Angles, Normals and the Protractor ലംബം
In this chapter almost every answer depends on an angle: the angle of incidence, the angle of refraction, the critical angle of 42°, the 45° angles inside a cycle reflector. And every one of them is measured from a special line called the normal – never from the surface. This page teaches you what an angle is, how to read a protractor without mistakes, how to draw the normal on a flat or a curved surface, and a few angle facts (90°, 45° + 45°, opposite angles) that the textbook uses without explaining.
- Rays drawn as lines with arrows (page 1).
- A protractor and a ruler from your geometry box, if you want to try the activities on paper.
1. What is an angle?
Two straight lines that start from the same point make an angle. The point is the vertex and the two lines are the arms. The angle tells how much you must turn one arm to bring it onto the other. We measure it in degrees (°). One full turn is 360°, a half turn is 180° and a quarter turn is 90°.
| Name | Size | Example in this chapter |
|---|---|---|
| acute angle | more than 0°, less than 90° | the critical angle of glass, 42° |
| right angle | exactly 90° | the normal and the surface; a refracted ray “along the surface” (r = 90°) |
| obtuse angle | more than 90°, less than 180° | the angle between an incident ray and its reflected ray when i = 60° (120°) |
| straight angle | exactly 180° | a straight line, such as the flat face of a glass slab |
2. Reading a protractor
A protractor has two rows of numbers: one starts with 0 on the right and the other starts with 0 on the left. Most mistakes come from reading the wrong row. Follow three steps:
- Put the centre mark of the protractor exactly on the vertex.
- Turn the protractor so that its base line lies along one arm.
- Start from the 0 on that arm and read along that row to the other arm.
The highlighted number is the correct reading. The other row at the same place gives 180° minus the answer – a very common wrong answer. Ask yourself: “Is my angle smaller or bigger than 90°?” That check catches the mistake at once.
3. Perpendicular lines and the normal
Two lines that meet at a right angle (90°) are perpendicular to each other. We mark the right angle with a small square. In light diagrams we draw a special perpendicular line wherever a ray meets a surface:
The normal is the line drawn perpendicular (at 90°) to the surface at the point where the ray meets it (the point of incidence, പതനബിന്ദു). We draw it as a dashed line on both sides of the surface. In the textbook it is called NN′.
The surface need not be level. A glass face can be slanted, as in a prism. The normal is always at 90° to that face at that point, so it tilts along with the face.
However the surface is tilted and wherever the point is, the normal makes 90° with the surface – look at the little square. To draw it on paper, put one edge of your set square along the surface with its corner at the point, and rule along the other edge.
A mason building a house checks that each wall is perpendicular to the floor with a plumb line (a weight hanging on a string) and a set square. A coconut palm that grows straight up is perpendicular to level ground; one leaning over a canal is not.
4. The normal to a curved surface
How do we draw the normal on a curved surface, like the round side of a semicircular glass slab (page 4)? For a circle there is an easy rule:
At any point on a circle, the radius drawn to that point (the line from the centre) is the normal to the curved surface there. It is perpendicular to the tangent – the straight line that just touches the circle at that point.
A ray aimed at the centre travels along a radius, so it meets the curved face along the normal and does not change direction there. It reaches the flat face exactly at the centre. That is why the textbook's critical-angle experiment (Figure 1.16, page 10 of this guide) uses a semicircular slab: only the flat face bends the ray.
5. Measure from the normal, not from the surface
The normal is at 90° to the surface. So if a ray makes some angle with the surface, the angle it makes with the normal is whatever is left of 90°:
Two angles that add up to 90° are called complementary angles. 30° and 60° are complementary; so are 45° and 45°. The angle of incidence (പതനകോൺ) and the angle of refraction (അപവർത്തനകോൺ) are always the angles with the normal.
A ray that skims along the surface makes 0° with the surface and 90° with the normal. A ray that falls straight down makes 90° with the surface and 0° with the normal: this is the ray “falling perpendicularly” (ലംബമായി പതിക്കുന്ന) in the textbook.
Measuring the angle of incidence or refraction from the surface. If a question says “a ray makes 30° with the surface of the glass”, the angle of incidence is 60°, not 30°.
Worked example 1 · From surface angle to angle of incidence
Q. A ray of light from a laser torch makes an angle of 25° with the surface of the water in a tank. What is its angle of incidence?
Draw the normal at the point of incidence. The angle with the normal and the angle with the surface together make 90°.
Worked example 2 · The 45° inside a right-angled prism
The angles of any triangle add up to 180°. A right-angled prism used in reflectors and periscopes has angles 90°, 45° and 45° (90 + 45 + 45 = 180). This example prepares you for page 11.
Q. Light enters the face PQ of a 45°–45°–90° glass prism PQR along the normal (straight in, without bending) and travels on until it meets the long face PR. Find the angle of incidence at PR.
The ray inside is perpendicular to PQ. Face PR makes 45° with PQ, so the ray makes 90° − 45° = 45° with PR. Then use “angle with the normal = 90° − angle with the surface”.
6. The full circle and opposite angles
When two straight lines cross, they make four angles. The two angles opposite each other are always equal (they are called vertically opposite angles). So if a ray goes straight through the point where it crosses the normal, the angle it makes with N above the surface equals the angle it makes with N′ below.
In the textbook activity of Figure 1.16 you place two protractors edge to edge to draw a full circle on chart paper, and mark angles from the normal NN′ in every quarter: 0° at N and at N′, 90° along the surface on both sides. Try the virtual chart below.
Each quarter of the chart is numbered from 0 at the normal to 90 at the surface, so you can read an angle from the normal directly, on either side. Full circle = 4 × 90° = 360°; the two protractors give 180° + 180° = 360°.