14.1 Voltage
14.2 Current
14.3 Resistance
14.4 Energy, Power, and Heat
14.5 Circuits
14.6 Capacitors
14.7 Key Formulas
14.8 Practice Questions
14.9 Explanations
Capacitors rarely come up on SAT II Physics, but they do sometimes make an appearance. Because capacitance is the most complicated thing you need to know about DC circuits, questions on capacitors will usually reward you simply for knowing what’s going on. So long as you understand the basic principles at work here, you’re likely to get a right answer on a question most students will answer wrong.
A capacitor is a device for storing charge, made up of two parallel plates with a space between them. The plates have an equal and opposite charge on them, creating a potential difference between the plates. A capacitor can be made of conductors of any shape, but the parallel-plate capacitor is the most common kind. In circuit diagrams, a capacitor is represented by two equal parallel lines.
For any capacitor, the ratio of the charge to the potential difference is called the capacitance, C:
For a parallel-plate capacitor, C is directly proportional to the area of the plates, A, and inversely proportional to the distance between them, d. That is, if the area of the plates is doubled, the capacitance is doubled, and if the distance between the plates is doubled, the capacitance is halved. The proportionality constant between C and A/d is , called the permittivity of free space, which we encountered in the previous chapter in relation to Coulomb’s constant. In case you forgot, C2 /N · m2.
The unit of capacitance is the farad (F). One farad is equal to one coulomb per volt. Most capacitors have very small capacitances, which are usually given in microfarads, where 1 µF = 10–6 F.
To move a small amount of negative charge from the positive plate to the negative plate of a capacitor, an external agent must do work. This work is the origin of the energy stored by the capacitor.
If the plates have a charge of magnitude q, the potential difference is . If q = 0, and work is done to add charge until q = Q, the total work required is:
This is the energy stored by the capacitor. Manipulating this equation and the equation for capacitance, , we can derive a number of equivalent forms:
Equivalent Capacitance
Like resistors, capacitors can be arranged in series or in parallel. The rule for adding capacitance is the reverse of adding resistance:
Capacitors in series add like resistors in parallel, and capacitors in parallel add like resistors in series.
For two capacitors in series:
For two capacitors in parallel:
Given = 2 µF, = 6 µF, and = 3 µF, what is the total capacitance of the circuit in the figure above?
First, we find the equivalent capacitance of and . Since they are in parallel, = + = 8 µF. Then is given by:
One way to keep the plates of a capacitor apart is to insert an insulator called a dielectric between them. A dielectric increases the capacitance. There is an electric field between the plates of a capacitor. This field polarizes the molecules in the dielectric; that is, some of the electrons in the molecules move to the end of the molecule, near the positive plate:
The movement of electrons creates a layer of negative charge by the positive plate and a layer of positive charge by the negative plate. This separation of charge, in turn, creates an electric field in the dielectric that is in the opposite direction of the original field of the capacitor. This reduces the total electric field:
The Greek letter is called the dielectric constant, and it varies from material to material. For all materials, > 1.
For a parallel-plate capacitor, the reduction in E means that is also reduced by a factor of . Then, since C = Q/ , we find that:
If the potential difference across the capacitor is too large, then the electric field will be so strong that the electrons escape from their atoms and move toward the positive plate. This dielectric breakdown not only discharges the capacitor, but also burns a hole in the dielectric and ruins the capacitor.
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