doPhysics · Optics
Reflection: Plane and Spherical Mirrors
Light travels in rays. The laws of reflection, the image in a plane mirror, turned mirrors and turned rays, many images, moving images, spherical mirrors, the Cartesian sign convention, the mirror formula and magnification, and the speed of an image.
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Why this chapter
Why this chapter exists
A mirror is the simplest optical instrument, and it already raises every question of optics. Where is the image, how large is it, which way up, and how does it move? This chapter treats light as rays that travel in straight lines and obey two laws at every surface. Those laws explain the image in a plane mirror, the turning of rays by turned mirrors, and the corridor of images in a barber's shop. Spherical mirrors follow, with one sign convention and one formula for every case. The same method carries through the chapters on refraction and lenses.
You will learn
By the end of this chapter you will
- State the laws of reflection and locate the image in a plane mirror by drawing and by reasoning.
- Find the deviation of a ray by one mirror, by a turned mirror and by two inclined mirrors.
- Find the least mirror for a full-length image, count the images in inclined mirrors, and find the velocity of an image in a plane mirror.
- Use the Cartesian sign convention and the mirror formula for concave and convex mirrors, with ray diagrams as a check.
- Find the linear, areal and longitudinal magnification, and read the graphs of v and m against u.
- Find the velocity of an image in a spherical mirror, along and across the axis.
Before you begin
Before you begin
You should know that light is a wave, and the words wavelength and frequency from the unit on Waves. You should be comfortable with similar triangles, the angles of a triangle, and the sine and cosine of an angle. For the last lesson you need to differentiate a function of time, as in the chapter Motion in One Dimension. Vectors in two dimensions help in two examples.
Lessons
1
Rays, Reflection and the Plane Mirror
Light travels in straight lines when openings are large compared with its wavelength. Rays and beams, real and virtual images, the laws of reflection, and the image in a plane mirror.
Light travels in raysReal and virtual objects and imagesLaws of reflectionImage in a plane mirror
2
Turning Mirrors and Turned Rays
The deviation of a ray by a plane mirror, \(180° - 2i\). A mirror turned through \(\theta\) turns the reflected ray through \(2\theta\). Two mirrors at an angle \(\theta\) turn a ray through \(360° - 2\theta\), whatever the angle of incidence.
Deviation by one mirrorA turned mirror turns the ray twiceDeviation by two mirrors
3
Many Images and Moving Images
The least mirror for a full-length image, the images formed by two mirrors at an angle and by parallel mirrors, and the velocity of an image in a plane mirror.
Least mirror for a full length imageImages in inclined mirrorsVelocity of an image in a plane mirror
4
Spherical Mirrors and the Sign Convention
Concave and convex mirrors and their pole, centre of curvature and focus. Why only rays near the axis meet at F. The Cartesian sign convention, stated once in full, and the four rays of a ray diagram.
Concave and convex mirrorsParaxial rays and the focusThe sign conventionRay diagrams for spherical mirrors
5
The Mirror Formula and Magnification
The mirror formula \(1/v + 1/u = 1/f\) from similar triangles, the linear magnification \(m = -v/u\), Newton's formula \(xy = f^2\), areal and longitudinal magnification, and the graphs of \(v\) and \(m\) against \(u\).
The mirror formulaLinear magnification of a mirrorNewtons formula and areal magnificationGraphs of v and m against u
6
Moving Objects, Moving Images
The velocity of an image in a spherical mirror, along the axis (\(v_I = -m^2 v_O\)) and across it (\(v_I = m v_O\)). Why an image races away near F, and why the car in a scooter's mirror is closer than it appears.
Image velocity along the axisImage velocity across the axisThe scooter rear view mirror
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- angle of incidence
- The angle \(i\) between the incident ray and the normal at the point of incidence.
- angle of reflection
- The angle \(r\) between the reflected ray and the normal at the point of incidence. By the law of reflection, \(r = i\).
- aperture
- The width of the reflecting surface of a mirror, across its face.
- areal magnification
- The ratio of the area of the image of a small flat object to its area, equal to \(m^2\).
- beam
- A bundle of rays: parallel, diverging from a point, or converging towards a point.
- Cartesian sign convention
- The NCERT's rule for signed distances, used in this book. Distances are measured from the pole, positive along the incident light and negative against it; heights above the axis are positive.
- centre of curvature
- The centre C of the sphere of which a spherical mirror is a part.
- corner reflector
- Three plane mirrors at right angles to one another, which send every ray back parallel to its incoming direction. Bicycle reflectors and road studs are made of tiny ones.
- deviation
- The angle between the direction a ray would have had without the mirror and the direction it has after reflection. For one plane mirror, \(\delta = 180° - 2i\).
- diffuse reflection
- Reflection from a rough surface, such as paper, which scatters a parallel beam in all directions. Each tiny patch still obeys \(r = i\).
- focal length
- The distance \(f\) from the pole to the principal focus, \(f = R/2\) for paraxial rays; negative for a concave mirror and positive for a convex one.
- geometrical optics
- The study of light as rays that travel in straight lines and obey the laws of reflection and refraction. It works when every opening is much wider than the wavelength of light.
- glancing angle
- The angle \(g = 90° - i\) between a ray and the surface of a mirror. One plane mirror turns a ray through \(2g\).
- lateral inversion
- The swap of left and right in the image of a plane mirror, which comes from the reversal of front and back along the normal.
- linear magnification
- The ratio \(m = h'/h\) of the height of the image to the height of the object, with signs. For a mirror \(m = -v/u\).
- longitudinal magnification
- The ratio of the length of the image of a short object lying along the axis to its length, \(dv/du = -m^2\).
- marginal rays
- Rays that meet a mirror far from its principal axis, near the edge of a wide aperture. They cross the axis nearer the mirror than F.
- Newton's formula
- The mirror formula written with distances measured from the focus: \(xy = f^2\).
- normal
- The line perpendicular to a surface at the point where a ray meets it. Angles of incidence and reflection are measured from the normal.
- paraxial rays
- Rays close to the principal axis and at small angles to it. The focal length and the mirror formula hold for them.
- pinhole camera
- A closed box with a tiny hole in one face. The hole passes one straight ray from each point of an object, so an inverted image forms on the opposite face.
- plane of incidence
- The plane that contains the incident ray and the normal. The reflected ray lies in it too.
- pole
- The centre P of the reflecting surface of a spherical mirror, the origin of the sign convention.
- principal axis
- The straight line through the pole and the centre of curvature of a spherical mirror.
- principal focus
- The point F where paraxial rays parallel to the principal axis meet after reflection from a concave mirror. For a convex mirror, the reflected rays seem to come from F.
- radius of curvature
- The radius \(R\) of the sphere of which a spherical mirror is a part. On the Cartesian convention it is negative for a concave mirror and positive for a convex one.
- real image
- An image formed where rays actually meet after reflection or refraction. It can be caught on a screen.
- real object
- A point from which rays actually diverge, such as a lamp or a lit face.
- regular reflection
- Reflection from a smooth surface, such as a mirror, which keeps a parallel beam parallel.
- spherical aberration
- The failure of a wide spherical mirror to bring a parallel beam to one point. Marginal rays cross the axis nearer the mirror than paraxial rays.
- spherical mirror
- A mirror whose surface is part of a sphere: concave if it reflects on the hollow side, convex if on the bulging side.
- virtual image
- An image formed where rays only seem to meet, at the meeting point of their backward extensions. It cannot be caught on a screen, but it can be seen.
- virtual object
- The point towards which a converging beam is heading when a mirror or a lens intercepts it before its rays meet.