Curved Mirrors

Curved Mirrors

Focus Question

• What are some advantages and disadvantages of curved mirrors compared to plane mirrors?

New Vocabulary

• principal axis

• focal point

• focal length

• concave mirror

• real image

• spherical aberration

• convex mirror

• magnification

Review Vocabulary

• ray: a line drawn at a right angle to a wavefront; represents the direction of wave travel

Properties of Curved Mirrors

• The properties of curved mirrors depend on the shape of the mirror.

• The mirror has the same geometric center (C) and radius of curvature (r) as a sphere of radius r.

• The line that passes through line segment CM is the principal axis, which is the straight line perpendicular to the surface of the mirror that divides the mirror in half.

• Focal Point (F): The point where incident light rays that are parallel to the principal axis converge after reflecting from the mirror. F is at the halfway point between M and C.

• Focal Length (f): The distance between the mirror and the focal point and is equal to r/2.

• When drawing ray diagrams, assume all reflection takes place at the mirror’s principal plane.

• The principal plane is perpendicular to the principal axis.

Real Images with Concave Mirrors

• Concave Mirror: A mirror with a reflective surface, the edges of which curve toward the observer.

• If the object is more than twice the focal length (f) from a concave mirror, the image formed will be:

  • Located between F and C

  • Reduced

  • Inverted

  • Real

• Real Image: An image formed by converging light rays.

• If the object is at the center of curvature (C), the image will be the same size and located at C.

• If the object is between the focal point (F) and the center of curvature (C), the image formed will be:

  • located beyond C

  • enlarged

  • inverted

  • real

• In ray diagrams, rays are shown as reflected from the principal plane.

• In reality, rays are reflected off the mirror itself, and only parallel rays close to the principal axis (paraxial rays) are reflected through the focal point. Other rays converge at points closer to the mirror.

• The image formed by parallel rays reflecting off a spherical mirror with a large mirror diameter and a small radius of curvature is a disk, not a point.

• Spherical Aberration: This effect makes an image look fuzzy, not sharp.

• A mirror ground to the shape of a parabola suffers no spherical aberration.

• Spherical aberration is reduced as the ratio of the mirror’s diameter to its radius of curvature is reduced.

Virtual Images with Concave Mirrors

• If the object is at the focal point (F), no image will be formed.

• If the object is between the focal point (F) and the mirror, the image formed will be:

  • Located farther from the mirror than the object

  • Enlarged

  • Upright

  • Virtual

Convex Mirrors

• Convex Mirror: A reflective surface with edges that curve away from the observer.

• Regardless of where the object is placed, the image formed will be:

  • Located between the mirror and F

  • Reduced

  • Upright

  • Virtual

Mirror Equation

• Using the spherical mirror model, a simple formula can relate the focal length, object position, and image position.

• When using this equation, remember that it is only approximately correct and does not predict spherical aberration because it uses the paraxial ray approximation.

Magnification

• Magnification: A property of spherical mirrors referring to how much larger or smaller an image is relative to the object.

• The magnification of an object by a spherical mirror, defined as the image height divided by the object height, is equal to the negative of the image position divided by the object distance. m = \frac{hi}{ho} = -\frac{xi}{xo}

• Value of m relates to the properties of the image:

  • m is positive: image is upright

  • m is negative: image is inverted

  • 0 < |m| < 1: image is reduced

  • |m| > 1: image is enlarged

Example Problems

• Problem: A concave mirror has a radius of curvature of 24.0 cm. A 6.4-cm-tall object is held 26.0 cm from the mirror. Where is the image and how tall is the image?

  • Solution:

    • x_o = 26.0 cm

    • h_o = 6.4 cm

    • f = r/2 = 12.0 cm

    • Using the mirror equation to find the image position: The image is 22.3 cm in front of the mirror.

    • Using the definition of magnification to find the image height: The image is 5.5 cm tall and inverted.

• Problem: A convex mirror has a radius of curvature of 28.4 cm. If you hold a 16.0-cm-long pencil 23.5 cm in front of the mirror, what will the image position and image height be?

  • Solution:

    • x_o = 23.5 cm

    • h_o = 16 cm

    • f = r/2 = -14.2 cm

    • Using the mirror equation to find the image position: The image is 8.85 cm behind the mirror.

    • Using the definition of magnification to find the image height: The image is 6.0 cm tall and upright.

Mirror Comparison

Quiz

1. Which is the straight line perpendicular to the surface of the mirror that divides the mirror in half?

   • A. principal axis

2. Which is the point where incident light rays that are parallel to the principal axis converge after reflecting from the mirror?

   • C. focal point

3. Which is the distance between the mirror and the focal point and is equal to r/2?

   • C. focal length

4. Which has a reflective surface with edges that curve toward the observer?

   • C. concave mirror

5. Which has a reflective surface with edges that curve away from the observer?

   • D. convex mirror