Light and shadow

What is it about?

Light travels in straight lines. Behind an opaque object there is a region that receives no light – its shadow. Edge rays from the rim of the light source past the rim of the object mark the borders of the shadow.

A point-like source gives a sharp shadow: umbra only.
An extended source gives an umbra (no part of the lamp is visible) surrounded by a penumbra (part of the lamp is visible) – the edge becomes blurred.
Two lamps give two shadows; where they overlap there is umbra. A red and a green lamp light the screen yellow; the shadow of the red lamp is green and vice versa.

For a point-like lamp the size of the shadow follows from similar triangles:

\frac{S}{G} = \frac{b}{a}

with $a$ the distance lamp–object and $b$ the distance lamp–screen.

Eclipses: at new moon the Moon’s umbra can reach the Earth (total solar eclipse, partial in the penumbra, annular if the umbra falls short). At full moon the Moon can enter the Earth’s shadow (lunar eclipse); in the umbra it glows copper red.

Try it

Move the disc towards the lamp and away again. How does the shadow change? Disc close to the lamp
Switch to the extended lamp and move the screen close to the disc. What happens to the penumbra? Extended lamp, screen close
A big lamp and a small disc – where does the umbra end? Follow the edge rays. Big lamp
Two lamps: move one lamp up or down until the two shadows no longer overlap. Two lamps
Coloured shadows: what colour is the shadow of each lamp? Red and green lamp
Solar eclipse: drag the observer across the day side of the Earth. Where is the eclipse total, where partial? Total solar eclipse · Annular solar eclipse
Press Play: the Moon moves around the Earth. Watch the phases of the Moon and when it enters the Earth’s shadow. Lunar eclipse

Tasks

Task 1: Size of the shadow

A point-like lamp is at $10\,\text{cm}$ , a disc with $G = 8\,\text{cm}$ at $30\,\text{cm}$ and the screen at $90\,\text{cm}$ . How large is the shadow? Load task

Show solution

$a = 20\,\text{cm}$ , $b = 80\,\text{cm}$ , so $S = 8\,\text{cm} \cdot \frac{80}{20} = 32\,\text{cm}$ . The shadow is four times as large as the disc. Show in the simulation

Task 2: Twice as large

The lamp is at $0\,\text{cm}$ and the screen at $80\,\text{cm}$ . Where must the disc be for a shadow twice its size? Load task

Show solution

$\frac{b}{a} = 2$ gives $a = 40\,\text{cm}$ – halfway between lamp and screen. Show

Task 3: Umbra and penumbra

A frosted bulb with $d = 8\,\text{cm}$ is at $10\,\text{cm}$ , a disc with $G = 10\,\text{cm}$ at $40\,\text{cm}$ and the screen at $100\,\text{cm}$ . How large are the umbra and the whole shadow? Hint: draw the four edge rays. Load task

Show solution

With $a = 30\,\text{cm}$ and $b = 90\,\text{cm}$ : the edge rays “top → top” and “bottom → bottom” bound the umbra, the crossed edge rays bound the penumbra.

$S_\text{umbra} = \frac{G \cdot b - d \cdot (b-a)}{a} = \frac{900 - 480}{30}\,\text{cm} = 14\,\text{cm}$ , $\quad S_\text{outer} = \frac{900 + 480}{30}\,\text{cm} = 46\,\text{cm}$ .

So the penumbra is a ring $16\,\text{cm}$ wide.

Task 4: Coloured shadows

A red and a green lamp light up a disc. Explain why the screen is yellow, why there is a green and a red shadow and why the middle is black. Load task

Show solution

Where both lamps shine, red and green light mix to yellow. In the shadow of the red lamp only the light of the green lamp arrives, so it is green – and vice versa. In the middle both lamps are blocked: umbra, it is dark there.

Task 5: Who sees the eclipse?

Why can a total solar eclipse only be seen in a narrow area of the Earth, but a lunar eclipse from the whole night side of the Earth? Load task

Show solution

The Moon’s umbra only just reaches the Earth and covers a small area there; only there is the Sun completely hidden. In a lunar eclipse the Moon itself gets dark – everyone who can see the Moon above the horizon sees it. Show lunar eclipse

Task 6: Why not every month?

In the simulation every new moon causes a solar eclipse. Why are real eclipses much rarer?

Show solution

The Moon’s orbit is tilted by about $5^\circ$ against the Earth’s orbit, so the Moon’s shadow usually passes above or below the Earth. Only when new moon (or full moon) happens near the points where the two orbits cross is there an eclipse. The simulation shows the orbits from above and leaves out the tilt.

Ideas for the classroom

Lamp and object are modelled as discs parallel to the screen; holders cast no shadow. The eclipse model is schematic and not to scale (the Sun and the Moon look about equally large from the Earth, as in reality).
Common stumbling blocks: “The shadow gets smaller when the object moves towards the lamp.” – It gets larger. “The phases of the Moon are caused by the Earth’s shadow.” – Only a lunar eclipse is.