The Shadow That Moved
A student places a 10 cm tall pencil upright 30 cm from a lamp and measures its shadow on a screen 60 cm behind the pencil. The shadow is 20 cm tall. The student then moves the pencil 10 cm closer to the lamp and finds the shadow grows. A classmate claims the shadow grew because the lamp got brighter, but the lamp was not adjusted. The student also notices that a second, dimmer lamp nearby casts a faint second shadow with soft edges. The student wants to explain the shadow size change, account for the second shadow, and predict what happens if the pencil is moved to 10 cm from the lamp.
1. Explain why the shadow grew when the pencil moved closer, and predict the shadow height at 10 cm from the lamp.
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The shadow grows because moving the pencil closer to the lamp makes the angle subtended by the pencil larger, so the light rays spread farther by the time they reach the screen. The brightness of the lamp is irrelevant; the classmate's explanation fails because the lamp was unchanged. Using similar triangles, shadow height divided by pencil height equals screen distance from lamp divided by pencil distance from lamp. At 30 cm, the screen is 30 + 60 = 90 cm from the lamp, so 20/10 = 90/30, which checks. At 10 cm from the lamp, the screen is 70 cm away, so shadow height = 10 × 70/10 = 70 cm. The prediction assumes the screen stays 60 cm behind the pencil and the lamp remains a small source; a larger source would blur the edge and change the measurement.
2. Account for the faint second shadow with soft edges and explain what it reveals about the light sources.
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The faint second shadow is cast by the second, dimmer lamp, which acts as a separate light source from a different position, so it throws the pencil's shadow onto a different part of the screen. Its soft edges occur because that lamp is an extended source rather than a point: light from different parts of the lamp arrives at slightly different angles, so the shadow's edge is a gradual transition between full shadow and full light, called the penumbra. The brighter lamp produces the darker, sharper shadow because it dominates the illumination. This reveals that shadows depend on the number, position, and size of light sources, not only on the object. The student should treat the two shadows as evidence of two sources and keep the dim lamp out of the size measurement to avoid confusion.