Premed · Premed · Physics 2
Lecture 16: Lenses and Mirrors
Physics II — Electromagnetism, Optics & Modern Physics
Learning Objectives
By the end of this lecture, students will be able to:
- Apply the mirror equation and magnification formula for concave and convex mirrors
- Construct ray diagrams to locate images formed by mirrors and thin lenses
- Apply the thin lens equation to converging and diverging lenses
- Distinguish between real and virtual images and calculate magnification
- Analyze systems of multiple lenses
Lecture Content
I. Plane Mirrors
A plane (flat) mirror produces a virtual, upright image that is the same size as the object. The image distance equals the object distance (measured behind the mirror), and the magnification is +1, indicating same size and upright. The image is virtual, meaning that light rays appear to diverge from the image location but do not actually pass through it. A plane mirror also produces left-right reversal, known as lateral inversion.
II. Curved Mirrors
A concave (converging) mirror has a reflecting surface that curves inward, like the inside of a spoon. Its center of curvature C is located at a distance R from the mirror, and its focal point F is at a distance f = R/2, where parallel rays converge after reflection. The focal length is positive for concave mirrors.
A convex (diverging) mirror has a reflecting surface that curves outward. Its focal point is behind the mirror (a virtual focal point), and its focal length is negative. A convex mirror always produces virtual, upright, and diminished images, which is why convex mirrors are used as wide-angle mirrors in vehicles and stores.
The mirror equation relates object distance, image distance, and focal length: 1/d_o + 1/d_i = 1/f = 2/R. The object distance d_o is always positive for real objects. The image distance d_i is positive for real images (formed in front of the mirror) and negative for virtual images (formed behind the mirror). The focal length f is positive for concave mirrors and negative for convex mirrors.
The magnification is m = -d_i / d_o = h_i / h_o. When |m| > 1, the image is enlarged; when |m| < 1, it is diminished. A positive m indicates an upright image, while a negative m indicates an inverted image.
III. Ray Diagrams for Mirrors
Three principal rays are used to locate images in mirrors, though any two are sufficient. The parallel ray arrives parallel to the principal axis and reflects through F (for a concave mirror) or appears to come from F (for a convex mirror). The focal ray passes through F and reflects parallel to the principal axis. The central ray is directed toward C and reflects back on itself because it strikes the mirror at normal incidence. The image is located where reflected rays intersect (real image) or where they appear to diverge from (virtual image).
For a concave mirror, the image properties depend on the object distance. When d_o > 2f, the image is real, inverted, and diminished. At d_o = 2f, the image is real, inverted, and the same size (located at C). When f < d_o < 2f, the image is real, inverted, and enlarged. At d_o = f, the image forms at infinity (reflected rays are parallel). When d_o < f, the image is virtual, upright, and enlarged, which is the configuration used in magnifying and shaving mirrors.
<image>Ray diagrams for a concave mirror showing three cases. Panel A: Object beyond C (d_o > 2f) — three principal rays are drawn from the tip of the object arrow. The parallel ray reflects through F, the focal ray reflects parallel to the axis, and the central ray reflects back through C. They converge to form a real, inverted, diminished image between F and C. Panel B: Object between F and C — the image is real, inverted, and enlarged, formed beyond C. Panel C: Object inside F — reflected rays diverge; extending them behind the mirror shows a virtual, upright, enlarged image. Each panel labels the object (O), image (I), focal point (F), center of curvature (C), and the mirror surface.</image>
IV. Thin Lenses
A thin lens refracts light to form images, with the assumption that its thickness is negligible compared to its focal length. A converging (convex) lens is thicker at the center and causes parallel rays to converge to a focal point; its focal length is positive. A diverging (concave) lens is thinner at the center and causes parallel rays to diverge as if they came from a virtual focal point; its focal length is negative. A diverging lens always produces virtual, upright, and diminished images.
The thin lens equation is identical in form to the mirror equation: 1/d_o + 1/d_i = 1/f. The sign conventions are as follows: d_o is positive for real objects on the incoming side; d_i is positive for real images on the outgoing side and negative for virtual images on the same side as the object; f is positive for converging lenses and negative for diverging lenses. The magnification formula is also the same: m = -d_i / d_o.
The lensmaker's equation connects the focal length to the lens geometry and material: 1/f = (n - 1)(1/R_1 - 1/R_2), where R_1 and R_2 are the radii of curvature of the two surfaces and n is the index of refraction of the lens material.
V. Ray Diagrams for Thin Lenses
For a converging lens, three principal rays locate the image. The parallel ray refracts through the far focal point F'. The focal ray passes through the near focal point F and refracts parallel to the axis. The central ray passes straight through the center of the lens without deviation.
For a diverging lens, the rays follow analogous paths. The parallel ray refracts as though it came from the near focal point F. The focal ray is directed toward the far focal point F' and refracts parallel to the axis. The central ray passes through the center undeviated.
The image properties for a converging lens mirror those of the concave mirror. When d_o > 2f, the image is real, inverted, and diminished. At d_o = 2f, the image is real, inverted, and the same size. When f < d_o < 2f, the image is real, inverted, and enlarged. When d_o < f, the image is virtual, upright, and enlarged, which is the magnifying glass configuration.
VI. Combinations of Lenses
For two thin lenses in contact, the combined focal length satisfies 1/f_total = 1/f_1 + 1/f_2. The lens power P = 1/f is measured in diopters (D = 1/m), and powers add for lenses in contact: P_total = P_1 + P_2.
For lenses separated by a distance d, the image formed by the first lens serves as the object for the second lens. The thin lens equation is applied sequentially, and the total magnification is the product of the individual magnifications: m_total = m_1 x m_2.
When chaining lenses, sign conventions must be tracked carefully. If the image from the first lens falls beyond the second lens, the object distance for the second lens is negative (a virtual object).
<image>A ray diagram showing a two-lens system. Lens 1 (converging, focal length f_1) forms a real, inverted image I_1 of the object O. This intermediate image I_1 serves as the object for Lens 2 (converging, focal length f_2), which forms the final image I_2. Principal rays are traced through both lenses. Object and image distances d_o1, d_i1, d_o2, d_i2 are labeled. The total magnification m_total = m_1 x m_2 is noted. The diagram shows the intermediate image between the two lenses.</image>
