Chapter 5: Electric circuits

The chapter problem: the defibrillator

A defibrillator stores energy in a capacitor. When the paddles touch a patient's chest, the capacitor discharges through the chest in a short, controlled pulse.

These are the facts to use all the way through the chapter.

• The capacitor has a capacitance of 100 µF.
• It is charged to a potential difference of 2000 V.
• The paddles and the patient's chest have a total resistance of 50 Ω.
• The connecting leads are copper. Their total length is 1.2 m and their diameter is 2.0 mm.
• The resistivity of copper is 1.7 × 10⁻⁸ Ω m.
• The battery in the unit has an emf of 12 V and an internal resistance of 0.15 Ω. While the capacitor charges, the unit draws a current of 2.0 A from the battery.
• The unit is tested on a load of two 100 Ω resistors in parallel, which stands in for a patient.

By the end of the chapter you will find the charge stored on the capacitor and the energy it holds. You will find the resistance of the leads and the total resistance of the discharge path. You will work out the time constant of the discharge and follow the potential difference as it falls. At the end you will judge whether the leads matter and how the energy is delivered.

First contact: no working yet. Estimate. The capacitor stores about $200\,\text{J}$. Does it deliver that energy in about a microsecond, about a millisecond, or about a second? Write down your choice and one sentence of reasoning.

By the end of this chapter you will be able to

• define electric current and use $I = \dfrac{\Delta Q}{\Delta t}$;
• find the charge that flows in a circuit, and the number of charge carriers;
• explain the difference between emf and potential difference;
• use the emf of a source, its internal resistance and its terminal potential difference;
• calculate energy transferred with $W = QV$ and $W = VIt$;
• define resistance and use Ohm's law;
• describe and compare the $I$–$V$ characteristics of a resistor, a filament lamp and a diode;
• use resistivity, and $R = \dfrac{\rho L}{A}$;
• find the equivalent resistance of resistors in series and in parallel;
• use Kirchhoff's laws in a circuit;
• use the potential divider equation;
• define capacitance and use $C = \dfrac{Q}{V}$;
• find the energy stored in a charged capacitor;
• describe the exponential discharge of a capacitor and use the time constant $\tau = RC$;
• solve the defibrillator problem, stage by stage.

Where it is used

A circuit is the working part of almost every device you use. A phone charger holds the current steady while the battery fills. A camera flash stores energy in a capacitor and releases it in a fraction of a millisecond. The wiring in a house is designed so that the current in each cable does not overheat it. An engineer choosing a cable uses resistivity, not length alone. A pacemaker and a defibrillator both deliver charge to the body, but at very different rates and energies.

Your turn to connect it

Think of a device that holds charge and then releases it. It could be a camera flash, a phone, a computer power supply or an electric car. Write two examples. For one of them, write a sentence about what the charge is used for.

Roadmap

1. 5.1 Current, charge and emf
2. 5.2 Resistance and resistivity
3. 5.3 Direct-current circuits
4. 5.4 Capacitance
5. Review