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5.6: Extending the Electrical Model

• • Contributed by Physics 7
• Physics at UC Davis

Household Electricity

Utility companies such as PG&E and SMUD supply our workplaces and houses with alternating current (AC) electricity. The current varies sinusoidally, at 60 Hz, switching directions 120 times each second. The fundamental ideas we have developed to understand direct current circuits (as from a battery) can also be applied to AC. In fact, the values of the voltages and currents used when dealing with 60 Hz AC electricity are typically the root mean square values, which makes all the algebraic relationships we have developed applicable. Therefore, unless stated otherwise, values of voltages and currents for 60 Hz AC can be treated as DC values.

The primary reason AC is used in power distribution systems is the ease with which voltages can be changed. The large round “cans” hanging on power poles are transformers. In a typical residential power distribution system, the wires at the top of the pole are often at 12,000 to 22,000 V. The transformer steps this voltage down to 120 V and 240 V, the voltage of wires entering most apartments and houses. We will study how transformers work in Part 3 when we get into the fascinating world of the interaction of electricity and magnetism.

You probably use several small appliances everyday that make use of a “step-down” transformer. Many small computer peripherals have a small black box with two prongs sticking out that plugs into the 120 V power-outlet strip. The 120 V from the power strip is dropped down to 6 to 12 V in the isolation transformer. This is an excellent safety feature. Voltages in the 10 to 20 V range are relatively harmless to humans, if contact is limited to skin and not to internal organs. The 120 V in the wall outlet can cause sufficient currents through a person’s body when contact is made through skin and is definitely considered dangerous. (The major health risk from 120 V shocks are currents in the chest region, which can cause the heart to go into fibrillation. Much larger currents cause burns, and ironically, can also be effective in stopping fibrillation of the heart. This is exactly what an AED, automated external defibrillator, does.)

Every outlet in our homes can be considered a source of constant voltage of about 120 Volts. Each appliance has a characteristic resistance $$R$$, and this determines how much power is used by this appliance when it is turned on

$P = \dfrac{($$\Delta$$V)^2}{R}.$

The table in the margin gives some typical values. When you pay your PG&E or SMUD electrical bill, they charge you for energy, not power. The unit they use, however, sounds like a power unit: kilowatt-hour. Can you explain why this is an energy unit?

Internal Resistances of Power Sources Most wires used in house wiring, for appliances, and in lab, have very low resistance. In particular, the voltage drop IR is small compared to other voltage changes in the circuits. Therefore, we typically model them as having zero resistance. Similarly, new batteries have resistances that are also small compared to the resistances of other components in the circuit to which the battery is attached. However, the internal resistance r of batteries does increase over time, as the reactant chemicals inside turn into by-products that impede the flow of electrons through the battery. A 1.5 volt battery that is almost “used up” still provides charges that pass through it with almost 1.5 joules per coulomb. That is, it might still have an emf of nearly 1.5 V. But due to its high internal resistance r, most of this voltage is converted to thermal energy inside the battery itself when the battery is connected into a circuit. A battery that is very warm when it is being used is probably almost completely depleted.

If batteries are hooked up in series, their internal resistances (if any) will add, just as do their emfs. For example, four "AA" cells of 1.5 volts each used in series in a portable radio would have a total $$\varepsilon$$ of 6 volts, but will also have four times the internal resistance of one cell. Because of its larger size, a “D” cell, while still having an $$\varepsilon$$ of 1.5 V, will have considerable less internal resistance than will a “AA” cell.

When similar size batteries are wired in parallel in a circuit, the voltage across each battery will be the same (since they are hooked together with low resistance wire. The potential drop across each is the same. But their internal resistances and actual values of $$\varepsilon$$, will determine how much current gets into the rest of the circuit and how much is “wasted” “going around in circles” among the batteries. In general, it is not a good idea to hook up batteries in parallel.