If the bulb's filament is broken, the whole circuit becomes open. If the bulb is in a series circuit or is the only bulb, the electricity would no longer flow. If the bulb is in a parallel circuit, along with other bulbs, then only the blown bulb would go out, and the other bulbs would still work.
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It depends on the circuit. If it is a constant-current circuit, any light bulbs connected in parallel with it will become brighter. If it is a constant-voltage circuit like a typical household circuit, nothing will happen. Any connected in series with it will go out.
No power is used. The lamp in the light fixture is the load of the circuit. The load resistance is what makes the current flow. When the current flows through the filament's resistance, heat and light are generated. With no lamp in the fixture, the circuit's voltage potential is still at the socket contact points and if touched could cause you a shock. This is a good reason to shut the fixture off at the switch when removing the burnt out bulb and replacing it with a new one.
A small gauge filament is a thin wire with a high melting point, just like the filament in light bulbs. That filament will heat up when electricity will pass through it. If you have enough battery power, I would suggest using a piece if pencil lead.
An electric current passes through a thin filament, heating it until it produces light. The enclosing glass bulb prevents the oxygen in air from reaching the hot filament, which otherwise would be destroyed rapidly by oxidation.
The total current in the circuit will decrease.