34-2 Spherical Mirrors#

Prompts

Distinguish concave and convex spherical mirrors. Where is the center of curvature? What are the signs of \(r\) and \(f\)?
Sketch how parallel rays form the focal point for (a) concave and (b) convex mirrors. Which has a real focus and which a virtual focus?
For a concave mirror: object inside vs outside the focal point—what type of image (real or virtual) and what orientation?
For a convex mirror: what type of image can it form? Where is it located?
State the mirror equation and the lateral magnification formula. What sign convention applies to \(i\) and \(m\)?
Name two principal rays used to locate an image in a ray diagram for a spherical mirror.

Lecture Notes#

Overview#

Spherical mirrors: small section of a sphere—concave (caved in, \(r > 0\)) or convex (flexed out, \(r < 0\)).
Focal point \(F\): parallel rays (near the central axis) reflect to a common point (concave: real) or appear to diverge from it (convex: virtual).
Focal length \(f = r/2\); \(f > 0\) for concave, \(f < 0\) for convex.
Concave: real image if object outside \(F\); virtual if inside \(F\). Convex: always virtual image.
Mirror equation and magnification relate \(p\), \(i\), \(f\), and image size.

Concave vs convex mirrors#

Type	Shape	\(r\)	\(f\)	Center of curvature \(C\)
Concave	Caved in	\(> 0\)	\(> 0\)	In front of mirror
Convex	Flexed out	\(< 0\)	\(< 0\)	Behind mirror
Plane	Flat	\(\to \infty\)	\(\to \infty\)	At infinity

Applications: Makeup/shaving mirrors are concave (magnified image). Store surveillance mirrors are convex (wider field of view).

Focal points#

Concave mirror: Parallel rays (parallel to the central axis) reflect through a common point \(F\) in front of the mirror—a real focal point. An image can form on a card placed at \(F\).

Convex mirror: Parallel rays diverge after reflection. Backward extensions of the reflected rays meet at a point \(F\) behind the mirror—a virtual focal point. No real image forms there.

(253)#\[ f = \frac{r}{2} \]

Images from concave mirrors#

Object location	Image type	Image location	Orientation
Inside \(F\) (between mirror and \(F\))	Virtual	Behind mirror	Same as object
At \(F\)	—	At infinity	—
Outside \(F\)	Real	In front of mirror	Inverted

Important

Real images form on the same side of the mirror as the object; virtual images form on the opposite side.

Images from convex mirrors#

A convex mirror forms only virtual images—always behind the mirror, with the same orientation as the object, and smaller. Regardless of object position, \(i < 0\).

Poll: Cosmetic mirror—type and focal length

You see an upright, magnified image of your face in a cosmetic mirror. Which best describes the mirror and its focal length relative to your distance \(s\) from the mirror?

(A) Convex (diverging) mirror, \(|f| > s\)
(B) Convex (diverging) mirror, \(|f| < s\)
(C) Concave (converging) mirror, \(|f| > s\)
(D) Concave (converging) mirror, \(|f| < s\)

Mirror equation and magnification#

For small angles with the central axis (paraxial rays):

(254)#\[ \frac{1}{p} + \frac{1}{i} = \frac{1}{f} = \frac{2}{r} \]

Sign convention: \(p > 0\); \(f > 0\) (concave), \(f < 0\) (convex); \(i > 0\) (real image), \(i < 0\) (virtual image).

Lateral magnification:

(255)#\[ m = -\frac{i}{p} \]

(256)#\[ |m| = \frac{h'}{h} \]

where \(h\) = object height, \(h'\) = image height. \(m > 0\): same orientation; \(m < 0\): inverted.

Ray diagram: principal rays#

To locate an image, draw any two of these rays from an off-axis point:

Parallel to axis \(\to\) reflects through \(F\).
Through \(F\) \(\to\) reflects parallel to axis.
Through center of curvature \(C\) \(\to\) reflects back along itself.
To center \(c\) of mirror \(\to\) reflects symmetrically about the axis.

The image is at the intersection of the two rays (or their extensions). For convex mirrors, rays 2 and 3 use extensions toward \(F\) and \(C\) (both behind the mirror).

Example: Operating room lamp—lamp position

An operating room light uses a concave mirror to focus an image of a bright lamp onto the surgical site. The mirror has radius of curvature 30 cm. The mirror is 1.2 m from the patient. How far should the lamp be from the mirror?

Solution: The patient (surgical site) is at the image location: \(i = 1.2\) m = 120 cm. For a concave mirror, \(f = r/2 = 15\) cm. From \(1/p + 1/i = 1/f\): \(1/p = 1/15 - 1/120 = 7/120\) \(\Rightarrow\) \(p \approx 17\) cm. The lamp should be about 17 cm in front of the mirror.

Summary#

Concave: \(f > 0\), real focus; object outside \(F\) \(\to\) real inverted image; object inside \(F\) \(\to\) virtual upright image.
Convex: \(f < 0\), virtual focus; always virtual, upright, smaller image.
Mirror equation: \(1/p + 1/i = 1/f\); magnification: \(m = -i/p\).
Ray diagram: Use two principal rays to locate the image.