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Capacitance and Energy Storage: Series, Parallel, and Dielectrics, Slides of Physics

A comprehensive study on capacitance, a physical attribute that describes electrical energy stored on an object. It delves into the calculation of capacitance, the energy stored in a capacitor, and the use of dielectrics in capacitors. The document also covers the molecular model of a dielectric and the effects of adding dielectrics to capacitors. It includes examples, schematics, and formulas for calculating capacitance in various scenarios.

Typology: Slides

2023/2024

Uploaded on 03/18/2024

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Chapter 8
Capacitance
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Chapter 8

Capacitance

Outline

8.1 Capacitors and Capacitance

8.2 Capacitors in Series and in Parallel

8.3 Energy Stored in a Capacitor

8.4 Capacitor with a Dielectric

8.5 Molecular Model of a Dielectric

Capacitance

Capacitance is a physical attribute that describes electrical energy stored on an object. It is the ratio of charge to electrical potential:

C = QV

Capacitance is measured in farads [F], where 1 F = 1 C/V.

The amount of capacitance an object has only depends on material selection, and geometry. It is therefore intrinsically positive (i.e., no negative capacitance).

Capacitors

Capacitors are objects with a geometry designed to yield a specific amount of capacitance.

Capacitors are typically constructed by separating two parallel plates with an insulator, or dielectric.

Parallel Plate Capacitor (1)

+Q

-Q

                    • + + + + + +

E

A

d

Consider two parallel plates with charge +Q and -Q. The potential difference is simply E · d. 7

Parallel Plate Capacitor (2)

The electric field is determined using Gauss’s law as

E = (^) εσ 0 The surface charge density is simply Q / A. Thus, the capacitance is:

C =

Q

V =^

Q

Qd ε− 0 1 A −^1 =^ ε^0

A

d

Note here that the units for ε 0 are also [F/m]. Also note that Q is really charge separated here.

Spherical Capacitor

Consider the spherical capacitor with Q placed on the inner sphere. Potential between spheres is described by a point charge. The potential difference is given by:

V = (^4) πε^1 0

Q ( R^1

1

− R^1

2

Thus, Capacitance is given by:

C = 4 πε (^0) R^ R^1 R^2 2 −^ R 1

Cylindrical Capacitor (1)

Consider the “cylindrical” capacitor. Or more commonly a coaxial cable. 11

Cylindrical Capacitor (3)

| V | = (^2) πε Q 0 L

∫^ R^2

R 1

r

ˆ r · (ˆ rdr )

| V | = Q

2 πε 0 L

∫^ R^2

R 1

dr r

| V | = Q

2 πε 0 L

[ Ln ( r )] R R^21

Thus capacitance is given by:

C = (^) ln^2 ( R πε^0 L 2 / R 1 )

+Q

-Q E

r

Outline

8.1 Capacitors and Capacitance

8.2 Capacitors in Series and in Parallel

8.3 Energy Stored in a Capacitor

8.4 Capacitor with a Dielectric

8.5 Molecular Model of a Dielectric

Combining Capacitors

When several capacitors are connected together, they combine to have an equivalent capacitance.

They can be combined in series (top to bottom), or in parallel (side by side).

Series

Parallel

Capacitors in Parallel (1)

Distributing charge lowers the voltage across all capacitors.

The conductive places come in contact so they are brought to the same electrical potential.

Charge then distributes itself between the two plates.

Parallel

Capacitors in Parallel (3)

More generally, capacitors in parallel are given by: Ceff =

i

Ci

A good way to think of parallel capacitors is that the area adds. Suppose you have two parallel plate capacitors attached in parallel:

Ceff = ε 0^ A^1 + d^ A^2

Parallel

Example 8.

Equivalent Capacitance of a Parallel Network

Find the net capacitance for three capacitors connected in parallel, given their individual capacitance are 1.0, 5.0 and 8.0μ F