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Real DC cells and Real Wires

An ideal voltage source would maintain a given potential across its terminals regardless of the circuit. Real DC cells are not so good, and we find that any current through the cell reduces the potential across the terminals. We can model a real cell as an ideal potential source in series with a resistor, as if there were a resistor inside the cell. The potential across the ideal source is the electromotive force or emf, the resistance of the internal resistor is the internal resistance Rcell, while the actual potential difference across the whole cell is the terminal potential.

In the simplified circuit of Figure 15-18, the circuit outside of the cell is a represented by a single resistor and dashed lines enclose the cell. The potential jump across the ideal source is Vemf. The current flowing through the circuit is I, so the potential drop across the internal resistor is IRcell. Thus from the illustration we can give an expression for the terminal potential:

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Figure 15-18



In addition to knowing this equation, you should understand the discussion that leads to it.



A battery has a measured potential difference of 6.0 volts if no circuit is connected to it. When it is connected to a 10-Ω resistor, the current is 0.57 amps. What is the internal resistance of the battery?



First let’s DRAW A DIAGRAM (Figure 15-19). We label the negative terminal of the potential source 0 volts; the other side, 6 volts. The potential drop across the external resistor (10 Ω) is given by Ohm’s law:

ΔVext = IRext
= (0.57A)(10 Ω)
= 5.7V

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Figure 15-19

The potential on the other side of the external resistor is 0.3 volts. The potential difference across the internal resistor is 0.3 volts, and Ohm’s law gives us the resistance

Rint = ΔVint/I
= 0.3V/0.57A
= 0.53Ω

Another way to do this problem is to combine resistances. Notice that equation (4) does not automatically give the answer in this example, but if we draw a diagram and apply the methods of Section B, then we obtain the answer in two steps.

In addition to idealizing DC cells, we have been assuming that wires have no resistance at all. In fact, they have a small resistance that is proportional to their length and inversely proportional to their cross-sectional area. A given material has a resistivity ρ, so the resistance of the wire is given by

where p has the units [Ohm meters). / is the length of the wire (in [m]) and A is its cross-sectional area (in [m']).

Some resistivities are given in the table below.
substance resistivity (Ωm)
silver 1.5 X 10–8
copper 1.7 X 10–8
gold 2.4 X 10–8


You can assume wires have zero resistance unless the passage tells you otherwise.

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