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Summary
Lecture 5
Electric flux through area A:
Gauss’s law: the total electric flux through
a closed surface is equal to the total electric
charge inside the surface, divided by ε0.
Electric Potential
Potential Difference
Voltage
Gauss’s law, electric field,
and surface charge density:
Φ E = E⊥ A = EA⊥
Φ E = EA cos θ
Φ E = EA =
Qencl
ε0
ΦE =
∑E
⊥
A=
closed
surface
=
σA
ε0
E=
The work done by the electric field on the electric charge:
σ
ε0
Qencl
ε0
σ=
Q
A
Wa→b = Fe d = qEd
The change in electric potential energy equals the
PEb − PEa = −Wa→b = −qEd
negative of the work done by the electric field:
Physics 112, Spring 2010, Jan 22, Lecture 5
Electric Potential
Uniform electric field.
Electric Potential Energy and Potential
Work done by the electric field is similar
to that done by gravity (by conservative
force!).
+
near positive
plate
+
near negative
plate
low potential energy
low potential
-
near positive
plate
low potential energy
high potential
-
near negative high potential energy
plate
low potential
ΔPE = PEb − PEa = −Wa →b = −qEd
The work is done by the electric field to
move the charge and it changes the
potential energy of the charge.
The electric potential or simply the potential:
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Physics 112, Spring 2010, Jan 22, Lecture 5
Va =
PEa
q
Vb =
PEb
q
-
PEa and PEb are the potential energies of some electric charge in the electric filed.
Low PE
for negative
charge
Unit for electric potential: volt (1V = 1J/C).
high potential energy
high potential
-
Only the difference in potentials of potential difference is measurable!
A positive charge has high potential energy when potential V is high.
Hence, if we want to speak about Va, we need to know where is zero potential.
Zero potential can be chosen at the ground (the Earth) or at infinite distance.
Physics 112, Spring 2010, Jan 22, Lecture 5
A negative charge has high potential energy when potential V is low.
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Physics 112, Spring 2010, Jan 22, Lecture 5
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1
Electric Potential Difference = Voltage
Electric Field and Electric Potential Difference
PEb − PEa
W
= − ba
q
q
Again: The work done by the electric field to move positive charge from a to b:
The difference in potential
or the potential difference:
Vba = Vb − Va =
W = Fe d = qEd
Unit for potential difference: volt (1V = 1J/C).
Another version for work:
W = −q(Vb − Va ) = −qVba
electric potential difference = voltage
If the charge q moves through a potential difference Vab, its potential energy
changes by an amount qVba.
PEa
Va =
PEa = qVa
PEb − PEa = q(Vb − Va ) = qVba
q
The electric force accelerates the charge toward the negative plate.
PE of the charge decreases while KE increases.
PE + KE
E=−
The minus sign shows that
20 cm
= constant
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6
An electron in TV tube accelerates from rest
through a potential difference Vab= 5000 V.
1. What is the change in electric potential
energy of the electron?
ΔPE = qVba = (−1.6 ×10−19 C )(+5000V ) = −8 ×10−16 J
V⎞
⎛
V = Ed = ⎜ 200 ⎟(0.2 m) = 40 V
m⎠
⎝
The minus sign shows that PE of the charge
decreases.
⎛ N ⎞ ⎛V ⎞
Note : the unit of the electric field can be : ⎜ ⎟ or ⎜ ⎟
⎝C ⎠ ⎝m⎠
2. What is the speed of the electron as a result of this acceleration?
m = 9.1x10-31kg
or 20,000 N / C
Physics 112, Spring 2010, Jan 22, Lecture 5
V.
Physics 112, Spring 2010, Jan 22, Lecture 5
Voltage:
V
V
120 V
=
= 20,000
d 0.6 ×10 − 2 m
m
points in the direction of decreasing potential
Acceleration of Electron
2. There is voltage of 120 V between two plates. What is the electric field if the
distance between these two plates is 0.6 cm?
E=
V
d
The electric field also can be measure in N/C or V/m.
1. There is the electric field of 200 N/C between two metallic plates. What is the
voltage between these plates if the distance is 20 cm?
-
E=
Unit for electric field:
Electric Field and Voltage
+
r
E
Vba
d
1 N / C = 1N ⋅ m / C ⋅ m = 1J / C ⋅ m = 1 V / m
Physics 112, Spring 2010, Jan 20, Lecture 4
E
Vba = − Ed
− qVba = qEd
The potential of the electric field shows its ability do work on electric charge
placed in this electric field.
v= −
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ΔKE = −ΔPE
ΔKE + ΔPE = 0
1 2
mv = −q(Vb − Va ) = −qVba
2
2qVba
(2)(−1.6 ×10−19 C )(5000 V )
= −
= 4.2 ×107 m / s
m
9.1×10−31 kg
Physics 112, Spring 2010, Jan 22, Lecture 5
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Equipotential Lines
Electric Potential and Equipotential Lines
Equipotential lines or, in three dimensions, equipotential surfaces, represent
constant electric potential.
-
-
-
-
+
+
The electric potential difference:
d
Equipotential lines (the green dashed
lines) for electric dipole are
perpendicular to the electric field.
q
−
Wb (#1) a
PEa
=−
q
q
Wa→ # 2 = Wa→ #1
+
+
+
Vba (# 2) = Vba (#1)
Potential difference (voltage) between equipotential lines is constant.
Low and High Voltage Examples
1.5 V
PEb (#1)
The work done by the electric force (conservative) is independent of the path taken.
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Physics 112, Spring 2010, Jan 22, Lecture 5
Vb (#1) a = Vb (#1) − Va =
E
+ a
+
+
PEb (#1) − PEa = −Wa →b (#1) = −qEd
b(#2)
b(#1)
Equipotential lines (the green
dashed lines) between two charged
parallel plates are perpendicular to
the electric field.
-
Physics 112, Spring 2010, Jan 22, Lecture 5
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Spark in Air: Electric Breakdown
The dielectric breakdown strength of dry air is
~33,000 V/cm = 3,300 000 V/m
at Standard Temperature and Pressure (STP):
- temperature: 00C, 273 K, or 320F;
- pressure 100 kPa, 14.5 psi, or 1 atm.
200,000 V
This is only as a rough value since the actual
breakdown voltage is highly dependent upon
the electrode shape and size.
~ 5,000,000 V
12 V
Spark to car door in winter can involve
voltage as high as 20,000 V.
A high voltage is not necessarily dangerous
if it cannot deliver substantial current.
Physics 112, Spring 2010, Jan 22, Lecture 5
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Physics 112, Spring 2010, Jan 22, Lecture 5
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Electric Field Between Clouds and Earth (#1)
Charged clouds
+
+
-
-
-
+
+
+
+
-
-
-
-
Charged clouds
+
-
Electric Field Between Clouds and Earth (#2)
-
-
If electric field is
uniform, breakdown
is equally likely to
happed anywhere.
Electric breakdown is
more likely to happed
here.
Is it true?
108
d
Why you should not
seek shelter under a
tree during a storm?
+
V
E=
All this is one
conductor
E
V = Ed
+
+
d
The electric field is stronger
between the clouds and
conducting rod: smaller
distance!
+
+
+
Wet tree
V
(V/m)
d
Conducting (metal) rod
+
Ground
Ground
Physics 112, Spring 2010, Jan 22, Lecture 5
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Physics 112, Spring 2010, Jan 22, Lecture 5
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Potential Difference at Power Line
How can a bird stand on a high
voltage power line and not get
zapped?
The potential difference between its
feet is so small that very little current
flows through the bird’s body.
Can somebody survive if touching
the wire under 100,000 V as shown
in the picture?
1) no
2) yes
Physics 112, Spring 2010, Jan 22, Lecture 5
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