After Chapter 6.3, you will be able to:
Aside from batteries and resistors, the other major circuit element tested on the MCAT is the capacitor. Capacitors are characterized by their ability to hold charge at a particular voltage. There are excellent real-world examples of capacitors. Perhaps the most important capacitor you’ll encounter in the clinics is the defibrillator. While a defibrillator is charging, a high-pitched electronic tone sounds as electrons build up on the capacitor. When the defibrillator is fully charged, that charge can be released in one surge of power (after the operator yells Clear!). The clouds and the ground during a lightning storm also act as a capacitor, with the charge building up between them eventually discharging as a bolt of lightning. The MCAT focuses on a particular type of capacitor called a parallel plate capacitor, and all of our discussion will center on capacitors of this type.
When two electrically neutral metal plates are connected to a voltage source, positive charge builds up on the plate connected to the positive (higher potential) terminal, and negative charge builds up on the plate connected to the negative (lower potential) terminal. The two-plate system is a capacitor because it can store a particular amount of charge at a particular voltage. The capacitance of a capacitor is defined as the ratio of the magnitude of the charge stored on one plate to the potential difference (voltage) across the capacitor. Therefore, if a voltage V is applied across the plates of a capacitor and a charge Q collects on it (with +Q on the positive plate and –Q on the negative plate), then the capacitance is given by
The SI unit for capacitance is the farad
. Because one coulomb is such a large quantity of charge, one farad is a very large
capacitance. Capacitances are usually given in microfarads (1 μF = 1 × 10–6 F) or picofarads (1 pF = 1 × 10–12 F). Be careful not to confuse the farad with the Faraday constant from electrochemistry,
F, which is the amount of charge in one mole of electrons
.
The capacitance of a parallel plate capacitor is dependent upon the geometry of the two conduction surfaces. For the simple case of the parallel plate capacitor, the capacitance is given by
where ε0 is the permittivity of free space
A is the area of overlap of the two plates, and d is the separation of the two plates. The separation of charges sets up a uniform electric field between the plates with parallel field vectors, the magnitude of which can be calculated
as
If you look back to the equations discussed in Chapter 5 of MCAT Physics and Math Review, you can see that the equation for E here can be derived from the other fundamental electrostatics equations. If
and
, then V = E × r. r in this setup is the distance between the plates, d, so we can rewrite this as V = Ed.
The direction of the electric field at any point between the plates is from the positive plate toward the negative plate. If we imagine placing a positively charged particle between the oppositely charged plates, we would expect the particle to accelerate in that same direction. This should not be surprising, as electric field lines always point in the direction that indicates the direction of a force exerted on a positive charge.
Regardless of the particular geometry of a capacitor (parallel plate or otherwise), the function of a capacitor is to store an amount of energy in the form of charge separation at a particular voltage. This is akin to the function of a dam, the purpose of which is to store gravitational potential energy by holding back a mass of water at a given height. The potential energy stored in a capacitor is
The term dielectric material is just another way of saying insulation. When a dielectric material, such as air, glass, plastic, ceramic, or certain metal oxides, is introduced between the plates of a capacitor, it increases the capacitance by a factor called the dielectric constant (κ). The dielectric constant of a material is a measure of its insulating ability, and a vacuum has a dielectric constant of 1, by definition. For reference, the dielectric constant of air is just slightly above 1, glass is 4.7, and rubber is 7. These numbers need not be memorized; any relevant dielectric constants will be given on Test Day.
The capacitance due to a dielectric material is
where C′ is the new capacitance with the dielectric present and C is the original capacitance.
A dielectric material can never decrease the capacitance; thus, κ can never be less than 1.
When a dielectric material is placed in an isolated, charged capacitor—that is, a charged capacitor disconnected from any circuit—the voltage across the capacitor decreases. This is the result of the dielectric material shielding the opposite charges from each other. By lowering the voltage across a charged capacitor, the dielectric has increased the capacitance of the capacitor by a factor of the dielectric constant. Thus, when a dielectric material is introduced into an isolated capacitor, the increase in capacitance arises from a decrease in voltage.
When a dielectric material is placed in a charged capacitor within a circuit—that is, still connected to a voltage source—the charge on the capacitor increases. The voltage must remain constant because it must be equal to that of the voltage source. By increasing the amount of charge stored on the capacitor, the dielectric has increased the capacitance of the capacitor by a factor of the dielectric constant. Thus, when a dielectric material is introduced into a circuit capacitor, the increase in capacitance arises from an increase in stored charge.
The stored energy in a capacitor is only useful if it is allowed to discharge. The charge can be released from the plates either by discharging across the plates or through some conductive material with which the plates are in contact. For example, capacitors can discharge into wires, causing a current to pass through the wires in much the same way that batteries cause current to move through a circuit. The paddles of the defibrillator machine, once charged, are placed on either side of a patient’s heart that has gone into a life-threatening arrhythmia (such as ventricular fibrillation). The reason the doctor yells Clear! before discharging the paddles is because the current needs to travel through the patient’s heart—not through any other people who might be touching the patient and creating a parallel pathway. On a much larger scale, lightning occurs when a very, very large amount of charge exceeds the capacitance of the Earth’s surface and the underside of the cloud (the two serving, approximately, as a parallel plate capacitor). The large rapid discharge across the plates of a capacitor is termed a failure of the capacitor, while creating a current through the attached wires is the normal function of a capacitor.
Just like resistors, capacitors can be arranged within a circuit either in parallel or in series. They can also be arranged with resistors, although this is beyond the scope of the MCAT in most cases.
When capacitors are connected in series, the total capacitance decreases in similar fashion to the decreases in resistance seen in parallel resistors, as shown in Figure 6.3.
This is because the capacitors must share the voltage drop in the loop and therefore cannot store as much charge. Functionally, a group of capacitors in series acts like one equivalent capacitor with a much larger distance between its plates (in fact, with a distance equal to those of each of the series capacitors added together). This increase in distance, as seen earlier, means a smaller capacitance.
Rather than memorizing the following equations independently, understand the conceptual basis for the mathematics of resistors in series and in parallel, and then simply reverse that mathematical approach for capacitors. The equation for calculating the equivalent capacitance for capacitors in series is
which shows that Cs decreases as more capacitors are added. Note that for capacitors in series, the total voltage is the sum of the individual voltages, just like resistors in series.
Capacitors wired in parallel, shown in Figure 6.4, produce a resultant capacitance that is equal to the sum of the individual capacitances.
Therefore, Cp increases as more capacitors are added:
Just as we saw with resistors in parallel, the voltage across each parallel capacitor is the same and is equal to the voltage across the source.