What is a Capacitor?

  • A capacitor is an electric device used to store electrical energy

    • Two conducting plates

    • Insulating material between (dielectric)

  • Place opposite charges on each plate

  • Develop a potential difference across the plates

  • Energy is stored in the electric field between the plates

    Conductive Parallel Plates Dielectric Voltage Vc Electrical Charge


  • Capacitance (C) is the ratio of the charge separated on the plates of a capacitor to the potential difference between the plates

  • Units of capacitance are coulombs/volt, or farads (F)

    • A farad is a very large amount of capacitance

    To calculate capacitance, need to determine the E field between the
plates. We use Gauss' Law, with one end of our gaussian surface closed
inside one plate, and the other closed in the region between the
plates Gauss: so q=eoEA or COA Potential = force/q x distance:
Capacitance for Parallel Plates V=Ed separation d E-field Tota charge
q on inside of plate eoEA Ed

Calculating Capacitance

  1. Assume a charge of +Q and -Q on each conductor

  2. Find the electric field between the conductors (Gauss's Law)

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Example 1: Parallel Plates


  • Determine the capacitance between identical parallel plates of area A separated by a distance d
  1. Assume +Q and -Q

  2. C:\\266298A5\\73477446-49B2-471B-AFDD-BCD03931DCDD\_files\\image184.png

  3. C:\\266298A5\\73477446-49B2-471B-AFDD-BCD03931DCDD\_files\\image185.png



  4. C:\\266298A5\\73477446-49B2-471B-AFDD-BCD03931DCDD\_files\\image188.png

Example 2: Cylindrical Capacitor

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  1. Assume +Q and -Q
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  2. C:\\266298A5\\73477446-49B2-471B-AFDD-BCD03931DCDD\_files\\image193.png



  3. C:\\266298A5\\73477446-49B2-471B-AFDD-BCD03931DCDD\_files\\image195.png

Example 3: Spherical Capacitor


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  1. Assume +Q and -Q
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  2. C:\\266298A5\\73477446-49B2-471B-AFDD-BCD03931DCDD\_files\\image200.png



  3. C:\\266298A5\\73477446-49B2-471B-AFDD-BCD03931DCDD\_files\\image203.png

Energy Stored in a Capacitor

  • Work is done charging a capacitor, allowing the capacitor to store energy.

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    charge (Q) slope capacitance the ability Of the plates to hold a
certain amount Of charge for a given voltage Area energy stored in the
electric field between the plates voltage (V)

Field Energy Density

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    1 ЕОА


  • Insulating Materials

    —Oind Figure 26.24 (a) When a dielectric is polarized, the dipole
moments of the molecules in the dielectric are partially aligned with
the external field Ea. (b) This polarization causes an induced
negative surface charge on one side of the dielectric and an equal
induced positive surface charge on the opposite side. This separation
of charge results in a reduction in the net electric field within the

    Area A Area A Dielectric (Insulator)

Dielectric Constant (κ)

Capacitance = C Capacitance = C' vacuum permittivity = Eo dielectric
 permittivity = g

Forair, polarization — kE The capacitance is increased by the C
 factor k.

Example 4: Capacitors in Series


  • Determine the equivalent capacitance of two capacitors in series

  • C:\\266298A5\\73477446-49B2-471B-AFDD-BCD03931DCDD\_files\\image218.png

  • C:\\266298A5\\73477446-49B2-471B-AFDD-BCD03931DCDD\_files\\image219.png

    。긕曰卜-•曰曰卜刁 1 1 Cl C2 Tot Ⅳ-1

Example 5: Capacitors in Parallel


  • Determine the equivalent capacitance of two capacitors in parallel

  • C:\\266298A5\\73477446-49B2-471B-AFDD-BCD03931DCDD\_files\\image222.png

    تن تنذ نذ ذ Tot ٤٣٣ 2٠٠٠٣٤٨ 1٣٣٨

Example 6: Capacitance

  • A capacitor stores 3 microcoulombs of charge with a potential difference of 1.5 volts across the plates. What is its capacitance? How much energy is stored in the capacitor?

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  • C:\\266298A5\\73477446-49B2-471B-AFDD-BCD03931DCDD\_files\\image225.png

Example 7: Charge on a Capacitor

How much charge sits on the top plate of a 200 nF capacitor when
 charged to a potential difference of 6 volts? — How much energy is
 stored in the capacitor when it is fully charged? — How much energy is
 stored in the capacitor when the voltage across its plates is 3 volts?
 Q-zcv = l. 2•lö C u4cvtg

Example 8: Designing a Capacitor

How far apart should the plates of an air-gap capacitor be if the
 area of the top plate is 5 x 10-4 m2 and the capacitor must store 50
 mJ of charge at an operating potential difference of 100 volts?
 Us.osø- VS 100 v zC.os) zu

Example 9: Calculating Capacitance

Find the capacitance of two parallel plates of length 1 mm and width
 2 mm if they are separated by 3 micrometers of air. What would the
 device's capacitance be if the 3 micrometers of air were replaced by 3
 micrometers of Si02 (glass) which has a dielectric constant (relative
 permittivity) of 3.9? 3-10-6 44 z 3.q - 2.300 3•10-6

Example 10: Two Conducting Spheres

Two conducting spheres, each with charge Q, are connected by a wire
 as shown. Do any charges flow between the spheres? How do their
 potentials compare? Mfré»r g.\! Q

Example 11: Inserting a Dielectric

An air-gap parallel plate capacitor is connected to a source of
 constant potential difference. Inserting a dielectric between the
 plates of the capacitor increases which of the following quantities?
 (Choose all that apply) JA) Charge on the capacitor KB) Voltage across
 the capacitor VC) Capacitance of the capacitor JD) Energy stored in
 the capacitor. •->c- X E) Electric field strength between the plates

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