Transcript capacitance

20.4 Obtaining Electric Field
from Electric Potential

Assume, to start, that E has only an x
component
dV
E  ds becomes E x dx and E x  
dx

Similar statements would apply to the y and z
components
Equipotential surfaces must always be
perpendicular to the electric field lines
passing through them
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For Three Dimensions
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In general, the electric potential is a
function of all three dimensions
Given V (x, y, z) you can find Ex, Ey and
Ez as partial derivatives
V
Ex  
x
V
Ey  
y
V
Ez  
z
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Electric Field and Potential of a
Dipole
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The equipotential lines
are the dashed blue
lines
The electric field lines
are the brown lines
The equipotential lines
are everywhere
perpendicular to the
field lines
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20.5 Electric Potential for a
Continuous Charge Distribution
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Consider a small
charge element dq
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Treat it as a point
charge
The potential at
some point due to
this charge element
is
dq
dV  ke
r
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V for a Continuous
Charge Distribution, cont
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To find the total potential, you need to
integrate to include the contributions
from all the elements
dq
V  ke 
r
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This value for V uses the reference of
V = 0 when P is infinitely far away from
the charge distributions
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V for a Uniformly
Charged Sphere
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A solid sphere of
radius R and total
charge Q
Q
For r > R, V  k e
r
For r < R,

keQ 2
2
VD  VC 
R

r
2R 3
keQ 
r2 
VD 
3  2 
3R 
R 

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V for a Uniformly
Charged Sphere, Graph

The curve for VD is
for the potential
inside the curve
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It is parabolic
It joins smoothly with
the curve for VB
The curve for VB is
for the potential
outside the sphere
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It is a hyperbola
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20.6 V Due to a
Charged Conductor
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Consider two points on
the surface of the
charged conductor as
shown
E is always
perpendicular to to the
displacement ds
Therefore, E  ds = 0
Therefore, the potential
difference between A
and B is also zero
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V Due to a
Charged Conductor, cont
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V is constant everywhere on the surface of a
charged conductor in equilibrium
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DV = 0 between any two points on the surface
The surface of any charged conductor in
electrostatic equilibrium is an equipotential
surface
Because the electric field is zero inside the
conductor, we conclude that the electric
potential is constant everywhere inside the
conductor and equal to the value at the
surface
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E and V of a sphere conductor
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The electric potential is a
function of r
The electric field is a function
of r2
The effect of a charge on the
space surrounding it

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The charge sets up a vector
electric field which is related to
the force
The charge sets up a scalar
potential which is related to the
energy
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Two charged sphere conductors
connected by a conducting wire
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The charge density is high
where the radius of
curvature is small
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And low where the radius of
curvature is large
The electric field is large
near the convex points
having small radii of
curvature and reaches very
high values at sharp points
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Cavity in a Conductor
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Assume an
irregularly shaped
cavity is inside a
conductor
Assume no charges
are inside the cavity
The electric field
inside the conductor
is must be zero
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Cavity in a Conductor, cont
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The electric field inside does not
depend on the charge distribution on
the outside surface of the conductor
For all paths between A and B,
VB  VA    E  ds  0
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A cavity surrounded by conducting walls
is a field-free region as long as no
charges are inside the cavity
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20.7 Capacitors
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Capacitors are devices that store
electric charge
The capacitor is the first example of a
circuit element
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A circuit generally consists of a number of
electrical components (called circuit
elements) connected together by
conducting wires forming one or more
closed loops
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Makeup of a Capacitor
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A capacitor consists of two
conductors
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When the conductors are
charged, they carry charges of
equal magnitude and opposite
directions
A potential difference exists
between the conductors due
to the charge
The capacitor stores charge
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Definition of Capacitance

The capacitance, C, of a capacitor is
defined as the ratio of the magnitude of
the charge on either conductor to the
potential difference between the
conductors
Q
C
DV
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The SI unit of capacitance is a farad (F)
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More About Capacitance
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Capacitance will always be a positive quantity
The capacitance of a given capacitor is
constant
The capacitance is a measure of the
capacitor’s ability to store charge
The Farad is a large unit, typically you will
see microfarads (mF) and picofarads (pF)
The capacitance of a device depends on the
geometric arrangement of the conductors
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Parallel Plate Capacitor
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Each plate is connected
to a terminal of the
battery
If the capacitor is
initially uncharged, the
battery establishes an
electric field in the
connecting wires
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Capacitance – Parallel Plates
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The charge density on the plates is
s = Q/A
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A is the area of each plate, which are equal
Q is the charge on each plate, equal with
opposite signs
The electric field is uniform between the
plates and zero elsewhere
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Parallel Plate Assumptions
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The assumption that the electric field is uniform is
valid in the central region, but not at the ends of the
plates
If the separation between the plates is small
compared with the length of the plates, the effect of
the non-uniform field can be ignored
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Capacitance –
Parallel Plates, cont.

The capacitance is proportional to the
area of its plates and inversely
proportional to the plate separation
o A
Q
Q
Q
C



DV Ed Qd /  o A
d
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A parallel-plate Capacitor
connected to a Battery
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Consider the circuit to be a system
Before the switch is closed, the
energy is stored as chemical energy
in the battery
When the switch is closed, the energy
is transformed from chemical to
electric potential energy
The electric potential energy is
related to the separation of the
positive and negative charges on the
plates
A capacitor can be described as a
device that stores energy as well as
charge
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Capacitance –
Isolated Sphere
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Assume a spherical charged conductor
Assume V = 0 at infinity
Q
Q
R
C


 4 o R
DV keQ / R ke
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Note, this is independent of the charge
and the potential difference
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Capacitance of a
Cylindrical Capacitor
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From Gauss’ Law,
the field between
the cylinders is
E = 2 ke l / r
DV = -2 ke l ln (b/a)
The capacitance
becomes
Q
C

DV 2ke ln b

a

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20.8 Circuit Symbols
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A circuit diagram is a
simplified
representation of an
actual circuit
Circuit symbols are
used to represent the
various elements
Lines are used to
represent wires
The battery’s positive
terminal is indicated by
the longer line
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Capacitors in Parallel
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When capacitors are
first connected in
the circuit, electrons
are transferred from
the left plates
through the battery
to the right plate,
leaving the left plate
positively charged
and the right plate
negatively charged
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Capacitors in Parallel, 2
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The flow of charges ceases when the voltage
across the capacitors equals that of the
battery
The capacitors reach their maximum charge
when the flow of charge ceases
The total charge is equal to the sum of the
charges on the capacitors
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Q = Q 1 + Q2
The potential difference across the capacitors
is the same
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And each is equal to the voltage of the battery
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Capacitors in Parallel, 3
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The capacitors can
be replaced with
one capacitor with a
capacitance of Ceq
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The equivalent
capacitor must have
exactly the same
external effect on the
circuit as the original
capacitors
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Capacitors in Parallel, final
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Ceq = C1 + C2 + …
The equivalent capacitance of a parallel
combination of capacitors is the
algebraic sum of the individual
capacitances and is larger than any of
the individual capacitances
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Capacitors in Series
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When a battery is
connected to the
circuit, electrons are
transferred from the
left plate of C1 to the
right plate of C2
through the battery
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Capacitors in Series, 2
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As this negative charge accumulates on
the right plate of C2, an equivalent
amount of negative charge is removed
from the left plate of C2, leaving it with
an excess positive charge
All of the right plates gain charges of
–Q and all the left plates have charges
of +Q
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Capacitors in
Series, 3
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An equivalent capacitor
can be found that
performs the same
function as the series
combination
The potential differences
add up to the battery
voltage
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Capacitors in Series, final
Q  Q1  Q2 
DV  V1  V2 
1
1
1



Ceq C1 C2

The equivalent capacitance of a series
combination is always less than any
individual capacitor in the combination
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Summary and Hints
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Be careful with the choice of units
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In SI, capacitance is in F, distance is in m and the
potential differences in V
Electric fields can be in V/m or N/c
When two or more capacitors are connected
in parallel, the potential differences across
them are the same
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The charge on each capacitor is proportional to its
capacitance
The capacitors add directly to give the equivalent
capacitance
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Summary and Hints, cont
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When two or more capacitors are
connected in series, they carry the
same charge, but the potential
differences across them are not the
same
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The capacitances add as reciprocals and
the equivalent capacitance is always less
than the smallest individual capacitor
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Equivalent
Capacitance, Example
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The 1.0mF and 3.0mF are in parallel as are the 6.0mF and
2.0mF
These parallel combinations are in series with the
capacitors next to them
The series combinations are in parallel and the final
equivalent capacitance can be found
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