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**Capacitance and Dielectrics**

Chapter 26 Capacitance and Dielectrics

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Intro In this chapter we will introduce the first of the 3 simple electric circuit elements that we will discuss in AP Physics Capacitor Resistor Inductor Capacitors are commonly used devices often as form of energy storage in a circuit.

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26.1 Capacitor- two conductors separated by an insulator called a dielectric. Often times the two conductors of a capacitor are called plates. If the plates carry a charges of equal magnitude and opposite sign, there will exist a potential difference (ΔV) or a voltage between the two.

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**26.1 How much charge can we store on the plates?**

Experiments have shown that the amount of charge stored increases linearly with voltage between the conductors. We will call the constant of proportionality Capacitance .

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26.1 Capacitance is defined as the ratio of the magnitude of the charge on either conductor to the magnitude of the potential difference between the conductors. Capacitance is a constant for a given capacitor and has units of C/V or F (farad)

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26.1 1 Farad is a Coulomb of charge per Volt which is a huge capacitance. More commonly in the 10-6 to range or microfarads (μF) to picofarads (pF) Actual capacitors may often be marked mF (microfarad) or mmF (micromicrofarad)

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26.1 Consider two parallel plates attached to a battery (potential difference source) The battery creates an Electric field within the wires, moving electrons onto the negative plate. This continues until the negative terminal, wire and plate are equipotential.

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26.1 The opposite occurs with the positive terminal, pulling electrons from the plate until the plate/wire/+ terminal are equipotential. Example- A 4 pF capacitor will be able to store 4 pC of charge for every volt of potential difference between the plates. If we attach a 1.5 V battery, one of the conductors will have a +6 pC charge, the other will have -6 pC. 12 V battery?

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26.1 Quick Quiz p 797

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**26.2 Calculating Capacitance**

We can derive expressions for the capacitance of pairs of oppositely charged conductors by calculating ΔV using techniques from the previous chapters. The calculations are generally straightforward for simple capacitors with symmetrical geometry.

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**26.2 A single conductor Sphere (w/infinite imaginary shell)**

Since at the sphere, V = kQ/R, and at ∞, V = 0

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**26.2 Parallel Plate Capacitors**

For plates whose separation is much smaller than their size. From earlier (Ch 24) the E field between the plates is So the potential difference is

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**26.2 And the capacitance is therefore**

The capacitance is proportional to area and inversely proportional to the plate separation. True in the middle of the plates, but not near the edges.

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**26.2 The capacitor stores electrical potential energy as well as**

charge due to the separation of the positive and negative charges on the plates. Quick Quiz p. 800 Examples

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26.2 Example 26.2 Cylindrical Capacitor

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26.2 Example 26.3 Spherical Capacitor

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**26.3 Combinations of Capacitors**

Often two or more capacitors are combined in electric circuits. We can calculate the equivalent capacitance of a circuit, based on how the capacitors are connected. We will use circuit diagrams (schematics) as pictorial representations of the circuit.

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**26.3 Connecting wires- straight lines Capacitors- parallel lines**

of equal length Batteries- parallel lines of unequal length Switch- swinging line representing “open” or “closed” circuits

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26.3 Parallel Combination- two capacitors connected with their own conducting path to the battery. The potential difference across each capacitor is the same, and its equal to the potential across the combination.

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26.3 When the battery is connected electrons are removed from the positive plates and deposited on the negative plates. This flow of charge ceases when the potential across the plates reaches that of the battery. The capacitors are then at maximum charge Q1 and Q2, with a total charge given by

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26.3 Because the voltages across the capacitors are the same the charges are If we wanted to replace the two capacitors with a single equivalent capacitor the total charge stored must be

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**26.3 Therefore the equivalent capacitance must be**

The equivalent capacitance for any number of parallel combination of capacitors is

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26.3

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26.3 Series Combination- two or more capacitors connected along the same conducting path

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26.3 As the battery charges the capacitors the electrons leaving the positive plate of C1 end up on the negative plate of C2. The electrons from the positive plate of C2 move to the negative plate of C1. All capacitors hold the same charges.

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**26.3 The voltage of the battery is split across the capacitors.**

The total potential difference across a series combination of capacitors is the sum of the potential difference across each individual capacitor.

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26.3 If we wanted to find one capacitor equivalent to the series combination, the total potential difference is And each individual is

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26.3 So from We get And finally

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26.3 The inverse of the equivalent capacitance is equal to the sum of the inverses of the individual capacitances in series combination.

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26.3 Quick Quizzes p. 805 Example 26.4 p. 806

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**26.4 Energy Stored in a Charged Capacitor**

To determine the energy in a capacitor, we’re going to look at an atypical charging process. We’re going to imagine moving the charge from one plate to the other mechanically, through the space in between.

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26.4 Assume we currently have a charge q on our capacitor, giving the current potential difference to be ΔV = q/C. The work it will take to move a small increment of charge across the gap is

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26.4 The total work W, required to charge the capacitor from q = 0 to q = Q is

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26.4 The work done in charging the capacitor is the Electrical Potential Energy stored and applies to any capacitor regardless of geometry. Energy increases as both Charge and Voltage increase, within a limit. At high enough potential difference, discharge will occur between the plates.

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26.4 We can describe the energy as being stored in the electric field between the plates. For parallel plate caps Therefore

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26.4 We use this expression to derive a new quantity called Energy Density (uE) Since the volume occupied by the field is Ad, the energy U per unit volume is (U/Ad) The energy density of an E-field is proportional the square of the magnitude of the E-field at a given point.

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26.4 Quick Quizzes p 808 Example 26.5

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26.4 Defibrillation- Capacitors store 360 J of energy at a potential difference that will deliver the energy in a time of 2 ms. Circuitry allows the capacitor to be charged (to a much higher voltage than the battery) over several seconds. Similar technology to camera flashes

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**26.5 Capacitors and Dielectrics**

A dielectric is a non-conducting material (rubber/glass/waxed paper) that can be placed between the plates of a capacitor to increase its capacitance. If the space is entirely filled with the dielectric material, C will increase by a dimensionless factor κ, the dielectric constant of the material.

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**26.5 The new capacitance voltage and charge will be given by**

(for constant charges) (for constant voltage) Or specifically for a parallel plate capacitor

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26.5

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**26.5 Again we see that capacitance still increases with decreasing d.**

In practice though, there is a lower limit to d before discharge across the plates will occur for a given voltage. So for any given capacitor of separation d, there is a maximum voltage limit.

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**26.5 This limit depends on a factor called the Dielectric Strength.**

This is the maximum Electric Field (V/m) that the material can withstand before its insulating properties break down and the material becomes a conductor. Similar in concept to the spark touching a doorknob and also corona discharge.

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26.5 For parallel plates, the maximum voltage (AKA “working voltage,” “breakdown voltage,” and “rated voltage” is determined by Where Emax is the dielectric strength

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26.5 Table of Dielectric Constants/Strengths p. 812

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**26.5 Types of Commercial Capacitors**

Tubular Capacitors- metallic foil interlaced with wax paper/mylar, rolled into a tube.

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26.5 High voltage Capacitors- interwoven metallic plates immersed in an insulating (silicon) oil.

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**26.5 Electrolytic Capacitors**

Designed for large charges at low voltages. One conducting foil immersed in an conducting fluid (electrolyte) The metal forms a thin insulating oxide layer when voltage is applied.

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26.5 Variable Capacitors- interwoven sets of plates with one set fixed and one set able to be rotated. Typical used for tuning dial circuits (radios, power supplies etc)

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26.5 Quick Quizzes p Examples 26.6, 26.7

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