Look into the hollow of a shiny steel spoon held at arm's length. Your face appears small and upside down. Turn the spoon over and look at its back. Your face is now small and the right way up, and a wide view of the room surrounds it.
Mirrors cut from a sphere
A spherical mirror is a small part of the surface of a sphere, polished on one side. If the inner, hollow side reflects, the mirror is concave. If the outer, bulging side reflects, the mirror is convex. The hollow of a steel spoon is a concave mirror, and its back is a convex mirror.
A few names describe every spherical mirror. The centre of the mirror's surface is the pole P. The centre of the sphere of which the mirror is a part is the centre of curvature C. The radius of that sphere is the radius of curvature \(R\). The straight line through P and C is the principal axis. The width of the mirror across its face is the aperture.
Rays parallel to the principal axis and close to it meet at one point after reflection from a concave mirror. That point is the principal focus F. For a convex mirror the reflected rays spread out. They only seem to come from a point F behind the mirror, where their backward extensions meet. The distance PF is the focal length \(f\). In both cases F lies halfway between P and C, so \(f = R/2\) in size.
The principle of reversibility gives a second use of F. A small bulb placed at the focus of a concave mirror sends out rays that leave the mirror parallel. That is how the reflector of a torch or a car's headlight throws a narrow beam far ahead. A solar cooker works the other way round: parallel sunlight gathers at F, where the pot stands.
A beam of light parallel to the principal axis falls on a convex mirror. Where is the principal focus?
Why F is only nearly a point
Take a concave mirror of radius \(R\) and a ray parallel to the axis at a height \(h\). It meets the mirror at M, where the radius CM is the normal. The angle of incidence \(i\) is the angle between the ray and CM, with \(\sin i = h/R\). The reflected ray crosses the axis at a point X. Where X lies depends on \(h\).
\[ \angle MCX = i \]
Why. the ray is parallel to the axis, so the angle between the axis and CM equals the angle between the ray and CM (alternate angles).
\[ \angle CMX = i \]
Why. the reflected ray makes the angle \(r = i\) with the normal CM.
\[ CX = XM = \dfrac{R}{2\cos i} \]
Why. two equal angles make triangle CMX isosceles. Drop a perpendicular from X to CM; it bisects CM, so \(CX\cos i = R/2\).
\[ PX = R - \dfrac{R}{2\cos i} \]
Read it. the pole is a distance \(R\) from C along the axis, and X is \(CX\) from C on the same side.
For a ray close to the axis, \(i\) is small and \(\cos i \approx 1\). Then \(PX \approx R - R/2 = R/2\), the same for every such ray. These rays are paraxial rays, and their common crossing point is F. For a ray far from the axis, \(\cos i\) is smaller and X moves towards the mirror. With \(R = 40\,\text{cm}\) and \(h = 20\,\text{cm}\), \(i = 30°\), and the ray crosses at \(40 - 40/(2 \times 0.866) = 16.9\,\text{cm}\), not at \(20\,\text{cm}\).
So a wide spherical mirror does not bring a parallel beam to a sharp point. The outer rays, the marginal rays, cross nearer the mirror than the paraxial ones, and the focus is smeared into a small patch. This defect is spherical aberration. Everything that follows in this chapter assumes paraxial rays: small apertures and rays close to the axis. The results \(f = R/2\) and the mirror formula hold only there.
A mirror shaped as a paraboloid has no spherical aberration for rays parallel to its axis. Every such ray, however far out, passes through one focus. That is why a large telescope, a satellite dish and a good car headlight use parabolic mirrors. Near its pole a parabola and a sphere of radius \(2f\) are almost the same curve, which is why a small spherical mirror works well.
Two rays parallel to the principal axis fall on a wide concave mirror, one \(2\,\text{cm}\) from the axis and one \(25\,\text{cm}\) from it. The radius is \(40\,\text{cm}\). Which statement is correct?
The telescopes of Newton's time used lenses, and their images were fringed with colour. Newton believed the colours could never be cured in a lens. In 1668 he built a telescope that gathered light with a concave mirror instead, ground by his own hands from a metal alloy. A mirror reflects every colour alike, so the fringes were gone. Nearly every large telescope today is a reflector.
One sign convention for every mirror
A single formula for all mirrors needs distances with signs, measured by one rule. This book uses the Cartesian sign convention of the NCERT. It treats the principal axis as an \(x\) axis with its origin at the pole. It is stated here once, in full, and every formula of this chapter and the next two follows it.
Draw the incident light travelling from left to right. Measure every distance along the principal axis from the pole. A distance in the direction of the incident light, to the right of the pole, is positive. A distance against the incident light, to the left of the pole, is negative. A height above the principal axis is positive, and a height below it is negative.
The rule has quick consequences. A real object stands in front of the mirror, against the light, so \(u\) is always negative. A concave mirror has C and F in front of it: \(R < 0\) and \(f < 0\). A convex mirror has them behind it: \(R > 0\) and \(f > 0\). A real image forms in front of a mirror, so \(v < 0\), while a virtual image forms behind it, so \(v > 0\). An erect image has a positive height and an inverted one a negative height.
Signs go in only once. Write every known quantity with its sign, put the numbers into the formula exactly as they are, and read the sign of the answer. Never add a sign inside the formula as well. Other books use a "real is positive" convention with different formulas; do not mix the two.
A convex mirror of radius of curvature \(40\,\text{cm}\) forms an image of a real object. The light comes from the left. On the Cartesian sign convention, which statements are true?
Four rays that locate an image
An image point is where the rays from an object point meet, or seem to meet. Any two rays are enough. Four rays are easy to draw, because the laws of reflection fix their paths at once. A ray parallel to the axis reflects through F, or seems to come from F for a convex mirror. A ray through F, or aimed at F behind a convex mirror, reflects parallel to the axis. This follows from the first ray by reversibility.
A ray through C, or aimed at C, meets the mirror along a radius, which is the normal. It returns along its own path. A ray to the pole P meets a normal that is the axis itself. It reflects at the same angle on the other side of the axis. In drawings the rays bend at the plane through the pole, as the NCERT draws them, because paraxial rays meet the mirror almost there.
For a concave mirror the image depends on where the object is. Here is the whole list, from far to near.
- Object at infinity: the image is at F, real, inverted, a point.
- Object beyond C: the image is between F and C, real, inverted, smaller.
- Object at C: the image is at C, real, inverted, the same size.
- Object between C and F: the image is beyond C, real, inverted, larger.
- Object at F: the reflected rays are parallel, so the image is at infinity.
- Object between F and P: the image is behind the mirror, virtual, erect, larger.
A convex mirror is simpler. Wherever the real object is, the image is virtual, erect and smaller, and it lies behind the mirror between P and F. The mirror also shows a wide field of view in a small area. That is why a convex mirror is used for the rear-view mirror of a scooter, which is the subject of the last lesson.
An object is placed at the centre of curvature of a concave mirror. Where is the image, and what is it like?
A man uses a concave mirror to shave and sees an enlarged, upright image of his face. Where must his face be?
Now back to the steel spoon. At arm's length your face is beyond C of the hollow side, a small concave mirror. The rays meet in front of the spoon in a real, inverted, smaller image, and you see yourself upside down. The back of the spoon is a convex mirror. It always gives a virtual, erect, smaller image, set in a wide view of the room.
- Giving a concave mirror a positive focal length.With the light from the left, F of a concave mirror is in front of the pole, against the light: \(f < 0\). A convex mirror has \(f > 0\).
- Using \(f = R/2\) for a wide mirror and far-out rays.\(f = R/2\) holds for paraxial rays only. A ray far from the axis crosses at \(R - R/(2\cos i)\), nearer the mirror.
- Expecting a convex mirror to give a real image of a real object.A convex mirror spreads rays out. The image of a real object is always virtual, erect and smaller.