positive/negative based on location.
Object Distance (U): Always negative.

Concave Lenses

Focal Length (F): Negative.
Image Distance (V): Always positive.
Object Distance (U): Always negative.

Lens Maker's Formula

Formula: (1/f = 1/v - 1/u)

Magnification for Lenses

Formula: Magnification (M) = Image Distance (V) / Object Distance (U) = Image Height ((h_i)) / Object Height ((h_o))
Real Image: M is negative.
Virtual Image: M is positive.

Worked Example: Convex Lens

Object Height ((h_o)): 4 cm
Object Distance (U): -30 cm
Focal Length (F): +10 cm

Steps to Solve

Position of Image (V): Using lens formula, find (V = 15 cm) (positive, so on the opposite side of lens)
Image Size ((h_i)): Magnification (M = V / U = 15 / -30 = -1/2)
- Therefore, (h_i = M \times h_o = -1/2 \times 4 = -2 cm) (negative, so inverted)
Nature of Image: Real, inverted, and diminished.

Reflection and Refraction Laws

Understanding of Snell's Law: Ratio of sine of angle of incidence to sine of angle of refraction is constant for a given pair of media.

Applications in Lenses and Mirrors

Convex and Concave Characteristics: Remember distinct image forms both create.
Refraction at Plane Surfaces: Understanding bending angles and movements.
Lenses in Real Life: Application of medians and object placements for practical understanding.

These notes cover spherical mirrors, image formation with mirrors and lenses, key terminologies, and lens maker's formula detailed with sign conventions. Review concepts of real vs. virtual images and position/direction notations carefully as they are significant for exams.

Thanks for attending the lecture! Keep practicing and good luck!