Reflection and Images in Plane Mirrors
You brush your teeth and see yourself standing in the bathroom mirror. But the wall is only a few inches behind the glass. So where is that "other you" really, and why does it seem to be on the far side of the mirror? By the end of this page you will be able to explain it with a ray diagram.
The law of reflection
Light travels in straight lines called rays. When a ray hits a smooth, shiny surface such as a mirror, it bounces off. To describe the bounce, we draw a dashed line perpendicular to the mirror at the point where the ray lands. That line is the normal.
Why does nature do this? For a smooth surface, the bounce is symmetric. The surface pushes the ray back as if it were a ball hitting a wall, and the ball leaves at the same angle it came in.
A ray of light strikes a flat bathroom mirror. The angle between the incident ray and the normal (the line perpendicular to the mirror) is $35^\circ$. State the law of…
A ray of light strikes a flat bathroom mirror. The angle between the incident ray and the normal (the line perpendicular to the mirror) is $35^\circ$. State the law of reflection and give the angle of reflection measured from the normal.
A laser pointer on a lab table sends a ray toward a horizontal plane mirror lying flat on the table. The ray makes an angle of $30^\circ$ with the surface of the mirror…
A laser pointer on a lab table sends a ray toward a horizontal plane mirror lying flat on the table. The ray makes an angle of $30^\circ$ with the surface of the mirror (not with the normal). Find the angle of incidence, the angle of reflection, and the angle between the incident ray and the reflected ray.
The image in a plane mirror
A plane mirror is a flat mirror. When you look into one, light from your face bounces off the glass and reaches your eyes. Your brain assumes light always travels in straight lines. So it traces the rays straight back and "sees" a copy of you behind the mirror. That copy is the image.
Three facts describe every plane-mirror image:
The image is on a line perpendicular to the mirror, so the distance between you and your image is $d_o + d_i = 2d_o$.
Sofia stands $2.0\ \text{m}$ in front of a plane mirror in her school's dance studio. How far behind the mirror does her image appear to be, and how far is she from her…
Sofia stands $2.0\ \text{m}$ in front of a plane mirror in her school's dance studio. How far behind the mirror does her image appear to be, and how far is she from her image?
Classify the image formed by a plane mirror by answering each question: Is it real or virtual? Is it upright or inverted? Is it larger than, smaller than, or the same…
Classify the image formed by a plane mirror by answering each question: Is it real or virtual? Is it upright or inverted? Is it larger than, smaller than, or the same size as the object?
Marcus is $1.80\ \text{m}$ tall and stands $3.0\ \text{m}$ from a vertical plane mirror. He walks toward the mirror at a constant speed of $1.2\ \text{m/s}$ for…
Marcus is $1.80\ \text{m}$ tall and stands $3.0\ \text{m}$ from a vertical plane mirror. He walks toward the mirror at a constant speed of $1.2\ \text{m/s}$ for $1.5\ \text{s}$. (a) How far is he from the mirror at the end? (b) How far is he from his image at the end? (c) At what speed does his image approach him while he walks?
Locating an image with ray diagrams
You can prove where the image is with the law of reflection. Choose a point on the object, such as the tip of an arrow. Draw two rays leaving that point and hitting the mirror. Reflect each with $\theta_r = \theta_i$. Then extend the reflected rays backward behind the mirror as dashed lines. The place where the dashed lines cross is the image of that point.
Because every ray obeys the law of reflection, the geometry always gives the same result. The image tip sits on the perpendicular from the object tip to the mirror, as far behind as the object is in front. So you can skip the rays once you trust the pattern: drop a perpendicular to the mirror and go the same distance beyond it. Repeat for the bottom of the object, then join the points.
Aaliyah holds a small arrow-shaped object $4.0\ \text{cm}$ tall, pointing up, $10\ \text{cm}$ in front of a plane mirror. Describe how to locate the image of the arrow…
Aaliyah holds a small arrow-shaped object $4.0\ \text{cm}$ tall, pointing up, $10\ \text{cm}$ in front of a plane mirror. Describe how to locate the image of the arrow tip with two rays drawn from the tip. Then state the image's height, its distance behind the mirror, and the direction it points.
How tall must a mirror be?
Do you need a mirror as tall as you are to see your whole body? Let's find out with the law of reflection.
Look at the figure. Your eyes are at the point marked "eyes." To see your feet, a ray from your feet must hit the mirror and reflect into your eyes. The law of reflection puts the spot on the mirror exactly halfway between the height of your eyes and the height of your feet. To see the top of your head, the spot is halfway between your eyes and the top of your head.
So the mirror only needs to span from the halfway point for the head to the halfway point for the feet. Those two points are separated by half of the full height, so:
Ethan is $1.70\ \text{m}$ tall, with his eyes $1.60\ \text{m}$ above the floor. He wants to see his whole body in a vertical plane mirror hung on a wall. (a) Using the…
Ethan is $1.70\ \text{m}$ tall, with his eyes $1.60\ \text{m}$ above the floor. He wants to see his whole body in a vertical plane mirror hung on a wall. (a) Using the law of reflection and ray diagrams, find the minimum height of the mirror. (b) Find how high above the floor the bottom edge of the mirror must be. (c) Ethan then steps farther back from the mirror. Does the minimum mirror height change? Justify your answer.
Two mirrors at a right angle
Put two plane mirrors together in a corner and you see more than one copy of yourself. This is how the images come out. Each mirror forms an image of the object. Then each of those images can itself act as an object for the other mirror.
On a coordinate grid, take the corner at the origin, one mirror along the positive $x$-axis and the other along the positive $y$-axis. A plane mirror reflects a point to the opposite side of the mirror line, with the same distance from it.
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Reflect in the $x$-axis mirror: $y$ changes sign, $x$ stays. $(x, y) \to (x, -y)$.
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Reflect in the $y$-axis mirror: $x$ changes sign, $y$ stays. $(x, y) \to (-x, y)$.
The third image comes from reflecting twice, once in each mirror (in either order). That flips both signs: $(x, y) \to (-x, -y)$. It is the same as rotating the object $180^\circ$ about the corner, which explains why this image does not look "mirrored" as the others do.
Back to the opening question: the "other you" behind the bathroom mirror is a virtual image. Light bounces off the mirror obeying $\theta_i = \theta_r$, and your brain traces the reflected rays straight back to where they seem to cross, a distance behind the mirror equal to your distance in front.
Two plane mirrors meet at a right angle ($90^\circ$) to form a corner. A small object sits inside the corner, $0.20\ \text{m}$ from one mirror and $0.30\ \text{m}$ from…
Two plane mirrors meet at a right angle ($90^\circ$) to form a corner. A small object sits inside the corner, $0.20\ \text{m}$ from one mirror and $0.30\ \text{m}$ from the other. Treat the mirrors as meeting along a vertical line and look at the setup from above. Using coordinates with the corner at the origin, place the mirrors along the positive $x$-axis and positive $y$-axis, so the object is at $(0.30, 0.20)$ in meters. (a) Find the coordinates of the two images formed by a single reflection. (b) Find the coordinates of the third image, formed by reflection in both mirrors. (c) Explain with a ray diagram why a person looking into the corner sees three images.