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Touch-sensitive lamps almost always use a property of the human body -- its capacitance. The word "capacitance" has as its root the word "capacity" -- capacitance is the capacity an object has to hold electrons. The lamp, when standing by itself on a table, has a certain capacitance. This means that if a circuit tried to charge the lamp with electrons, it would take a certain number to "fill it." When you touch the lamp, your body adds to its capacity. It takes more electrons to fill you and the lamp, and the circuit detects that difference. It is even possible to buy little plug-in boxes that can turn any lamp into a touch-sensitive lamp. They work on the same principle.
Many touch-sensitive lamps have three brightness settings even though they do not use three-way bulbs. The circuit is changing the brightness of the lamp by changing the "duty cycle" of the power reaching the bulb. A bulb with a normal light switch gets "full power." Imagine, however, that you were you were to rapidly turn the power to the bulb on and off (say 100 times per second) -- then the bulb would only burn half as brightly because its duty cycle is 50 percent (half on, half off). "Rapidly switching the bulb on and off" is the basic idea used to change the brightness of the lamp -- the circuit uses zero percent (off), 33 percent, 66 percent and 100 percent duty cycles to control the lamp's brightness.