Mirrors

Mirrors, unlike lenses, are not transparent materials, but instead are polished surfaces that reflect incoming light rays. Mirrors can be plane (flat) or spherical (curved).

All mirrors obey the laws of reflection:

1. The incident ray, the reflected ray and the normal at the point of incidence all lie on the same plane.

2. The angle of incidence is always equal to the angle of reflection.

Spherical mirrors are classified as concave when the reflecting surface is curved inward and convex when the reflecting surface is curved outward.

The ray diagram shows the image formation, A', of an object placed in front of a plane mirror, A. ^[1]

Plane mirrors are flat mirrors. Most common mirrors, such as those in a restroom or dressing room, are plane mirrors.

The rules of plane mirrors are straight-forward and only explicitly stated so they may be modified for spherical mirrors.

1. The image formed has the same size as the object.

2. The image is placed at the same distance behind the mirror as the distance between the object and the mirror.

3. The image formed is virtual and erect.

Faruq is standing in the middle of a $6\text{-m-}$wide room with mirrors on opposite walls. How far away does the reflection of the back of his head appear to be?

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Since Faruq is in the middle of the room, he is $3\text{ m}$ away from each mirror. In the mirror he is looking into, he will see a reflection of his face $3\text{ m}$ behind the mirror, so the image is apparently $6\text{ m}$ away from him.

$6\text{ m}$ behind the mirror Faruq is looking into, he sees the mirror that is behind him, and $3\text{ m}$ into that mirror is the reflection of the back of his head.

So a photon from the back of his head travels $3\text{ m}$ to the back mirror, then $6 \text{ m}$ to the front mirror, and finally $3\text{ m}$ to his eye.

$$\text{image distance} = 3+6+3 = 12\text{ m}$$

Guiding rays are guidelines that help demonstrate the behavior of light as it interacts with a mirror. There are three main rays.

Ray parallel to principal axis:

If a ray of light is parallel to the principal axis, it passes through the focal point after reflecting (in a concave mirror), or reflects away from the focal point (in a convex mirror).

Ray passing through focus:

Any ray which passes through the focal point of the mirror prior to reflection will travel parallel to the principal axis after reflection. With a convex mirror, the light ray was traveling toward the focal point before reflection.

Ray passing through center of curvature:

In the above images, the ray of light passes through the center of curvature in both the type of mirrors. It does not deviate, but instead reflects back along the path it was on initially.

To locate an image, simply draw the guiding rays and determine their intersection.

The following rules govern the sign on heights and distances used in mathematically solving mirror problems.

• Heights above the principal axis are positive and those below the principal axis are negative.

• Distances to the left of the mirror are positive and those to the right of the mirror are negative.

• The focal length of a convex mirror is negative and that of a concave mirror is positive.

• The object is always placed to the left of the mirror, and the object distance is always positive (for single mirror systems).

The distance of an object from the pole of a mirror is known as the object distance. Object distance is denoted by the letter $p.$The distance of an image from the pole of a mirror is known as the image distance. Image distance is denoted by letter $q.$ The distance of the focal point from the pole of a mirror is known as the focal length. Focal length is denoted by letter $f.$

There is a relationship between object distance, image distance, and focal length of spherical mirror (concave mirror or convex mirror).This relationship is given by the mirror formula.

$$\dfrac 1p + \dfrac 1q = \dfrac 1f$$

$\text{From the above diagram: } PF = -f\\ PB = -u\\ PB' = -v\\ \text{As } \Delta ABC \sim \Delta A'B'C\\ \dfrac{A'B'}{AB} = \dfrac{B'C}{BC} \left[\text{Equation }I \right]\\ \text{Similarly } \Delta A'B'F \sim \Delta FMP\\ \dfrac{A'B'}{MP} = \dfrac{B'F}{FP}\\ \text{In case of large concave mirror, } MP = AB\\ \dfrac{A'B'}{AB} = \dfrac{B'F}{FP} \left[\text{Equation }II \right]\\ \text{From Equation I \& II}\\ \dfrac{B'C}{BC} = \dfrac{B'F}{PF}\\ \implies \dfrac{PC - PB'}{PB - PC} = \dfrac{PB' - PF}{PF}\\ \implies \dfrac{-2f - (-v)}{-u - (-2f)} = \dfrac{-v -(-f)}{-f}\\ \implies \dfrac{-2f+v}{-u+2f} = \dfrac{v-f}{f}\\ \implies -2f^2 + vf = -uv + uf + 2fv - 2f^2 \\ \implies uv = uf + vf\\ \text{Divinding by }uvf\\ \implies \dfrac{uv}{uvf} = \dfrac{uf}{uvf} + \dfrac{vf}{uvf} \\ \implies \dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}$.

Object at $\infty$:

Image formation when the object is placed at infinity

When the object is placed at $\infty$, the image formed is not formed by the actual intersection of rays, but imaginary rays, i.e. the image is virtual and erect. The image is formed at the focal point, and is the size of a point.

Object anywhere on the principal axis:

Image formation when the object is placed anywhere on the principal axis

In this case, the object is placed anywhere on the principal axis, which results in the formation of an image only between the focal point and the mirror. The image formed is also a virtual image, erect and diminished in size.

$$\begin{array} {c|c|c|c} \text{Position of the Object} & \text{Position of the Image} & \text{Size of the Image} & \text{Nature of the image}\\ \hline \text{at } \infty & \text{at F} & \text{Point-sized} & \text{Virtual and Erect}\\ \hline \text{Anywhere on the principal axis} & \text{Between F} \ce{and P} & \text{Diminished} & \text{Virtual and Erect}\\ \hline \end{array}$$

An object is placed $10\text{ cm}$ in front of a mirror of focal length $-5\text{ cm}.$ Where does the image show up?

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The object distance is given as $p = 10$ and the focal length is given as $f = -5.$

$$\dfrac 1p + \dfrac 1q = \dfrac 1f$$

$$\frac{1}{10} + \dfrac 1q = \frac{1}{-5}$$

$$\dfrac 1q = \frac{-3}{10}$$

$$q = -3.33\text{ cm}$$

So the image is virtual and appears $3.33 \text{ cm}$ behind the mirror.

Object at $\infty$:

Image formation when the object is at infinity.

When the object is nearly infinitely far away, the incident rays are parallel. Thus, the image is formed at the focal length, is highly diminished, and is real and inverted.

Object beyond $\ce{C}$:

Image formation when the object is placed beyond the center of curvature.

If the object is placed at $\ce{C},$ the image is formed between $\ce{F}$ and $\ce{C}.$ The image is smaller and it is real and inverted.

Object at $\ce{C}$:

Image formation when the object is placed at the center of curvature.

When the object is located at $\ce{C}$, the image is formed exactly at $\ce{C}.$ It is of the same size as the object and again, real and inverted.

Object between $\ce{C}$, and $\ce{F}$:

Image formation when the object is placed between the center of curvature and the focus.

If the object is placed between $\ce{C}$ and $\ce{F},$ the image is formed beyond $\ce{C},$ but this time it is magnified, real, and inverted.

Image formation when the object is placed at the focus.

Object between $\ce{F}$, and $\ce{P}$:

Image formation when the object is placed between the focus and the pole.

If the object is between $\ce{F}$ and $\ce{P},$ the image is formed on the same side. It is magnified and virtual and erect.

$$\begin{array} {c|c|c|c} \text{Position of the Object} & \text{Position of the Image} & \text{Size of the Image} & \text{Nature of the image}\\ \hline \text{At infinity} & \text{At Focus} & \text{Highly Diminished} & \text{Real and Inverted}\\ \hline \text{Beyond 2F}_1 & \text{Between }\ce{2F2}\text{ and }\ce{F2} & \text{Diminished} & \text{Real and Inverted}\\ \hline \text{At 2F}_1 & \text{At 2F}_2 & \text{Same size} & \text{Real and Inverted}\\ \hline \text{Between }\ce{2F1}\text{ and }\ce{F1} & \text{Beyond 2F}_2 & \text{Magnified} & \text{Real and Inverted}\\ \hline \text{At F}_1 & \text{At infinity} & \text{Highly magnified} & \text{Real and Inverted}\\ \hline \text{Between }\ce{F1}\text{ and }\ce{O} & \text{On the same side as the object} & \text{Magnified} & \text{Virtual and Erect}\\ \hline \end{array}$$

An object is placed $3\text{ cm}$ in front of a concave mirror of focal length $1\text{ cm}.$ Describe the image.

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Since the focal length is $1\text{ cm},$ the center of curvature is at $2\text{ cm}.$ Hence, the object is located beyond the center of curvature.

The $\text{"Beyond 2F"}$ case means the image is diminished, real, and inverted.

1. Image from wikipedia.org under the creative commons licensing for reuse and modification, W. Mirror. Retrieved from https://upload.wikimedia.org/wikipedia/commons/5/5f/Plane_mirror.png