Water Flow Analog of Electrical Current
The flow of water through a pipe has enough in common with the flow of electrons in a wire that examining their connection can help us understand both.

Flow Connection:
Water flow in a pipe is measured by the amount of water that flows past a point per second – the mass flow rate. Electrical flow in a wire is measured by the amount of charge that flow past a point per second – the charge flow rate or current I.

Note that (ρn q) is the charge per unit volume analogous to mass per unit volume ρ for fluid flow. Thus both the above equations have the same similar structure. All other thing being equal, the greater the flow rate the larger will be the velocity of fluid or the drift velocity of the charges.

Conservation Connection:
Both the mass of water and the amount of electrical charge are conserved quantities. For water this is expressed by the equation of continuity. For charge this is expressed by Kirchhoff’s current rule.

This means that the amount of water/charge that flows into a junction must equal to the amount that flows out.

For a two pipes/wires connected in series we know the flow rates in both pipes/wires must be the same. If the two series water pipes have different sizes, then the speed of the water is faster in smaller diameter pipe.

Similarly, the drift velocity of conduction electrons is larger in the smallest diameter wire for wires connected in series provided the wires are made out of the same material. If electrical wires are made out of different materials then the charge density (ρn q) can be different which cannot happen for an incompressible fluid where ρ = constant.

Fluid Pressure-Electrical Potential Connection:
Fluid pressure makes water flow in a pipe. Pressure is produced by either by a pump or by gravity. Electrical potential makes charges flow. A battery is one way to give charges an electrical potential. A fluid pump creates a pressure difference across the pipes connected to the pump. A battery creates a voltage difference across the ends of the wires connected to the battery.

Water pressure is also equal to the fluid’s energy per unit volume – hydrostatic potential energy density - see Bernoulli's Equation. Electrical potential is the current’s energy per unit charge. Both pressure and electrical potential are directly related to potential energy of the fluid and the charge. Thus you can think of a battery as a kind of “pump” which increases a current's electrical potential energy.

The fluid-pressure, electrical-potential connection has one very important difference. Connecting a battery to close a circuit is like turning on a pump. Before a circuit is closed with say a switch, no current flows because the pump/battery is effectively not turned on. Any time a battery is connected in a closed circuit is “turned on” unlike a water pipe system in which the pump can be turned on and off even if the pipes make a complete circuit.

If only gravity is causing the fluid to flow then water can flow out of the end of open pipe where as electrical charge will not flow out of an electrical socket. Gravity is always turned on where as the electrical potential is not turned on until the circuit makes a closed loop.

Resistance-Energy Loss Connection:
Both water pipes and electrical wires have an intrinsic resistance to the flow of water/charge through them. When we speak about an electrical resistor we usually mean a circuit component that has a large amount of resistance compared to metal wires of the same size connecting the components. Typical electrical resistors are made out of composition of carbon in a ceramic matrix. The semiconductor carbon has an intrinsic resistance that is 872 times larger than copper.

We could construct a water pipe that has a large amount of resistance by placing wire-mesh(s) inside the pipe. An alternate way would be to imagine that the water turns a paddle like a water wheel. Here energy is used up as the paddle is turned similar to the power dissipated when electrical current flows through a resistor.

Both of these systems will dissipate energy as the fluid/charge flows through them. The fluid/current enters with a higher potential energy and leaves with a lower potential. The energy difference generally goes off as heat.

Circuit Analogs:
A simple electrical circuit - consisting of a battery and a resistor - can be modeled by a pump to simulate a battery and a paddle to simulate electrical resistance.
Simple Circuit

In this water flow analog the vertical axis represents potential energy. This is completely reasonable since the gravitational potential energy, m g h , is directly proportional to the height h. As the current turns the paddle it does work and thus loses some energy similar to electrical current flowing through a resistor.

For two resistors in series this analog would look like the following.
Series Circuit

The battery “pumps” the charges up to a higher potential. The current then flows through the resistors losing energy, returning to the starting level. This starting level could be regard as the ground level. Any time you connect a circuit to ground at some point, you are forcing the potential at that point to be the zero reference level similar to the zero reference level of gravitational potential.

The potential of any one of the wires connecting the components are at the same potential any where along the wire. This happens because the resistance of the wires is assumed to be small compared to that of the resistors in the circuit.

For two resistors in parallel this analog looks like:
Parallel Circuit

Two resistors in parallel have the same potential drop. In this case both resistors are also in parallel with the battery so that potential drop across the resistors is same as the potential gain through the battery.

The amount of charge flowing through the two branches is not necessarily the sane unless both resistors have the sane value of resistance. More charge flows through the smaller resistor R2 – the water/charges take the path of least resistance. If the two resistors are made of the same material but have different values then the drift velocity through the smaller resistor is larger. This last observation is usually not stressed in electrical circuit analysis because resistors are normally not necessarily made of the same material – the carrier charge density is not necessarily the same as it would be for the water flow analogy.

Resistance is modeled by how hard it is to rotate the paddles. The slope of the analog for the resistors (or the battery) is not significant because gravitational or electrical energy is path independent. It takes the same amount of work to lift a ball through a height h regardless of the angle of the slope.

The water analog of a somewhat more complex circuit consisting of three identical resistor looks like:
More Complex Circuit

Here resistors R1 and R2 are in series with each other while their combination is in parallel with R3. The current through R12 drops by the same voltage as the current through R2 since they are in parallel. There is less current flowing through the R12 branch since this branch has twice the resistance as R3 branch.

One other difference between the water flow analog and electrical circuits is that the potential height of various connections is not fixed like it is for a water flow system. When you connect two different locations in an electrical circuit with a wire it force the potential/height of the two locations to have the same potential/height. See DC Circuit with Switch Analog QT Movie and observe the change in height of the switch before and after the switch is closed.

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