16.1 Reflection

Reflection and Mirrors

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Law of Reflection for Mirrors

The law of reflection is straightforward: the angle of incidence equals the angle of reflection, or $\theta_i = \theta_r$. Both angles are measured from the normal line, which is an imaginary line drawn perpendicular to the mirror surface at the point where light strikes it. The incident ray, reflected ray, and normal all lie in the same plane.

Two types of reflection show up depending on the surface:

• Specular reflection happens on smooth surfaces like polished mirrors. Parallel incoming rays all reflect in a single, predictable direction. This is what lets a flat mirror produce a clear image.
• Diffuse reflection happens on rough surfaces like paper or fabric. Parallel incoming rays scatter in many directions because the surface normals point every which way. You can still see the object, but no clear image forms.

Image formation depends on light rays from each point on an object reflecting off the mirror according to the law of reflection. After reflecting, those rays either physically converge at a point (forming a real image) or appear to diverge from a point behind the mirror (forming a virtual image, as in plane mirrors).

Image Calculations with Curved Mirrors

Curved mirrors come in two types:

• Concave (converging) mirrors curve inward, like the inside of a spoon. Parallel rays hitting the mirror reflect inward and converge at the focal point (F), which sits in front of the mirror surface. Telescope mirrors use this design.
• Convex (diverging) mirrors curve outward, like the back of a spoon. Parallel rays reflect outward and diverge, but if you trace them backward, they appear to originate from a focal point behind the mirror. Car side mirrors use this design.

For both types, the center of curvature (C) is the center of the sphere the mirror was cut from, and the focal length is half the radius of curvature: $f = \frac{R}{2}$.

Mirror equation:

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

This relates the focal length ($f$), the object distance ($d_o$), and the image distance ($d_i$). All distances are measured from the mirror surface.

Magnification equation:

$M = -\frac{d_i}{d_o} = \frac{h_i}{h_o}$

This compares the image height ($h_i$) to the object height ($h_o$). A negative magnification means the image is inverted; a positive magnification means it's upright. If $|M| > 1$, the image is enlarged; if $|M| < 1$, it's reduced.

Sign conventions matter here. For mirrors:

• $d_o$ is positive when the object is in front of the mirror (the usual case).
• $d_i$ is positive when the image forms in front of the mirror (real image) and negative when it forms behind the mirror (virtual image).
• $f$ is positive for concave mirrors and negative for convex mirrors.

Ray diagrams are the visual way to find where an image forms. You draw rays from the top of the object using specific rules, and the image forms where the reflected rays (or their extensions) intersect.

Concave mirror ray tracing rules:

1. A ray parallel to the principal axis reflects through the focal point.
2. A ray passing through the focal point reflects parallel to the principal axis.
3. A ray passing through the center of curvature reflects straight back along its incoming path.

Convex mirror ray tracing rules:

1. A ray parallel to the principal axis reflects as if it came from the focal point behind the mirror.
2. A ray aimed toward the focal point (behind the mirror) reflects parallel to the principal axis.
3. A ray aimed toward the center of curvature reflects back along the same path.

For convex mirrors, you'll always need to extend the reflected rays backward (as dashed lines) to find the virtual image behind the mirror.

Real vs. Virtual Mirror Images

The distinction between real and virtual images comes down to whether light actually reaches the image location.

• Real images form where reflected rays physically converge. They can be projected onto a screen and are always inverted. Concave mirrors produce real images when the object is placed beyond the focal point.
• Virtual images form where reflected rays only appear to diverge from. They cannot be projected and are always upright. Plane mirrors always produce virtual images. Concave mirrors produce virtual images when the object is between the focal point and the mirror.

Convex mirrors always produce virtual, upright, and reduced images regardless of object placement. That's why the warning on car side mirrors says "objects are closer than they appear."

Common applications of concave mirrors:

• Reflecting telescopes and satellite dishes use large concave mirrors to gather and focus light or signals from distant sources.
• Makeup mirrors and dental mirrors place the object inside the focal point to produce a magnified, upright virtual image.
• Car headlights and flashlights place a bulb at the focal point so reflected light emerges as a parallel beam.

Common applications of convex mirrors:

• Rearview and side mirrors on vehicles provide a wider field of view at the cost of making objects appear smaller.
• Security mirrors in stores give a panoramic view of large areas.

Wave Optics and Related Phenomena

Geometric optics (ray tracing, mirror equations) treats light as straight-line rays. Wave optics treats light as electromagnetic waves, which explains phenomena that ray models can't.

• Interference occurs when two or more light waves overlap. If their crests align, they add together (constructive interference, producing brighter regions). If a crest meets a trough, they cancel (destructive interference, producing darker regions).
• Polarization refers to the direction in which a light wave's electric field oscillates. Unpolarized light oscillates in all directions perpendicular to travel. A polarizing filter blocks all orientations except one, which is how polarized sunglasses reduce glare.
• Refraction is the bending of light as it passes from one medium to another with a different refractive index. Snell's law describes this quantitatively: $n_1 \sin\theta_1 = n_2 \sin\theta_2$, where $n$ is the refractive index and $\theta$ is the angle measured from the normal.