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Why or when does it occur in a circuit? What does it imply when you speak of a voltage drop across a resistor? (Obviously, it probably means that the current's voltage before the resistor is higher than the voltage after the resistor, but why does this drop occur?)

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Voltage is the unit of electric potential, the electric potential difference (in your case, the potential difference between the two ends of resistor in a circuit) can be called the voltage drop.

The potential difference produces an electric field $\vec{E}$, and the direction of $\vec{E}$ points from high potential to low potential. The electric field applies a force on charged particles (i.e. electrons in circuits) such that the electrons are driven by this force and move, thereby producing a current. So you can see the potential (voltage) difference is the reason why there is a current. By the way, you cannot say the "current's voltage", since the current is defined as $I = dQ/dt$. That is, it only describes the flow of charge per unit time.

When electrons move through a resistor they are scattered by the other electrons and nuclei, causing the electrons lose some of their kinetic energy. But the presence of the electric field will then accelerate the electrons again. We can calculate the average kinetic energy statistically, and assume the electrons are moving at a single average velocity.

Thus after each collision there is a loss of kinetic energy (it is converted to heat) but which is recovered due to the work done by the electric field. And this work is equal to the potential energy difference. You can see that the electrons have the same kinetic energies both when they enter and when they leave the resistor, but different potential energies. So we can say the voltage drop at the two ends of a resistor is caused by the potential energy difference.

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Good explanation! Sorry for my ignorance, but where does the lost energy from the voltage drop go to? –  wrongusername Sep 22 '11 at 4:58
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manifests as heat in the resistive element. –  Vineet Menon Sep 22 '11 at 6:52
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Or if you're using the circuit to power a motor, some of the energy goes into mechanical energy produced by the motor. Or if you're using it to charge a battery, some of it goes into chemical energy. –  David Z Sep 22 '11 at 17:28
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