28.5.1 A Short Circuit

In FIGURE 28.19 we’ve replaced the resistor with an ideal wire having $R_{wire} = 0 Ω$ . When a connection of very low or zero resistance is made between two points in a circuit that are normally separated by a higher resistance, we have what is called a short circuit. The wire in Figure 28.19 is shorting out the battery.

The figure illustrates the short-circuit current of a battery.

FIGURE 28.19 Full Alternative Text

If the battery were ideal, shorting it with an ideal wire $(R = 0 Ω)$ would cause the current to be $I = ε / 0 = \infty$ . The current, of course, cannot really become infinite. Instead, the battery’s internal resistance $r$ becomes the only resistance in the circuit. If we use $R = 0 Ω$ in Equation 28.17, we find that the short-circuit current is

I_{short} = \frac{ε}{r}

(28.20)

A 3 V battery with $1 Ω$ internal resistance generates a short circuit current of 3 A. This is the maximum possible current that this battery can produce. Adding any external resistance $R$ will decrease the current to a value less than 3 A.

EXAMPLE 28.6 A short-circuited battery

What is the short-circuit current of a 12 V car battery with an internal resistance of $0.020 Ω ?$ What happens to the power supplied by the battery?

SOLVE The short-circuit current is

I_{short} = \frac{ε}{r} = \frac{12 V}{0.02 Ω} = 600 A

Power is generated by chemical reactions in the battery and dissipated by the load resistance. But with a short-circuited battery, the load resistance is inside the battery! The “shorted” battery has to dissipate power $P = I^{2} r = 7200 W$ internally.

REVIEW This value is realistic. Car batteries are designed to drive the starter motor, which has a very small resistance and can draw a current of a few hundred amps. That is why the battery cables are so thick. A shorted car battery can produce an enormous amount of current. The normal response of a shorted car battery is to explode; it simply cannot dissipate this much power. Shorting a flashlight battery can make it rather hot, but your life is not in danger. Although the voltage of a car battery is relatively small, a car battery can be dangerous and should be treated with great respect.

Most of the time a battery is used under conditions in which $r ≪ R$ and the internal resistance is negligible. The ideal battery model is fully justified in that case. Thus we will assume that batteries are ideal unless stated otherwise. But keep in mind that batteries (and other sources of emf) do have an internal resistance, and this internal resistance limits the current of the battery.