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Capacitors

February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 1

Notes !  Exam 1 was a success

•  81% average score including correction set

!  First score update email was sent out over lon-capa •  HW 1 %, exam 1 % (including correction set), clicker % •  If your clicker score is zero, I don’t have you clicker number •  Email said HW1 and HW2, but actual HW score only included

HW1



Capacitors !  Capacitors are devices that store energy in an electric field !  Capacitors are used in many every-day applications •  Heart defibrillators •  Camera flash units •  Touch screens

!  Capacitors are an essential part of electronics !  Capacitors can be micro-sized on

computer chips or super-sized for high power circuits such as FM radio transmitters and exist in a variety of shapes


Capacitance !  A capacitor consists of two separated conductors, usually

called plates, even if these conductors are not simple planes !  If we take apart a typical capacitor, we might find two sheets

of metal foil separated by an insulating layer of Mylar

!  The sandwiched layers can be rolled up with another insulating layer into a compact form that does not look like parallel sheets of metal

Capacitance !  Assume a convenient geometry and then generalize the

results !  The geometry we choose is a parallel plate capacitor, which

consists of two parallel conducting plates, each with area A, separated by a distance d, in a vacuum

!  The capacitor is charged by placing a charge of +q on one plate and –q on the other plate !  The electric potential, ΔV, between the plates is proportional

to the amount of charge on the plates February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 5

Capacitor potential !  Because the plates are conductors, the

charge will distribute itself uniformly over the plates

!  We can use the techniques in Chapter 23 to calculate the potential numerically using a computer

!  The potential has a steep drop between the plates and a gradual drop outside the plates

!  Thus the electric field will be strong between the plates and weak outside the plates


Capacitor potential

!  We can take a slice through the x-y plane

!  The equipotential lines are close together between the plates and far apart outside the plates


Capacitor field

!  Here we show the electric field vectors at regularly spaced grid points in the x-y plane

!  The field between the plates is perpendicular to the plates and has a much larger magnitude than the field outside the plates

!  The field outside the plates is the fringe field


E r( ) = −

∇V r( )

Capacitance !  The potential difference between the two plates is

proportional to the amount of charge on the plates !  The proportionality constant is the the capacitance of the

device

!  The capacitance of a device depends on the area of the plates and the distance between the plates but not on the charge or potential difference

!  The capacitance tells how much charge is required to produce a given potential difference between the plates

!  We can rewrite this equation as


C = q

ΔV

q =CΔV

qCV

=

q =VC

Capacitance !  The units of capacitance are coulombs per volt !  A new unit was assigned to capacitance, named after British

physicist Michael Faraday (1791 – 1867) called the farad (F)

!  One farad represents a very large capacitance !  Typical capacitors have capacitances ranging from 1 pF to

1 μF !  With this definition, we can write the electric permittivity of

free space as


1 F = 1 C

1 V

ε0 = 8.85 ⋅10−12 F/m

Circuits !  An electric circuit consists of wires that connect circuit

elements !  These elements can be capacitors !  Other important elements include resistors, inductors,

ammeters, voltmeters, diodes, and transistors !  Circuits usually need a power source !  A battery can provide a fixed potential difference commonly

called voltage !  An AC power source provides a time-varying potential

difference


Circuit Symbols !  Circuit elements are represented by commonly used

symbols


Charging and Discharging a Capacitor

!  A capacitor is charged by connecting it to a battery to create a circuit

!  Charge flows from the battery to the capacitor until the potential difference across the capacitor is the same as the potential difference across the battery

!  If the capacitor is then disconnected, it will hold its charge and potential difference

!  We can use a circuit diagram to illustrate the charging/discharging process •  Switch position a charges the capacitor

•  Connects the battery across the plates •  Switch position b discharges the capacitor

•  Shorts the plates together February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 16

Parallel Plate Capacitor !  We will consider an ideal parallel plate capacitor !  Two parallel conducting plates in a vacuum with charge +q

on one plate and –q on the other plate

!  The field is constant between the plates and zero outside February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 17

Parallel Plate Capacitor !  We can calculate the field using Gauss’s Law

!  We use the Gaussian surface shown by the dotted red line

!  We add the contributions to integral from the top, the bottom, and the sides

!  The sides are outside the capacitor, so the field is zero !  The top is inside the conductor, so the field is zero !  The bottom part is in the constant field between the plates


E∫∫ •dA = q

ε0

Parallel Plate Capacitor !  The electric field is constant and points downward !  The vector normal to the surface also points downward !  So the integral over the Gaussian surface becomes

!  The electric potential difference across the two plates is !  The path of integration is chosen to be from the negatively

charged plate to the positively charged plate, which gives us !  Combining these equations gives us the capacitance of a

parallel plate capacitor


E∫∫ •dA = EdA∫∫ = E dA∫∫ = EA = q

ε0

ΔV = −

E•ds

i

f

∫

ΔV = − −Eds( ) =

0

d

∫ Ed = qdε0A

C = q

ΔV= ε0A

d

0ACdε=


Demo: Large Capacitance !  Energy stored in this particular capacitor: 90 J

!  This is equivalent to the kinetic energy of a mass of 1 kg moving at a velocity of 13.4 m/s!

E = 12 mv2

v = 2Em

= 2 ⋅90 J1 kg

=13.4 m/s

February 2, 2014 Physics for Scientists&Engineers 2 22

Example - Capacitance of a Parallel Plate Capacitor !  We have a parallel plate capacitor

constructed of two parallel plates, each with area 625 cm2 separated by a distance of 1.00 mm.

!  What is the capacitance of this parallel plate capacitor?

C =ε0Ad

A = 625 cm2 = 0.0625 m2

d = 1.00 mm = 1.00 ⋅10−3 m

C =8.85 ⋅10−12 F/m( ) 0.0625 m2( )

1.00 ⋅10−3 m= 5.53 ⋅10-10 F

C = 0.553 nFA parallel plate capacitor constructed out of square conducting plates 25 cm x 25 cm (=625 cm2) separated by 1 mm produces a capacitor with a capacitance of about 0.55 nF

February 2, 2014 Physics for Scientists&Engineers 2 23

Example 2 - Capacitance of a Parallel Plate Capacitor !  We have a parallel plate

capacitor constructed of two parallel plates separated by a distance of 1.00 mm.

!  What area is required to produce a capacitance of 0.55 F?

C =ε0Ad

d = 1.00 mm = 1.00 ⋅10−3 m

A =dCε0

=1.00 ⋅10−3 m( ) 0.55 F( )

8.85 ⋅10−12 F/m( ) = 0.62 ⋅108 m2

A parallel plate capacitor constructed out of square conducting plates 7.9 km x 7.9 km (4.9 miles x 4.9 miles) separated by 1 mm produces a capacitor with a capacitance of 0.55 F

capacitors - michigan state university€¦ · physicist michael faraday (1791 – 1867) called the...

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