Electric potential and capacitors
February 10, 2014
Physics for Scientists & Engineers 2, Chapter 24
1
Electric Potential Energy
!  For a conservative force, the work is path-independent
!  Electric potential energy, U, is defined in terms of the work
done by the electric field, We, when the system changes its
configuration from some initial configuration to some final
configuration
•  The change in the electric potential energy is the negative of the
work done by the electric field
ΔU = U f − U i = −We
U i is the initial electric potential energy
U f is the final electric potential energy
!  The work done on a charge q in an electric field is
We = ∫
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f
i
! !
f !
!
qE ⋅ds = q ∫ E ⋅ds
i
Physics for Scientists & Engineers 2, Chapter 23
2
Equipotential surfaces
!  A constant electric field has straight, equally spaced, and
parallel field lines, which produces equipotential surfaces in
the form of parallel planes
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Physics for Scientists & Engineers 2, Chapter 23
4
Single Point Charge
!  The electric field lines for point charges are radial
!  The equipotential surfaces are concentric spheres in 3D
!  The equipotential lines take the form of concentric circles in 2D
Positive charge
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Negative charge
Physics for Scientists & Engineers 2, Chapter 23
5
Electric Potential Difference ΔV
!  The electric potential difference between an initial point i
and final point f is
U f U i ΔU
ΔV = Vf −Vi =
− =
q
q
q
!  Work and potential are related through
We
ΔV = −
q
!  The units of electric potential are
1J
1V=
1C
!  We express the electric field as
[F] N J/m V
[E]=
= =
=
[q] C
C m
2/10/14
Physics for Scientists & Engineers 2, Chapter 23
6
Graphical Extraction of the Electric Field
!  Calculate the magnitude of the electric field at point P.
!  To perform this task, we draw a line through point P
perpendicular to the
equipotential line reaching
from the next line down to
the next line up in potential.
!  Measure the length Δs of
this line.
!  The magnitude of the
electric field can be
approximated as
ES = −
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ΔV
Δs
Chapter 23
7
System of Point Charges
!  We calculate the electric potential from a system of n point
charges by adding the potential functions from each charge
kqi
V = ∑Vi = ∑
i=1
i=1 ri
n
n
!  This summation produces an electric potential at all points
in space – a scalar function
!  Calculating the electric potential from a group of point
charges is usually much simpler than calculating the electric
field
•  It’s a scalar
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Physics for Scientists & Engineers 2, Chapter 23
9
Electric Potential Energy for a
System of Particles
!  Now we consider a system of point charges that produce the electric
potential themselves.
!  We begin with a system of charges that are infinitely far apart, U = 0,
by convention.
!  To bring these charges into proximity with each other, we must do
work on the charges, which changes the electric potential energy of the
system.
!  We then bring in point charge q1. Because there is no electric field and
no corresponding electric force, this action requires no work to be
done on the charge.
!  Keeping this charge (q1) stationary, we bring the second point charge
(q2) in from infinity to a distance r from q1.
!  This requires work q2V1(r).
!  We then bring in all other charges
!  The total energy of a system of 3 charges is
U123 = U12 +U13 +U 23
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k
U123 = (q1q2 + q1q3 + q2 q3 )
d
Chapter 23
10
Review: Capacitance
!  The definition of capacitance is
•  Capacitance is a property of the capacitor
q
C=
V
!  The capacitance of a parallel plate capacitor
depends on the plate area and the plate separation and is given by
•  A is the area of each plate
ε0 A
•  d is the distance between the plates
C=
Physics for Scientists & Engineers 2
d
11
Cylindrical and spherical capacitor
!  The capacitance of a
cylindrical capacitor
with length L,
inner radius r1 and
outer radius r2 is
2πε 0 L
C=
ln ( r2 / r1 )
!  The capacitance of a spherical capactitor
with inner radius r1 and outer radius r2
is
rr
C = 4πε 0
1 2
r2 − r1
!  The capacitance of an isolated sphere
is
C = 4πε 0 R
February 10, 2014
Physics for Scientists & Engineers 2, Chapter 24
12
Review
!  The electric potential energy stored in a capacitor
is given by
1
U = CV 2
2
!  The field energy density stored in a parallel plate capacitor is
given by
1 ⎛V ⎞
u = ε0 ⎜ ⎟
2 ⎝d⎠
2
!  The field energy density in general is
1
u = ε0E2
2
September 17, 2008
Physics for Scientists & Engineers 2, Lecture 14
13
Dielectric
!  Inserting a dielectric between the capacitor plates tends to
partially cancel the original electric field
Eair
q
q
E=
=
=
κ κε 0 A ε A
!  The capacitance goes up
qd
ΔV = Ed =
κε 0 A
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q κε 0 A
C=
=
ΔV
d
C = κ Cair
Physics for Scientists & Engineers 2, Chapter 24
14
Circuit Symbols
!  Circuit elements are represented by commonly used
symbols.
February 10, 2014
Chapter 24
15
Capacitors in series and in parallel
!  The equivalent capacitance for n capacitors in parallel is
n
Ceq = ∑ Ci
=
i=1
!  The equivalent capacitance for n capacitors in series is
n
1
1
=∑
Ceq i=1 Ci
=
!  For two capacitors
in series:
C1C2
Ceq =
C1 + C2
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Physics for Scientists & Engineers 2, Chapter 23
16
Supercapacitor / Ultracapacitor
!  Supercapacitors (ultracapacitors) are made using a material
with a very large surface area between the capacitor plates
!  Two layers of activated charcoal are given opposite charge
and are separated by an insulating material
!  This produces a capacitor with capacitance millions of times
larger than ordinary capacitors
!  However, the potential difference can only be 2 to 3 V
February 10, 2014
Physics for Scientists & Engineers 2, Chapter 24
19