Capacitors - Michigan State University€¦ · physicist Michael Faraday (1791 – 1867) called the...
Transcript of Capacitors - Michigan State University€¦ · physicist Michael Faraday (1791 – 1867) called the...
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
February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 2
February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 3
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
February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 4
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
February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 7
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
February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 8
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
February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 9
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
February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 10
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
February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 12
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
February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 14
Circuit Symbols ! Circuit elements are represented by commonly used
symbols
February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 15
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
February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 18
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
February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 19
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=
February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 20
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