Mirrors are among the most familiar optical devices in everyday life. Every morning you probably look into a bathroom mirror, and many healthcare professionals rely on specialized mirrors during examinations and procedures. Dentists use small concave mirrors to enlarge teeth, physicians use mirrors in head-mounted examination lights to focus illumination, and large convex mirrors improve visibility in hospitals, laboratories, and parking facilities by providing a wider field of view.

Like lenses, mirrors redirect light to produce images. Depending on their shape, mirrors can create images that are the same size as an object, magnified, reduced, upright, or inverted. In this section, we apply the law of reflection to understand how mirrors form images and show that many of the mathematical relationships for mirrors closely parallel those developed for thin lenses.

Figure 70.1 illustrates image formation by a flat mirror. Rays of light leaving the top of the bottle reflect from the mirror and enter the observer's eye. Although the reflected rays travel only in front of the mirror, the brain naturally assumes that light travels in straight lines. When the reflected rays are extended backward, they appear to originate from a point behind the mirror. This apparent point is where the image is perceived.

Applying the law of reflection, which states that the angle of incidence equals the angle of reflection, shows that the image forms the same distance behind the mirror as the object is in front of it. Because no light actually passes through the image location, the image cannot be projected onto a screen. Such an image is called a virtual image.

Virtual images are extremely useful even though they do not physically exist at their apparent location. The image you see in a bathroom mirror, a rearview mirror, or a dressing mirror is always virtual. Your eyes receive reflected light exactly as if it originated behind the mirror, allowing your brain to reconstruct a realistic three-dimensional image.

Flat mirrors are only one type of mirror used in science and medicine. Curved mirrors can either concentrate light or spread it apart, depending on their shape. To understand how this happens, we first examine the focal length of a spherical mirror.

Consider the concave spherical mirrors shown in Figure 70.2. Every incoming ray obeys the law of reflection, but the exact location where the reflected rays meet depends on the mirror's shape. If the mirror is large compared with its radius of curvature (Figure 70.2(a)), rays striking different parts of the surface do not converge at exactly the same point. This optical imperfection, known as spherical aberration, produces a blurred focus.

An ideal parabolic mirror would bring all parallel rays to a single focal point regardless of where they strike the surface. However, parabolic mirrors are much more difficult and expensive to manufacture than spherical mirrors. Fortunately, when only the central portion of a spherical mirror is used (Figure 70.2(b)), spherical aberration becomes very small, and the mirror behaves almost like a perfect focusing mirror. This approximation is analogous to the thin lens approximation introduced in the previous chapter.

For a concave mirror, parallel rays reflecting from the surface converge at a point called the focal point, labeled F in Figure 70.2. The distance from the mirror's center (its vertex) to this point is the focal length, [latex]f[/latex]. Because concave mirrors bring light rays together, they are called converging mirrors, and their focal length is defined as positive.

Just as with lenses, mirrors can be described by their optical power. A mirror that brings light to a focus over a shorter distance is considered more powerful than one with a longer focal length. Consequently, the power of a spherical mirror is also defined as

Thus, mirrors with smaller focal lengths have larger optical powers. Highly curved mirrors concentrate light more strongly than gently curved mirrors, making them useful in applications ranging from reflecting telescopes to medical examination mirrors.

For spherical mirrors, the focal length is directly related to the mirror's radius of curvature. Using the law of reflection together with simple geometry, it can be shown that

where [latex]R[/latex] is the radius of curvature of the mirror. A mirror with a smaller radius of curvature has a shorter focal length and therefore greater optical power.

Convex mirrors behave differently from concave mirrors. As illustrated in Figure 70.3, parallel rays reflecting from a convex mirror spread apart after reflection. When these reflected rays are extended backward, they appear to originate from a point behind the mirror known as the focal point. Because convex mirrors cause light rays to diverge, they are called diverging mirrors, and both their focal length and optical power are defined to be negative.

Ray tracing provides a simple graphical method for locating images formed by mirrors, just as it does for lenses. The following rules summarize the behavior of light rays reflecting from thin spherical mirrors.

Because we assume the mirrors are small compared with their radii of curvature, the same mathematical relationships developed for thin lenses also apply to spherical mirrors. As a result, the mirror equation and magnification equation have exactly the same form as the thin-lens equations, differing only in the sign conventions used for mirrors.

Consider the concave mirror shown in Figure 70.4. The object is located farther from the mirror than its focal length, so [latex]d_o>f[/latex]. This configuration is directly analogous to the first case studied for converging lenses. Using the ray-tracing rules, rays originating from the same point on the object reflect and meet at a common point in front of the mirror. Since the rays physically converge, the image can be projected onto a screen and is therefore a real image.

Like the corresponding image formed by a converging lens, this image is inverted and has a positive image distance. Consequently, its magnification is negative. This situation is known as a Case 1 mirror image. The only significant difference from the lens case is that both the object and the image lie on the same side of the reflecting surface.

Example 70.2: Concentrating Solar Energy with a Curved Mirror

A parabolic trough is a solar-energy device that uses a curved reflecting surface to concentrate sunlight onto a dark pipe containing a fluid. The heated fluid can then transfer energy to a heat exchanger, where it may be used to produce steam and generate electricity. Figure 70.5 shows a large solar-thermal installation that uses this principle.

For this simplified model, assume that the reflector is one-quarter of a cylindrical surface and that the fluid-carrying pipe lies along its focal line. Determine the following:

1. If the pipe is placed [latex]40.0~\text{cm}[/latex] from the mirror at its focal point, what is the mirror’s radius of curvature?
2. How much solar power is concentrated onto each meter of pipe if the incident solar intensity is [latex]0.900~\text{kW/m}^2[/latex]?
3. If the pipe has a diameter of [latex]2.00~\text{cm}[/latex], what is the ideal temperature increase of the mineral oil in one meter of pipe during [latex]1.00~\text{min}[/latex]? Assume that all the solar energy incident on the reflector is transferred to the oil.

Strategy

This example combines several areas of physics:

• Part (a) uses the relationship between the focal length and radius of curvature of a concave mirror.
• Part (b) uses geometry and the definition of intensity as power per unit area.
• Part (c) uses density, thermal energy, and specific heat capacity.

Solution for (a)

For a spherical mirror, the focal length is related to the radius of curvature by

[latex]f=\frac{R}{2}[/latex]

Solving for [latex]R[/latex] gives

[latex]R=2f=2(40.0~\text{cm})=80.0~\text{cm}[/latex]

Solution for (b)

The incident solar intensity is

[latex]I=0.900~\text{kW/m}^2=900~\text{W/m}^2[/latex]

The reflector is modeled as one-quarter of a cylindrical surface. For a reflector length [latex]L[/latex], its area is

[latex]A=\frac{1}{4}(2\pi R)L[/latex]

For a length of [latex]1.00~\text{m}[/latex] and radius [latex]R=0.800~\text{m}[/latex],

[latex]A = \frac{\pi}{2}(0.800~\text{m})(1.00~\text{m}) = 1.26~\text{m}^2[/latex]

The solar power intercepted by this section of reflector is

[latex]P=IA = (900~\text{W/m}^2)(1.26~\text{m}^2) = 1.13\times10^3~\text{W}[/latex]

Thus, approximately [latex]1.13~\text{kW}[/latex] of solar power is concentrated onto each meter of pipe.

Solution for (c)

The thermal energy absorbed during [latex]60.0~\text{s}[/latex] is

[latex]Q=Pt = (1.13\times10^3~\text{W})(60.0~\text{s}) = 6.78\times10^4~\text{J}[/latex]

The volume of oil in one meter of pipe is

[latex]V = \pi\left(\frac{d}{2}\right)^2L = \pi(0.0100~\text{m})^2(1.00~\text{m}) = 3.14\times10^{-4}~\text{m}^3[/latex]

Using a mineral-oil density of [latex]\rho=8.00\times10^2~\text{kg/m}^3[/latex], the mass is

[latex]m = \rho V = (8.00\times10^2~\text{kg/m}^3) (3.14\times10^{-4}~\text{m}^3) = 0.251~\text{kg}[/latex]

The temperature increase is found from

[latex]Q=mc\Delta T[/latex]

Solving for [latex]\Delta T[/latex] and using [latex]c=1670~\text{J/(kg}\cdot^\circ\text{C)}[/latex],

[latex]\Delta T = \frac{Q}{mc} = \frac{6.78\times10^4~\text{J}} {(0.251~\text{kg})(1670~\text{J/(kg}\cdot^\circ\text{C)})} = 162^\circ\ \text{C}[/latex]

Discussion

In this idealized calculation, the oil temperature rises by [latex]162^\circ\ \text{C}[/latex] in one minute. The result is intentionally an upper limit because it assumes that all incident solar energy is absorbed by the oil. A real system loses energy through reflection, imperfect absorption, thermal radiation, conduction, convection, and fluid flow.

This example illustrates why concentrating mirrors are useful: they collect solar energy over a large area and direct it toward a much smaller receiver. The same optical principle is used in solar furnaces, some medical illumination systems, reflecting telescopes, and other devices that require electromagnetic radiation to be concentrated at a focal point.

When an object is placed closer to a concave mirror than the mirror’s focal length, the reflected rays diverge rather than meeting in front of the mirror. However, when the reflected rays are extended backward, they appear to originate from a point behind the mirror. The image is therefore virtual and cannot be projected onto a screen.

This situation is analogous to the image formed by a converging lens when the object is inside the focal length. Since [latex]f>d_o[/latex] and the focal length is positive, the concave mirror produces an image that is upright and larger than the object. This is the principle used by makeup mirrors, shaving mirrors, and dental mirrors when they are positioned close to the face, skin, or teeth.

A convex mirror behaves differently. Because it causes reflected rays to spread apart, it is a diverging mirror and has a negative focal length. A convex mirror always forms the same type of image for a real object: the image is virtual, upright, and smaller than the object.

As shown in Figure 70.7, the reflected rays do not meet in front of the mirror. Their backward extensions appear to originate from a point behind the mirror, locating the virtual image. Because the image is reduced, the mirror can display a much wider field of view than a flat mirror of the same size.

Convex mirrors are commonly used in vehicle side mirrors, hospital corridors, parking structures, laboratories, and stores. Their wide field of view helps observers see around corners and monitor large areas. The tradeoff is that objects appear smaller and therefore seem farther away than they actually are.

The three main image types formed by mirrors are directly analogous to those formed by lenses:

• A concave mirror with the object beyond the focal point forms a real, inverted image.
• A concave mirror with the object inside the focal length forms a virtual, upright, enlarged image.
• A convex mirror always forms a virtual, upright, reduced image.

A useful way to remember the correspondence is that concave mirrors behave like converging lenses, while convex mirrors behave like diverging lenses.

Conceptual Questions

19. Describe the differences between real and virtual images. How can you determine, simply by observing an image produced by a single lens or mirror, whether it is real or virtual?
20. Can you see a virtual image? Can you photograph one? Can a virtual image be projected onto a screen using additional optical elements? Explain your reasoning.
21. Does a real image need to be projected onto a screen in order to exist? Explain.
22. An image is always located at which distance: the object distance [latex]d_o[/latex], the image distance [latex]d_i[/latex], or the focal length [latex]f[/latex]?
23. Under what conditions will the image formed by a lens or mirror be located exactly at the focal point?
24. What does a negative magnification indicate? What does it mean physically when the magnitude of the magnification is less than one?
25. A Case 1 image always has a negative magnification. Can such an image still be larger than the object? Explain.
26. Figure 70.8 shows a light bulb positioned between two mirrors. One mirror produces a nearly parallel beam of light, while the second mirror redirects light that would otherwise escape. Where must the filament be located relative to the focal point (or center of curvature) of each mirror?

27. Design an arrangement of mirrors that allows you to see the back of your head. What is the minimum number of mirrors required?
28. If you want to see your entire body in a flat mirror, how tall must the mirror be? Does the required mirror size depend on how far you stand from it? Support your answer with a sketch.
29. A flat mirror can be thought of as having an infinite focal length. If so, where does it form an image? How are the image distance [latex]d_i[/latex] and object distance [latex]d_o[/latex] related?
30. Why are convex mirrors commonly used as vehicle side mirrors and security mirrors? What is the principal disadvantage of using a convex mirror instead of a flat mirror?