Modelling a mirror

What the Activity is for

Making a mirror with string.

This is set up as a bench top experiment, but really it's a piece of practical mathematics. If run well, it can drive home the function of do like me, but later, trip time, and therefore geometry in the design and explanation of mirror action.

What to Prepare

• several long trip time rulers, marked out in nanoseconds, foldable
• a large piece of paper, suited to the scale of the rulers
• a demonstration parabolic mirror, perhaps in a smoke box

What Happens During this Activity

Start by marking out a set of points along one side of the paper as the start of a set of paths, representing a beam. Make the point that the vibrations in the beam crossing that edge of the paper could all be in step. Somewhere towards the centre of the paper, mark a detector. Now the challenge is to design a mirror so that all the contributions reach the detector still going up and down together – that is, in step.

Suppose you have a plain mirror at the back of the paper. The path from the centre of the paper, passing by the detector, bouncing off the mirror, and back to the detector, will cover the shortest distance. The path from the edge of the paper, passing straight back to the mirror, bouncing off the mirror, and then angling down to the detector will travel the greatest distance.

Start by developing the outlines of the argument.

Paths between these two extremes will travel distances somewhere between these two values. Since the light is travelling through air at all times, the trip times will depend only on the distances. The speed is constant. So the only way to make the trip times equal is to reduce the distances of the angled paths, coming from the extremities. One way to do this is to move the mirror at these extremities closer to the initial points, defining the start of the beam. The farther away from the centre line of the mirror – that is the line joining the mirror and the detector – the greater this shortening will need to be.

Now produce the rulers. Start with one in the middle, placing one end where the beam starts and folding the ruler where the path bounces off the mirror. Read off the trip time.

Use a new ruler, starting at another marked point farther away from the centre line. You know how long the trip time has to be – the same as the first one – so bend the ruler at an appropriate point. Repeat for all the remaining points that define the paths to be explored. Join the points where the rulers bend, and you'll have a parabolic mirror.

To go with this you'll want a good mirror demonstration, such as that provided by a smoke box, which you can find on practicalphysics.org. You might also have access to mirrors for 3 cm waves or for infrared: these can be impressive and emphatically make the point that the mechanism works across many different frequencies.