Physics 30 Lesson 7: Optics – Curved Mirrors - Detailed Notes

Physics 30 Lesson 7: Optics – Curved Mirrors

Refer to Pearson pages 656 to 665.

I. Plane Mirrors – Revisited

• Previous work on Reflection indicates that:

  • The image of an object in plane mirrors always exists within the mirror.

  • The image cannot be physically touched or projected since it does not exist outside the mirror.

• Definitions:

  • Virtual Images: Images existing “inside” the mirror, not real objects.

  • Real Images: Images that can be projected onto a screen outside the mirror.

  • Example: Image produced by an overhead projector can be projected onto a screen.

II. Spherical Mirrors

• Law of Reflection: For spherical mirrors, the law of reflection applies: $\theta<em>i = \theta</em>r$ (where $\theta<em>i$ is the angle of incidence and $\theta</em>r$ is the angle of reflection).

• Normal Line: The normal line is defined as the radius of the sphere.

• Types of Spherical Mirrors:

  • Converging (Concave) Mirrors:

  • Light rays reflect towards the focal point.

  • Behavior of parallel incident rays:

    • Rays parallel to the principal axis reflect toward a real focal point.

  • Diverging (Convex) Mirrors:

  • Light rays reflect away from the focal point.

  • Parallel incident rays reflect away from a virtual focal point.

• Image and Object Spatial Terms:

  • A real image exists in actual space, allowing for physical interaction.

  • A virtual image exists only within the mirror (e.g., one’s reflection in a plane mirror).

III. Image Formation – Spherical Mirrors

• Ray Diagrams:

  • Billions of light rays hit mirrors from objects. However, it is unnecessary to draw all these rays.

  • Three Key Rays to Determine Image Formation:

  1. Ray 1: The incident ray parallel to the principal axis reflects through (or away from) the focal point.

  2. Ray 2: The incident ray traveling through the focal point reflects parallel to the principal axis.

  3. Ray 3: The incident ray that passes through the center of curvature reflects straight back.

  • Factors Determining Image Formation:

  • Type of mirror (concave or convex)

  • Focal length ($f$)

  • Distance from mirror to object ($d_o$)

IV. Mirror Equations

• Key Equations:

  • General Formulas:

  • $\frac{1}{f} = \frac{1}{d<em>o} + \frac{1}{d</em>i}$ (mirror equation)

  • $M = \frac{h<em>i}{h</em>o} = -\frac{d<em>i}{d</em>o}$ (magnification)

  • Variable Definitions:

  • $R$ = radius of curvature of the mirror

  • $f$ = focal length

  • $d_o$ = distance from mirror to the object

  • $d_i$ = distance from mirror to the image

  • $h_o$ = height of object

  • $h_i$ = height of image

• Image Characteristics:

  • Real images exist in real space and are always inverted.

  • Virtual images exist within the mirror and are always erect.

• Sign Conventions:

  • $f$:

  • (+) for concave mirror

  • (-) for convex mirror

  • $d_o$: Always (+)

  • $d_i$:

  • (+) for real image

  • (-) for virtual image

  • $h_o$: Always (+)

  • $h_i$:

  • (+) for virtual image

  • (-) for real image

V. Example Problems

Example 1: Converging Mirror

• Given:

  • Height of object ($h_o$) = 5.0 cm

  • Distance from object to mirror ($d_o$) = 60 cm

  • Radius of curvature ($R$) = 80 cm

• Analysis:

  • Focal length: $f = \frac{R}{2} = \frac{80cm}{2} = 40 cm$

  • Using mirror formula:
    $\frac{1}{f} = \frac{1}{d<em>o} + \frac{1}{d</em>i}$

  • Calculate $d<em>i$ : $\frac{1}{d</em>i} = \frac{1}{40} - \frac{1}{60}$
    $d_i = +120 cm$ (real and inverted)

  • Height of image ($h<em>i$): $M = -\frac{d</em>i}{d<em>o} = -\frac{120}{60} = -2$ $h</em>i = M \cdot h_o = -2 \cdot 5.0 cm = -10 cm$

Example 2: Diverging Mirror

• Given:

  • Height of object ($h_o$) = 5.0 cm

  • Distance from object to mirror ($d_o$) = 60 cm

  • Radius of curvature ($R$) = 80 cm

• Analysis:

  • Calculate focal length as in Example 1: $f = -40 cm$

  • Using mirror formula results in a virtual image:
    $d_i = -3.68 cm$(virtual and erect)

  • Size of image: $h<em>i = M \cdot h</em>o$ (magnification calculated similarly)
    (1.32 cm)

Example 3: Object Size Consideration

• An object produces an erect image 1/3 its size when placed 20 cm from the mirror.

• Conjecture:

  • Mirror type is convex (diverging) since a virtual image cannot be larger than the object with concave.

• Calculations:

  • $d_o = 20cm$

  • $d_i = -6.67 cm$

  • Using mirror equation to find focal length, results indicate a diverging mirror.

VI. Practice Problems

1. **Concave Mirror:

   • Object height (5 cm) at distance 14 cm. A. Image distance = 7.8 cm; B. Image Description: real, inverted, smaller; C. Image size = -2.8 cm.**

2. **Convex Mirror:

   • Object height (5 cm) at distance 14 cm. A. Image distance =-3.68 cm; B. Virtual, erect, smaller; C. Image size = 1.32 cm.**

3. **Concave Mirror:

   • Object distance of 30 cm produces a quarter-sized image. A. Focal length = 6.0 cm.**

4. Concave Mirror with erect image 80 cm and object distance 40 cm indicates a radius of curvature of 160 cm.

5. Concave Mirror with inverted image at 120 cm, object at 40 cm yields a radius of curvature of 60 cm.

6. Convex Mirror produces 20 cm distance and erect image 1/6 of size, with object distance 300 cm.

VII. Laboratory Activity – Concave Mirrors

Purpose:

• To determine the focal length of a concave mirror.

Apparatus:

• Construct a setup based on provided diagrams, ensuring proper material organization post-experiment.

Procedure:

1. Place the object at around 60 cm in front of the concave mirror and record the object distance.

2. Move a white screen until a sharp image appears. Record the image distance.

3. Repeat to gather three different positions for accuracy.

Observations:

• Create a data table to organize findings.

Analysis:

1. Calculate focal lengths for each position, deriving an average.

2. Illustrate scale ray diagrams for each observation detailing image formation.