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I'm trying to understand how voltage works in circuits, and currently I'm stuck at Kirchhoff's Loop Law.

Suppose we have a simple circuit, just a battery and a resistor. Our battery supplies a voltage across the circuit $\Delta V_{battery}$. Now, according to Kirchoff's loop law, the resistor has a $\Delta V$ equal and opposite to that of the battery, causing the sum of the voltages across the loop to be 0.

The way I see it, the battery supplies energy in the form of a potential difference, and the resistor removes all that energy in the form of heat. Now our electrons have exited the resistor with no potential energy, and our current will cease to exist.

However, resistors only lower the current of a circuit (this I understand via Ohm's Law), not remove it entirely. So clearly there is a flaw in my thinking. Where am I going wrong?

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Your reasoning is flawed in thinking that no net energy change means that there is no motion of charges.

A classical mechanics example: Your car is moving at some non-zero constant speed. There are dissipative forces like air resistance and friction taking energy out of the system. You (or rather the processes of the car) is putting energy in. The net change in energy is $0$, yet the car still moves.

The power supplied and dissipated in the circuit might cancel out, but this doesn't mean nothing happens. Kirchoff's loop rule just says the net work done on a charge around the loop is $0$, as is required for electrostatic interactions.

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