Light – Reflection and Refraction

Laws of reflection

(i) The angle of incidence equals the angle of reflection. (ii) The incident ray, the reflected ray and the normal drawn at the point of incidence lie in one plane. The laws apply to all reflecting surfaces, spherical ones included.

Concave mirror

A spherical mirror whose reflecting surface curves inwards, facing the centre of the sphere. It is used in torches, search-lights and vehicle headlights for powerful parallel beams, by dentists to see enlarged images of teeth, and in solar furnaces to concentrate sunlight.

Convex mirror

A spherical mirror whose reflecting surface curves outwards. It always gives a virtual, erect and diminished image, and its outward curve gives a wider field of view, which is why it is used as a rear-view mirror in vehicles.

Principal focus of a spherical mirror

The point on the principal axis where rays parallel to the axis meet after reflection from a concave mirror, or from which they appear to diverge after reflection from a convex mirror. Its distance from the pole is the focal length, f.

Relation between radius of curvature and focal length of a spherical mirror

R = 2f, for mirrors of small aperture, where R is the radius of curvature and f the focal length. So the principal focus F sits halfway from the pole P to the centre of curvature C.

How do you locate the image formed by a spherical mirror using a ray diagram?

Draw two of these rays from a point on the object: parallel to the axis, reflecting through F (or seeming to come from F); through F, reflecting parallel to the axis; through C, returning along itself; or to the pole, reflecting at an equal angle. Where the reflected rays meet is the image.

Why is a concave mirror used as a shaving mirror?

When the face is between the pole and the principal focus, a concave mirror forms an enlarged, virtual and erect image behind the mirror, so the face appears larger. Placed farther away than the focus, the object gives real, inverted images instead.

New Cartesian sign convention

Distances are measured from the pole of a mirror, or the optical centre of a lens, with the object placed on the left. Distances to the right and heights above the principal axis are positive; those to the left and below are negative. A convex lens thus has a positive focal length and a concave lens a negative one.

Mirror formula

1/v + 1/u = 1/f, where u is the object distance, v the image distance and f the focal length, all measured from the pole. It is valid for all spherical mirrors and all object positions, provided New Cartesian signs are used.

Magnification produced by a spherical mirror

m = h'/h = -v/u, where h is the height of the object, h' the height of the image, u the object distance and v the image distance. A negative value of m means the image is real; a positive value means it is virtual.

How is the mirror formula applied to find the position and size of an image?

Write each quantity with its sign, solve 1/v = 1/f - 1/u, then use m = -v/u and h' = mh. In the chapter's example, a 4.0 cm object at u = -25.0 cm before a concave mirror of f = -15.0 cm gives v = -37.5 cm and h' = -6.0 cm: a real, inverted, enlarged image.

Refraction of light

The change in the direction of light when it travels obliquely from one transparent medium into another. It happens because light travels at different speeds in different media, which is why a pencil in water looks displaced and the bottom of a pond looks raised.

Why is the ray emerging from a rectangular glass slab parallel to the incident ray?

The ray bends towards the normal on entering the glass and away from the normal on leaving it. Because the two faces are parallel, the bending at each face is equal and opposite, so the emergent ray is parallel to the incident ray, though shifted slightly sideways.

Snell's law of refraction

sin i / sin r = constant, where i is the angle of incidence and r the angle of refraction, for light of a given colour and a given pair of media (0 < i < 90°). That constant is called the refractive index of medium 2 relative to medium 1.

Refractive index

n21 = v1/v2, the speed of light in medium 1 divided by the speed in medium 2. With respect to air or vacuum the absolute refractive index is n = c/v, where c is the speed of light in air and v its speed in the medium; water has n = 1.33.

For a given pair of media, does changing the angle of incidence change the speed of light in the second medium?

No. The angle of refraction changes along with the angle of incidence, but sin i / sin r stays constant because it equals the refractive index, which is fixed by the speeds of light in the two media. The speed depends on the medium, not on the angle.

Convex lens

A lens bounded by two spherical surfaces bulging outwards, thicker at the middle than at the edges. It makes rays parallel to its principal axis converge to a principal focus on the other side, so it is also called a converging lens.

Concave lens

A lens bounded by two spherical surfaces curving inwards, thicker at the edges than at the middle. It diverges light rays, and whatever the position of the object it always forms a virtual, erect and diminished image.

Does a convex lens always form a real, inverted image?

No. For object positions beyond the focus F1 the image is real and inverted, but when the object is between F1 and the optical centre, the image is virtual, erect and enlarged, formed on the same side of the lens as the object.

How do you locate the image formed by a lens using a ray diagram?

Draw two of these rays from a point on the object: parallel to the axis, passing through the far focus of a convex lens or seeming to come from the near focus of a concave lens; through or towards a focus, emerging parallel to the axis; or through the optical centre, going straight on.

Lens formula

1/v - 1/u = 1/f, where u is the object distance, v the image distance and f the focal length, all measured from the optical centre. It is valid for any spherical lens; the minus sign distinguishes it from the mirror formula.

Magnification produced by a lens

m = h'/h = v/u, where h and h' are the heights of the object and image and u and v their distances from the optical centre. Unlike the mirror expression it has no minus sign; a negative m still means a real, inverted image.

How is the lens formula applied to find the position and size of an image?

Sign each value, solve 1/v = 1/f + 1/u, then find m = v/u and h' = mh. In the chapter's example, a 2.0 cm object at u = -15 cm from a convex lens with f = +10 cm gives v = +30 cm and h' = -4.0 cm: real, inverted and twice as large.

Power of a lens

P = 1/f, the reciprocal of the focal length. With f in metres, P is in dioptres (D), so 1 D is the power of a lens of focal length 1 m; a convex lens has positive power and a concave lens negative. Lenses in contact add: P = P1 + P2 + P3 + ...

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