Download Equipotential Lines 17.1 Electric Potential Energy PE = energy

17.1 Electric Potential Energy
Electric Potential Energy, PE
Units: Joules
Electric Potential, V
Units: Volts
Equipotential Lines
The Electron Volt
Uniform E field
Assembling point charges
• Capacitance
• Dielectrics
• Storage of Electric Energy
Electric force is a conservative force and so we can
assign an electric potential energy (PE) to the system
PE = energy associated with an arrangement
of objects that exert forces on each other
PE = capacity or potential to do work
Change in Gravitational PE for a
mass, m, near the Earth’s surface
∆ PEba = qVba
∆ PEgravity = mgh
Why does lightning occur?
Electric Potential
Electric Potential
(Electric) Potential, Va
is a scalar quantity that is associated
with the charges that create the E field
Va is defined to be the electric potential energy per unit charge
In the real world only changes in PE are physically meaningful,
so rewrite 17.2a in terms of the potential difference.
Potential difference or ꞌVoltageꞌ
= change in the PE per unit charge
Potential energy of system
small (positive) test
charge placed at point a
Units = volts
1V = 1 J/C
Change in Electric PE for a charge,
q, that moves from point a to b
through a potential difference, Vba
∆
Units: volt (V)
(17-2a)
Potential at a
=
We define
where V=0
(17-3)
1 V = 1 J/C.
(V=0 at the “ground” in a circuit which is typically
located at the negative terminal of the battery
For a collection of point charges, V = 0 at r = ∞)
Va depends only on the charges that create the E field – not the test charge, q.
So, what does the electric potential (V) look like?
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Electric Potential
Electric Potential
It’s a scalar (a number) so problem solving is easier than with electric
fields (vectors) - but take care with the sign of the potential.
Potential at a distance r from a point charge Q is given by,
Think of potential as a topological map…
(derived using calculus and
assuming V = 0 at r = ∞).
V is like height
and q is like mass
in the gravitational case
[point charge] (17-5)
V
m
∆PEG = mgh
h
.. and a
negative
charge.
For a positive
pt. charge
q
V
+Q
V
-Q
+Q
-Q
r
r
V is HIGH near (+) charges
and LOW near (-) charges
∆PEba = qVba
Note, this formula gives the potential at r
AND the potential difference between ∞ and r
And we typically take V = 0 at r = ∞
17.3 Equipotentials
17.3 Equipotentials
We can represent the potential
near a point charge, Q using
spherical ‘equipotentials’
0 V infinitely far
away from Q
For a collection of point charges, simply add the
potentials from the individual charges.
10 V
V
20 V
+Q
30 V
40 V
50 V
r
+
E field is strongest where the
equipotentials are closer
All points on an Equipotential line or
surface are at the same potential
Equipotentials and Field lines for a ‘dipole’
(equal and opposite point charges)
E field lines are ⊥ to Equipotentials
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Question…
17.3 Equipotentials
In a capacitor, the potential changes
linearly from one plate to the other.
What is the potential at the center of the square ?
(assume V = 0 very far from the charges)
-Q
+Q
Equipotentials are equally spaced planes,
parallel to the capacitor plates.
l
The surface of a conductor
is an equipotential.
-Q
l
+Q
Electric field lines are
perpendicular to equipotentials.
Is E = 0 at the center too?
Can the equipotentials be
reversed (0 → 20V) in the figure?
Equipotentials and Field lines
between two capacitor plates
Remember…
Question
Potential and Potential Energy are different!
[Volts]
[Joules]
Look at the Capacitor again…
+Q
-Q
A + 4 C point charge starts from
rest and moves from a to b
+Q
-Q
high PE
low PE
4 C point
charge
a
b
q+
15 V
5V
0V
10 V
20 V
Potential is higher
at the positive plate
We choose
where V = 0 V.
Potential at a point is a scaler
(number) that depends on the
charge on the plates and where
we define zero volts to be.
q low PE
high PE
=
Potential energy of the system
when point charge, q is at point a,
depends on the potential and q.
5V
What is the change in PE when
it reaches the negative plate? The + 4C charge has a mass of 1x10-3 kg.
1.
2.
3.
4.
- 20 J
+ 20 J
No change
Infinite
What if it were a negative 4 C charge ?
How fast is it going when it reaches b ?
1. 4 m/s
2. 200 m/s
3. 40,000 m/s
4.
Very close to the speed of light
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17.4 The Electron Volt - a Unit of Energy
In atoms, 1 Joule is a VERY large amount of energy so we
often use a different unit:
The electronvolt, eV
One eV = energy acquired by a particle with charge of magnitude
e when it moves through a potential difference of 1 volt.
(13 cm ~ 390,000V)
A 1 cm spark corresponds to a
potential difference of 30,000 V
e
Eg: How much KE does an electron gain
when it is accelerated from rest through a
potential difference of 250 V ?
250 V
It gains 250 eV of KE and loses 250 eV of PE (energy conservation).
Work
Question
The work done by the electric force (Wba)
on a charge q as it moves in an electric
field is defined to be minus the change in
the electric potential energy
Wba = - ∆PEba = - qVba
Units = Joules
(17.2b )
E
q
b
a
(only if ∆KE=0)
C
(assume
∆KE = 0)
How much work is done by an external
force to move a 1 C charge from ..
The work done by an external force in
moving a charge q in an electric field is
Wext = qVba
B
A
∆PEba = PEb – PEa
Note the
different sign for
external work
15 V
10 V
5V
E
a
b
A → B?
B → C?
C → A?
(Phys111: = ∆ + ∆) ← use this if ∆KE ≠ 0
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The next three slides have been updated. I removed the derivation of equation 17-4a
Question…
17.2 Relation between electric potential and field
How does E relate to V?
The component of E in a given direction is equal to the rate at which
the electric potential decreases over distance in that direction
= −
∆
∆
[general E]
What is the electric field inside a capacitor if one plate is at +12 V and
the other plate is at -3 V? The plates are separated by 2 mm.
2 mm
New units for
electric field: V/m
E points in direction
of decreasing V
+12 V
-3 V
So for the uniform electric field inside a capacitor…
E inside a capacitor is equal to minus the potential difference
between the plates divided by the plate separation
=−
The potential is 20 V everywhere in this region…
What is E in this region?
[uniform E] (17-4a)
Example: Work required to assemble a number of point charges
-
To calculate the external work required
to assemble several point charges use
+
(let ∆KE = 0)
(assume
Wext = q Vba ∆ = 0)
+
Example: Work required to assemble a number of point charges
How much work is done by an external force to bring a 2nd charge (Q2)
from infinity and place it a distance r from Q1?
Q1
r
Q1
1
Zero ! There are no other charges present
so Vba
(17-5)
Wext = Q2V1 = Q2 kQ1
r
Use the above eqn. and bring the
charges in from infinity one at a time
How much work is done by an external force to bring
charge Q1 from infinity and place it at the origin?
1
Work done by external force to move Q2 is
Q1
V = 0 at r = ∞
Potential difference (due to Q1) between infinity and r, is
Q2
Q2
r13
Q3
Work done by an external force in moving Q3 is,
r23
Wext = Q3V1+Q3V2 = Q3(V1+V2)
= Q3 kQ1 + Q3 kQ2
r13
r23
For more charges,
just add more terms.
= 0, and Wext = 0
Total work needed to assemble the charges is the sum of all steps
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What can you say about the work done by an EXTERNAL FORCE to bring a
+1 µC charge from far away and place it on the third corner of the
equilateral triangle?
1.
2.
3.
4.
17.7 Capacitance
+ 1 µC
Its positive
Its negative
Its zero
Huh?
d
- 1 µC
A capacitor consists of two conductors
that are close but not touching.
d
d
+ 1 µC
How much WORK is done by the ELECTRIC FORCE when the – 2.0 µC
charge moves closer to the positive charge, as shown?
10.0 cm
+ 3.0 µC
− 2.0 µC
4.0 cm
Capacitor symbol
(the + 3 µC charge is fixed in space)
(1) 0 J
(2) - 0.12 J
(3) - 0.65 J
(4) + 0.81 J
What does a capacitor do?
17.7 Capacitance
For a charged capacitor..
charge on each plate
Units: farad (F)
1 F = 1 C/V
17.8 Dielectrics
A dielectric is an insulator, characterized
by a dielectric constant K.
voltage across capacitor
(17-7)
Capacitance depends on geometry
and capacitor materials.
Capacitance of a parallel-plate capacitor
INCREASES when it is filled with a dielectric:
C = K ε0 A/d
For a parallel-plate capacitor
(with air gap)
ε = K ε0
(17-9)
(17-8)
circuit diagram
Permittivity of free space
8.85 x 10-12 C2/Nm2
d
ε = permittivity of the material
Max field a dielectric can
experience without breaking down.
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17.8 Dielectrics
Eg: What happens when a
dielectric is inserted into a
capacitor connected to a battery?
C
17.9 Storage of Electric Energy
Question
What happens when a dielectric is
inserted into an isolated capacitor
that has charge Q on it?
A charged capacitor also stores electric energy
Energy stored (PE) = work done to charge the capacitor.
C
(17-10)
V
+Q -Q
V stays constant
(battery keeps V constant)
C increases
(due to dielectric)
Q increases (Q = CV,
more charge is pulled from battery)
1.
2.
3.
4.
5.
Q stays constant, C decreases, V decreases
Q stays constant, C decreases, V increases
Q stays constant, C increases, V decreases
Q stays constant, C increases, V increases
Q increases, C stays constant, V increases
Question...
A charged parallel-plate capacitor is
disconnected from a battery.
Suppose the capacitor plates are
pulled apart.
Energy density, uE = stored
energy per unit volume, is
the same no matter the
origin of the electric field:
Units
J/m3
uE =
(17-11)
Pointed conductors, ion winds, lightning and St. Elmo’s fire
+Q
-Q
In Ch. 16 we saw that E = 0 inside a conductor and is ⊥ to the outer surface.
In non-spherical conductors, surface charge densities and
electric fields are biggest where the curvature is greatest.
Equipotentials are closer together there too.
What happens to the potential energy stored in the capacitor ?
1. It increases
2. It decreases
3. It stays the same
Where does that energy come from?
If the E field is strong enough an “ion
wind” or “corona discharge” may occur
(St. Elmo Fire)
conductor
Equipotentials (dashed)
Big surface
charge density
and E field
Field lines (solid)
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Summary of Chapter 17
Work done by electric force on a charge that
moves in electric field is minus change in PE:
Change in potential energy when a charge q
moves through potential difference of V is,
E = - V/d
Wba = - ∆PE
∆PE = qV
(17-3)
Uniform field (17-4a)
Potential difference between
r and ∞ due to pt. charge Q
Point charge (17-5)
Equipotential = line or surface at constant potential.
Charge on a
capacitor:
Capacitance of a
parallel-plate capacitor:
A dielectric is an insulator. It can be used to increase the
capacitance of a capacitor by a factor K.
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