Electric Potential. Capacitors (Chapters 28,...
Transcript of Electric Potential. Capacitors (Chapters 28,...
Electric Potential. Capacitors
(Chapters 28, 29)
• Electric potential energy, U
• Electric potential energy in a constant field
• Conservation of energy
• Electric potential, V
• Relation to the electric field strength
• The potential of a point charge
• Calculating the potential of
• Multiple point charges
• Continuous distributions of charge
• Capacitors
• Capacitance, C
• Simple circuits
• Energy
• Dielectrics
Electric Potential Energy – Constant electric field
• The electrostatic force is a conservative force, so it is associated with an electrical
potential energy, U: decreasing from positive to negative charges
• To find a qualitative expression, recall that the work done by a conservative force is
always equal to the negative of the change in potential energy
• For simplicity, let’s start with the particular case of a uniform electric field E like
between two metallic parallel plates charged with unlike charges (that is, a capacitor)
• Consider a positive charge q moving parallel
with the field E from a to b: the field does
positive work on it
• Therefore, the change in electrical potential
energy when the field is uniform is
How is this useful in describing the motion of
the charged particle?
b aU U U W qE x
eW F x qE x
+
+
+
+
+
+
+
+
+
+
+
+
–
–
–
–
–
–
–
–
–
–
–
–
E
eF qE+
Δx
q
a b
Electric Potential Energy – Spontaneous Charge Movements
• In nature, systems allowed to evolve without constraints will have the tendency to
adopt configurations corresponding to a minimum of energy
• Accordingly, if a charge is allowed to move spontaneously in an electric field, the
motion will be such that the electric potential energy decreases, that is
• Therefore, if the field E is uniform and positive (that is, it points to the right, in
positive x direction), we have
E
0
0 0
0 x
U qE x
q
q x
x
Positive charge moves to the right
(same as the direction of the field)
• Notice that this behavior – derived by analyzing the spontaneous change in
potential energy – is consistent with the direction of motion expected based on the
electric force acted on the charge by the electric field
0f iU U U
F
Negative charge moves to the right
(against the direction of the field)
Fx
+
–
Electric Potential Energy – Energy conservation
• In conclusion, any electric charge released in the electric field experiences a force
and accelerates, gaining kinetic energy K on behalf of the electric potential energy of
the charge-field combination
Ex: The motion of a charge in a constant electric field can be
compared to the motion of a mass in the constant gravitational field:
• A positive charge q moving a vertical distance Δy loses electric
potential energy qEΔy, which gets converted into kinetic energy
• A mass m falling a vertical distance Δy loses gravitational
potential energy mgΔy, which gets converted into kinetic energy
K qE y
nonconservativeK U W
net potential energy including
gravitational, elastic and electric
Quiz 1: A positive charge of 1 C moves 1 m in a uniform field E.
What is the field E if the charge gains the same kinetic energy as a 1
kg mass freely falling 1 m?
a) 1.0 N/C b) 9.8 N/C c) 4.9 N/C
+
eF qE
electric gravitational
q m
gF mg
K mg y
Δy
• Hence, if there is no nonconservative force (like friction), we have K + U = const.
• Conservation of net energy still works as discusses
in PHY181, except that the potential energy U must
also include the electric potential energy (besides the
gravitational and elastic potential energies):
+ + + + + + + + + + + + + + +
– – – – – – – – – – – – – – – –
Electric Potential – Potential difference
• Hence, the potential difference ΔV between points a and b in an electric field is the
change in the potential energy of a test charge q moved from a to b divided by the
charge:
• Both electric potential energy and potential difference are scalar quantities
• Notice that, by multiplying the charge with the potential difference it moves through,
we obtain the energy exchanged between the charge and the field in the process
a b
UV V V U q V
q
SIVolts V J CV
Def: The electric potential V is the electric potential energy per unit charge
• In order to describe the energy of the electric field rather than the energy of a
particular charge in the field, we introduce
Ex: In a capacitor, the field E is uniform, so the
potential difference between any two points a-b
at a separation Δx along the field is given by
a bab V U q qVV E x q E x
d
Quiz 2: What is the potential difference between the positive plate and the point c inside the
capacitor represented above?
V+
V–
a
b
Δx EΔVab
a) E/2d b) Ed c) E2d d) Ed/2
c
+ + + + + + + + + + + + + + +
– – – – – – – – – – – – – – – –
• Since the potential is not absolute, in general we will
be working with potential differences:
• Later we’ll even denote it V, and we’ll call it voltage,
but we’ll still refer to a potential difference
• Then, if a charge moves through a difference of potential between two
points a and b, the work done on it by the electric field is
b aW q V q V V
Ex: A 1.5 V battery has a potential difference between its terminals of 1.5V, such
that the work done by the battery to move a unit positive charge across it is 1.5 J.
Electric Potential – Comments
b a ab
UV V V V
q
alternative notation
High V
Low V (ground)
0 0W q V V
0 0W q V V E
• A positive charge released in a field will move from a high potential to a
low potential, while a negative charge will move from low to high
+
–
• The electric potential is the potential energy per unit charge in the same way the
electric field is the electric force per unit charge: so, while the force and the potential
energy depends on the test charge q0, the field and the potential depend only on the
source of field:
• Hence, everywhere in the space surrounding a charge, each point is characterized by
a field E (vector) and a potential V (scalar). (However, the field strength is absolute
while the potential is relative to a “ground” where the potential is set to be zero)
• Based on the definition ΔU = qΔV = –W, we see that, if a particle moves between
two points in space with the same potential, the electric force will do zero work
• In conclusion:
0V U q0qE F
Electric Potential – Electric field and electric potential
1. When a positive charge is placed in an
electric field
• It moves in the direction of the field
• The electrical potential energy decreases
• It moves from a point of higher potential
to a point of lower potential (ΔV < 0)
• Its kinetic energy increases qU V
2. When a negative charge is placed in an
electric field
• It moves opposite to the field
• The electrical potential energy decreases
• It moves from a point of lower potential
to a point of higher potential (ΔV > 0)
• Its kinetic energy increases qU V
Problem:
1. Motion of a proton in an electric field: An proton moves 2.5 cm parallel to a uniform
electric field of E = 200 N/C. Assume the electric field in the positive direction.
a) How much work is done by the field on the proton?
b) What change occurs in the potential energy of the proton?
c) What potential difference did the proton move through?
d) If the proton is released from rest what is its final speed?
• Since “everywhere in the space surrounding a charge, each point is characterized by
a field E (vector) and a potential V (scalar)”, it is natural to ask: Is there a relationship
between the two? Yes, as following:
cos cosedU dW F dx qEdxq
dU qdV
cosEdx q
0,
maxdV dV Edx
Electric Potential – Relationship between E and V
1. Consider a positive test charge q in an electric
field: since the vector electric field is tangent to the
line, the electric force on q is also tangent
2. Say that q is moved a small step dx
a) perpendicular on the field line, the field won’t do
any work:
b) parallel with the field line, the field will do a
maximum work:
electric
field line
+
E
F qEdx
0dV
maxdV
3. Therefore, the vector field is oriented such that the change in potential is maximum.
q
0dV
0dV
θ
dVE slope
dx
4. Hence, if a field is given by its position dependent potential,
and V(x) is plotted along a certain x-axis, the field in every
point is given by the negative of the slope of the V(x) graph
maxdVdU = 0 → dV = 0
dU = max → dV = max
Demonstration: for an elementary step dx in the field making an angle θ with the force
• The fact that the electric field points in the direction corresponding to the fastest
decrement in potential can be based on the observation we made in PHY181 about
how any conservative force is given by the gradient of its potential energy
• In the case of electric forces we have:
such that
, , , ,e
dU dU dU dV dV dVF qE q
dx dy dz dx dy dz
, , x y zE dV dx E dV dy E dV dz
E V
r r
dVE dV E dr
dr
Quiz 3: How does the potential depend on the position in the interior of any statically charged
conductor?
Electric Potential – Potential gradient
gradient
a) It is always zero
b) It increases from center to the surface of the conductor
c) It is constant
• Conversely, when the field is integrated between two points, one obtains the
potential difference between the two points, if we know the potential, we can
calculate the field and vice-versa
Ex: For a radial field
Electric Potential – Charged Conductors
• In the previous chapter, we learned that the electric field inside a conductor in
electrostatic equilibrium is zero. What about the electric potential?
• All points on the surface of a charged conductor in electrostatic equilibrium are at
the same potential which can be taken by convention to be zero (ground)
• A volume, or surface, or line with points at the same electric potential is called as
having an equipotential
• As shown by the relationship between E and V, the electric field at every point on
an equipotential surface is perpendicular to the surface, and the lines of electric field
are everywhere perpendicular on equipotential lines
Ex: Consider an arbitrarily shaped conductor with an excess of positive
charge in electrostatic equilibrium
• All of the charge resides at the surface, predominantly on pointier sides
• The electric field is zero inside the conductor, and nonzero and
perpendicular on the surface just outside the surface
• The electric potential is a constant everywhere on the surface of the
conductor, so no work is necessary to move charges on the surface
between any two points A and B: consistent with the fact that the field – and
so the electric force on the moving charge – is perpendicular on the surface
• The potential everywhere inside the conductor is constant and equal to its
value at the surface so the bulk is an equipotential volume
Electric Potential – Potential energy of a pair of point charges
• The work done by a field created by a point charge q on a charge q0 moved
between two points is, for any path ℓ (since electric forces are conservative),
2 2
1 1
0 2cos
r r
e e e
r r
drW F d F d F dr kqq
r
0U rr
qqk
Def: The electric potential energy of a pair of point charges separated by a
distance r is equal to the work done by one of the charges to bring the other one
from infinity to the distance r.
• Recall that the potential energy doesn’t make sense if it not with respect to a certain
point, a “ground” where the potential energy is zero
• So, since the expression above is zero when r → ∞:
• Therefore, the potential energy of the
point charge q0 at distance r from q is
0 2 1
2 1
1 1W kqq U U
r r
path
E
+
eF d
+ dr
q
q0 r1
r2
V1 V2
θ
Electric Potential – Point Source
0
U rV r k
q r
q
Comments:
• This expression gives us the work that the
field would do in order to bring a unit charge
from infinitely far away to a distance r from q
• Since it is associated with the electric field, a
potential exists at some point in space
irrespective if there is a test charge at that point
• Unlike the electric field which decreases like
1/r2, the electric potential decreases like 1/r
• A positive charge creates a positive potential
and a negative charge a negative potential
q
0
1
4 r
qV
Ex: The field strength E
in the vicinity of a
positive charge at
position r = 0 decreases
faster than the
respective potential V
relative to a point at r = ∞
V kr
q
2E k
q
r
slope of the V(r) curve for every r
+
E
+ q
r1
r2
V1 V2
1
1
qV k
r
2
2
qV k
r
• Then, based on the definition of the potential, we can
find the potential produced by a point charge source q
at a distance r, taking the ground to be at an infinite
distance from the source charge:
Equipotential Maps – Another way to visually represent fields
• The equipotential surfaces and lines separated by equal potential steps can be used
to complement the electric field lines to picture out the electric field. For instance:
1. Point charges: the equipotential surfaces are a family of spheres centered at the
point charge: the potential decreases in magnitude on larger and larger spheres, and
the electric field lines are everywhere perpendicular to the equipotential surfaces
90°
• The stronger is the field, the faster the potential varies, and the closer are the
equipotential lines
• Notice that the equipotential lines can be compared with the elevation contours on a
topographic map: thus, the potential around a positive charge forms a peak of
potential and around a negative charge a well (or sink) of potential
equipotential spheres
V increasing (peak) V decreasing (sink)
The field is oriented
in the direction of
maximum V change
+ –
2. Dipole: if the map of potential is seen
from above, the contours of equipotential
lines appear projected on a plane (x-y)
• If a third axis perpendicular on the plane
of the dipole is considered – representing
the potential V –, the map has potential
peaks in the position of positive charges and
wells in the position of negative charges
• Notice that a profile projection of the map
shows the potential variation with position
in the V-x plane
V
x
Top view
Side view
Equipotential Maps – Two Point Charges
x
y
x
y
V
+ –
+ –
Quiz 4: An electron is released from rest at x = 2 m in the potential shown. What does the
electron do right after being released?
a) Stays at x = 2 m.
b) Move to the right (+x) at steady speed.
c) Move to the right with increasing speed.
d) Move to the left (–x) at steady speed.
e) Move to the left with increasing speed.
–
Quiz 5: Which of the sets of equipotential surfaces
below a horizontal electric field given on the right?
(c is a constant)
a) b) c)
d) e) f)
ˆE cxi
1. Multiple point charges: we can apply once again the Superposition Principle:
• The total electric potential at some point P due to n point charges at distances Ri
from P is the algebraic sum of the electric potentials due to the individual charges
• The algebraic sum is used because potentials are scalar quantities – simpler
calculations than in the case of net electric fields.
10
1
4
ni
net i
i i i
qV V
R
2. Continuous distribution of charges: we’ll consider only sources with very
simple shapes and uniform distribution of charges.
• Then, the potential in a point of coordinates (x,y,z) is:
where R is the distance to each element of charge at position (x’,y’,z’).
• Notice that the integration is simpler than in the case of the field strength since
there is no vector involved.
2 2 2
0 0
1 1
4 4
dq dqV
R x x y y z z
Electric Potential – Multiple point charges and continuous sources
3. Continuous linear charge - ring: Consider once again the uniformly charged ring in the
xy-plane. The ring has radius a and a charge q distributed evenly along its circumference.
a) Calculate the potential in point P (0,0,h) above its center.
b) Comment on what happens if the distance h is much larger than the radius a.
4. Continuous linear charge - line: Consider a charge q distributed with uniform density
along y-axis , in the interval (–a, a).
a) Write out the charge dq per element of length dy of the line in terms of q and a.
b) Calculate the potential in a point P (r,0) in terms of k, q, and radial distance r.
Useful integral:
1 22 2
1 2 1 22 2 2 2
ln
a
a
r a ady
r y r a a
Problems:
2. Net electric potential by superposition: Three charges with
magnitudes and signs given on the figure in terms of q = 3 μC are
located in three corners of a square of side a = 2 cm.
a) Find the net electric potentials V1,2 in the center of the square,
and at the upper right corner of a square
b) A test charge q/2 is moved between the center of the square and
the upper right corner. Calculate the work done by the net field of
the other charges on the test charge.
a
+3q
+q –2q
a V2 = ?
– +
+
V1 = ?
Problems:
5. Potential of a point charge: Knowing the electric field of a point charge in a point r, re-
obtain the potential at that position.
6. Potential inside a capacitor revisited: Knowing the electric field, calculate the potential in
between the plates of a parallel plate capacitor with a surface charge density σ.
8. Metallic sphere: Consider a charge q deposited on a metallic
sphere – see figure. Confirm the adjacent graphical
representations by calculating the potential inside and outside
the sphere using the electric field calculated using Gauss’s Law?
7. Metallic cylinder: Consider a charge deposited on a very long long metallic cylinder with
linear density λ.
a) Use the electric field calculated using Gauss’s law to find the difference of the potential
between two points outside the cylinder at distances r1 and r2 from its axis.
b) Where would be a suitable ground for this particular potential?
Capacitors – What are they?
• Any two closely separated conductors will form
a capacitor: in order to charge it, one can use a
battery to do work in order to transfer a charge Q
from one conductor to the other, such that one
conductor will have a deficit +Q and the other a
surplus –Q of electrons.
• As a result an electric field will appear between
the conductors: electric field means ability to do
work, or stored energy
• A capacitor is an electric device used in a variety of electric circuits
• Its functionality is based on the storage of energy associated with the electric field
between two symmetric distributions of unlike charges insulated from one another
• Capacitors come in various arrangements of conductors: parallel plates, concentric
spheres, coaxial cylinders etc
• While developing the generic ideas about any such capacitor architectures, in PHY
182 we’ll focus on the simplest (and most common) type: the parallel plate
capacitor
Capacitors – Capacitance
• The ability of a capacitor to store charge (that is electric field and energy) is given
by its capacitance:
QC
V
Comments:
• One Farad is a very large capacitance: so
most often we’ll see µF, nF or pF
• The capacitance of a capacitor is a
characteristic of the device, and does not
depend on the difference of potential applied
across the plates
• Thus, according to the definition, if a
difference of potential V is applied across the
plates of a capacitor of capacitance C, it will
store a maximum charge Q = CV
Def: The capacitance, C, of a capacitor is defined as the ratio between the amount
of electric charge Q it holds and the potential difference V between its plates
SI1 Farad (F) 1C 1VC
+ + + + + + + + + + + + + + +
– – – – – – – – – – – – – – – –
d
+Q
–Q
EV
Q CV C V V CEd
Problem:
9. Spherical capacitor: A spherical capacitor consists in an
interior sphere of radius ra in the center of a spherical shell
of inner radius rb. Calculate the capacitance in terms of ra,
rb and constants.
• For a parallel-plate capacitor filled with air, we can
easily derive the capacitance by applying the
definition to a capacitor as on the adjacent figure
• If under a potential difference Vab = Ed, the plates
will store a charge of density σ, such that Q = σA.
Then
Ex: Consider a parallel-plate capacitor of area A, of plate separation d
• When connected to the battery of voltage Vacross battery, charge is pulled
off one plate and transferred to the other plate
• The transfer stops when
• Then the charge stored on the capacitor plates will be
across capacitor across batteryV V
0 across capacitor
AQ CV V
d
0ab
Q Q AC
V Ed d
0
AC
d
Capacitors – The parallel-plate capacitor
• So, the capacitance depends on the geometric arrangement of the conductors and
the electric properties of the insulating material between them. But how?
+
+
+
+
+
• Since the potential difference across the plates is V =
Ed, if we apply a larger voltage V, a larger field is
produces corresponding to more charge Q deposited
• Since the field is constant, the equipotential surfaces
are equally spaced flat surfaces parallel with the plates
• The equipotential surface close to the positive plate
has the largest potential, and the potential decreases
uniformly to the surface on the negative plate:
Capacitors – Revisit the field between the parallel plates
1 1 2 2 3 3 4ab a b a bV V V V V V V V V V V E x Ed
Va > V1 > V2 > V3 > Vb
Δx
Ex: The potential differences Vi – Vi –1 between
equipotential surfaces at 4 equal steps Δx between the
plates of a capacitor are EΔx, such that potential difference
across the plates is
E
V
x
Potential
Va
V1
V2
V3
Vb
slope = |ΔV/Δx| = E
d
• If the plate separation is much smaller than the size of the plates, the electric field
inside is well approximated by the field of two infinite parallel sheets of charge
• However, the approximation works only close to the
center of the plates, not near the edges where the field
is not uniform
–
–
–
–
–
• A circuit is a network of electric devices usually containing a source of electrical
energy (such as a battery) connected to electric elements (such as capacitors)
• A circuit diagram can be used to show the path of the real circuit
• If a capacitor is connected in a circuit across two points with an electric potential
difference, the electrons are transferred through wires from one plate to the other
plate, leaving one plate positively charged and the other plate negatively charged
• The flow of charges ceases when the voltage across the capacitor equals that across
the two points in the circuits. Then, as long the potential difference remains
unchanged, the capacitor stays inactive: a storage of charge (that is, electric energy)
+ _
C
Electric Circuits – Capacitors in circuits
• The capacitors are represented in circuits using a symbol for
the two plates. The battery needed to produce potential
differences is represented by a similar symbol
• The similarity is due to the fact that both devices are sources
of electric charge; however, while the battery is ideally a
limitless source, the capacitor is limited by its capacity
• Capacitors can be combined in circuits: the simplest combinations are in parallel
and series. Let’s find the equivalent capacitance that performs the same function as
these elementary combinations
V
• Capacitors in parallel are all connected across the same two points.
For illustration, consider two capacitor in parallel
• Therefore, all capacitors will be connected across the battery, so
they will be under the same voltage V
• The total charge is equal to the sum of the charges on the capacitors
• This net charge can be considered as being stored on the parallel
combination seen as only one capacitor with an equivalent
capacitance Cp:
1 2netQ Q Q
pC V 1C V 2C V 1 2pC C C
1 2 netQ Q Q
Electric Circuits – Capacitors in parallel
• The result can be extrapolated to for n capacitors
in parallel:
• Notice that the parallel equivalent capacitance is
larger than any of the individual capacitances
1 2 3 ...p nC C C C C
• For n capacitors in series:
• Notice that the series equivalent capacitance is smaller than any of the individual
capacitances
• Capacitors in series chained negative plate to positive
plate, such that each plate holds the same charge, and the
charge on the combination is the same as on each capacitor
• However, the potential difference delivered by the battery
across the equivalent capacitance Cs is imparted across the
capacitors in series. Hence
1 2 V V V
1 2
s
Q Q Q
C C C
1 2
1 2 1 2
1 1 1s
s
C CC
C C C C C
1 2
1 1 1 1...
s nC C C C
Electric Circuits – Capacitors in series
1 2Q Q Q
Problems:
10. Mixed combinations of capacitors: A capacitive
circuit combines capacitors as in the figure (the
numbers are capacitances in μF).
a) Find the equivalent capacitance of the capacitive
circuit in the figure.
b) Say that a 12-V battery is connected between points
ab. What is the amount of charge stored on the
combination of capacitors?
11. Capacitive circuit analysis: Three capacitors are
connected across a 12-V battery as on the figure.
a) Find the equivalent capacitance of the circuit
b) Find the charge on each capacitor in the circuit and
the potential difference across it
Energy Stored in a Capacitor
• Therefore, we see that a capacitor can be seen a charge or energy storage device
• When connected across a conductive medium this energy is released. In general,
capacitors act as energy reservoirs that can slowly charged and then discharged
quickly to provide large amounts of energy in a short pulse
2
2
QU
C
21 12 2
U CV QV
+Q –Q
0 0
1Q Q
W U Vdq qdqC
2
12 021
2
Volume
AEd
CVU du
Ad Ad
2102
u E
q
V
• The energy stored in a capacitor is equal to the energy
necessary to increase the charge on the plates from zero to Q:
• From the definition of capacitance, this can be rewritten in
different forms
• We can define the energy density u as the energy per unit
volume. For a parallel plate capacitor (but with a result valid
for any capacitor) Vacuum
Dielectrics – Capacitors with dielectrics • A dielectric is an insulating material that, when placed between the plates of a
capacitor, increases the capacitance
Ex: Dielectrics can be rubber, plastic, or waxed paper
0 0 A A
C Cd d
Ex: A dielectric improves the performance of a capacitor:
a) Say that a capacitor without dielectric stores a certain
amount of charge Q0. The voltage across the plates is V0.
b) With the dielectric inside, the charge stays the same but C
increases and V decreases: the same charge is held with a
lower V.
• If the capacitance of a capacitor with air between the plates is C0, when a dielectric
completely fills the region between the plates, the capacitance increases by the factor
κ > 1 called dielectric constant:
ε =κε0 is called the electric permittivity of the dielectric
20 12
E
E u E
Quiz 6: Is the energy increasing or decreasing when a dielectric is inserted in between the
charged plates?
Dielectric
a) Increases b) Decreases c) Stays the same
• If we decrease V we also decrease E which is done by a polarization in the
dielectric resulting in an electric field opposite to the initial field.
• Polarization occurs when there is a separation between the
“centers of gravity” of negative and positive charge of the
molecules
• In a capacitor, the dielectric becomes polarized because it is
in an electric field that exists between the plates
• For any given plate separation, there is a maximum electric
field that can be produced in the dielectric before it breaks
down and begins to conduct
• This maximum electric field is called the dielectric strength
• The polarization results in an induced surface charge density σi which decreases the
net charge density σ – σi
• Therefore, for a high κ dielectric, the induced density can almost cancel out the
density on the plates, so a small potential difference will hold a large charge density
σ on the plates
0
0 0
11i
i
EE
Dielectrics – Polarization
Problems:
12. Dielectrics in series and parallel: A parallel plate capacitor of capacitance C0 has the
space between the plates filled with two slabs of dielectric, with constants κ1 and κ2. What is
the capacitance in terms of C0, d, κ1 and κ2 when the space is filled as in figure a) and then as
in figure b):
a) b)
13. Energy in capacitors: Three parallel-plate capacitors
are networked as in the figure with given geometrical
characteristics. Together they hold a net charge Q. A
particle of mass m and charge q hangs above a very long
wire of static charge density λ. The particle is released
from rest and pulls the dielectric out of a capacitor. Sketch
the steps necessary to calculate the speed of the particle at
the moment when the dielectric is completely out, using
conservation of energy.
+ + + + + + + + + + +
– – – – – – – – – – – –
½d d
κ1
κ2
+ + + + + + + + + + +
– – – – – – – – – – – –
d κ1 κ2