Some science students are initially baffled when studying electricity. This is not so surprising, as it is hard to relate to the concept of current, the rate of flow of charge. The charge that is transported through an electrical circuit is composed of electrons - many of them. I have never seen an electron, and while I have felt a charge by means of electric shock, I have trouble sensing exactly how many Coulombs are being transferred to me (or are they being taken from me? I can never tell.). I have an easier time visualizing, say, mass than I do visualizing charge, and I imagine this to be true for most people.

This is why it is appealing to explore the analogy between electrical current through a circuit and fluid flow rate through a system of pipes. As a student, I could never remember whether current was constant through resistors connected in series or parallel. The answer becomes obvious when one considers the flow rate of water through pipes rather than the current through wires.

When water flows through one continuous pipe, the mass that enters must also leave. For this reason, the flow rate of water (kg/s) through a single pipe is constant, even if the area of cross section of the pipe is changing. This is known as the principle of continuity. If the size of the pipe changes, the velocity of the fluid adjusts itself to ensure that the flow rate remains constant. Water flowing through a single pipe of variable cross-section is analogous to the charge flowing through resistors connected in series - if the flow rate of water remains constant in such a case, then so too must the current in series.

If a pipe were to fork into two separate paths, the total flow rate must be conserved across the junction. Similarly, when current splits at a junction within a circuit, the incoming current i1 splits into the two outgoing paths i2 and i3, such that i1 = i2 + i3; current splits in a parallel circuit.

If the flow rate of water links to the current of electricity, then the mass (kg) of a water system links to the charge (C) of an electrical circuit. Digging just a little deeper, we notice that the individual water molecules making up the mass of water relate to the individual electrons constituting the charge.

The analogy may be extended much further. If you want to stop the flow of water in a pipe, close a valve. If you want to stop the flow of current, open a switch (this makes the passage of current impossible). How can one inject additional pressure into a fluid system? One may use a mechanical pump. Correspondingly, a battery may be connected to a DC circuit to inject voltage. We can then correlate a fluid pressure gradient across a cross-section with an electrical voltage drop across a resistor.

The commonality between electrical voltage and fluid pressure do not end there. The first time one is faced with Kirchhoff's voltage law, one may not associate with it, as voltage itself seems quite abstract. Kirchhoff showed that the voltage drops throughout a circuit loop must equal the voltage gain. Fortunately, Bernoulli showed something very similar about steady in-compressible fluid flow. The sum of the internal pressure, hydro-static pressure, and dynamic pressure of a fluid at any location along a single path of pipe must be constant.

Both Kirchhoff's law and Bernoulli's equation are statements of conservation of energy - the first for electrical energy, and the second for mechanical energy. While voltage and pressure are both invisible entities, I can more easily relate to pressure. I can feel hydro-static pressure against my ear drums by descending a few meters under water. Voltage itself cannot be felt.

I would like to close with one final hydro/electric comparison that will give a clear illustration of exactly how electricity can become dangerous...

Imagine you are standing chest-deep in a river of flowing water. If millions of Liters of water flowed past you gradually over the course of an hour, you would be unharmed. However, if that same volume of water rushed by you in a matter of seconds, you would surely drown. It is not the volume of water that is dangerous, rather, it is the rate at which the volume flows by: the flow rate. The extreme example of this is a deadly tsunami.

It is the same with charge. While several Coulombs of charge represents a significant amount of electricity (a huge number of electrons), it is not dangerous in and of itself. It is the rate at which this charge flows, electrical current, that may represent a lethal danger.

Let us say that a charge of 0.02 Coulombs were to be transferred to your body. If this transfer were to occur gradually over a period of one second, that would translate to a non-lethal current of 0.02 Amps, since current

*I*=

*Q*/

*t*, where

*Q*is charge in Coulombs and

*t*is time in seconds. If the transfer of charge were to occur more rapidly, say, in one tenth of a second, the current passing through your body would be 0.2 Amps, which could cause your heart to stop beating (this biological pump is critical, as it causes the pressure difference required to cause your blood to circulate).

So, what determines the rate at which a given charge will be transferred to you? This is governed by the resistance of the resistor, measured in Ohms. When one is shocked, one becomes a resistor inside a circuit. A high resistance is good, since Ohm's law says that

*I*=

*V*/

*R*, where

*V*is voltage in volts and

*R*is resistance in Ohms. The human body has a resistance in the range of 100,000 Ohms when it is dry. As such, a person can withstand a fairly high voltage drop (though it may not be pleasant) under dry conditions. However, a wet human body has a resistance that is on the order of just 100 Ohms. As such, even low voltage household items, like a hair dryer, may become lethal under wet conditions.

I find it strange that few physics courses deal with both the subject matter of fluids and electricity. The theory behind each respective area of study is so complementary to the other. We all know that in practice, "Water and electricity don't mix." This does not mean that they cannot share a chalk board.

## 2 comments:

would this analogy work with compressible fluids?

The analogies still hold (eg, whether compressible or not, conservation of energy applies to all matter). The specific dynamic behavior is however not the same.

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