Rays through C and at the pole

The idea

A ray through (or directed towards) the centre of curvature falls along the normal and is reflected back along the same path; a ray striking the pole obliquely is reflected making an equal angle with the principal axis.

A ball and a wall

Imagine a ball thrown straight at a wall, along the line that is at right angles to the wall. Now think of a ray of light that falls on a mirror along the line that is at right angles to the mirror.

What do you expect the ray to do after it is reflected?

• Come straight back along the same path
• Go on parallel to the mirror
• Pass through the focus F

Two more rays

The green ray goes towards the centre of curvature C. The purple ray hits the pole P. Open every tab.

Along the normal

A radius of a sphere meets the surface at a right angle. So a ray along a radius, that is through C, falls along the normal and returns on itself. At the pole, the axis is the normal, so the angles on both sides of it are equal.

A ray through the centre of curvature (or directed towards it) falls along the normal, so it is reflected back along the same path. A ray striking the pole obliquely is reflected making an equal angle with the principal axis.

Notes

A ray through C (or towards C) falls along the normal and returns on its own path. A ray at the pole P is reflected at an equal angle with the principal axis.

Check yourself

A ray passes through the centre of curvature C of a concave mirror and falls on the mirror. After reflection it

Why does a ray through C come back along the same path?

A ray strikes the pole of a mirror, making 25° with the principal axis. What angle does the reflected ray make with the principal axis, in degrees?

Answer: 25 degrees

The principal axis is the normal at the pole, and the angle of reflection equals the angle of incidence, so both angles with the axis are 25°.

Which line is the normal to the mirror at its pole?

• passes through the focus F. A ray that passes through F after reflection is one that was parallel to the axis before it.
• goes back along the same path — correct. Yes!
• comes out parallel to the principal axis. That happens to a ray that passes through F before reflection, not to a ray through C.
• The ray is parallel to the principal axis. A ray through C is generally not parallel to the axis. It returns because it falls along the normal.
• The mirror surface is flat at that point. The surface is curved. The ray returns because it falls along the normal to the surface there.
• The ray falls on the mirror along the normal to the surface — correct. Yes!
• The principal axis — correct. Yes!
• The line along the surface of the mirror at P. The normal is at right angles to the surface. At the pole that line is the principal axis.
• The line joining the object to its image. That line has nothing to do with the normal. The normal at the pole is the principal axis.