Download 9. Electric potential - McMaster Physics and Astronomy

Concept 9.1: The electric potential is the
electric potential energy per unit charge.
9. Electric potential
electric potential
in Volts (V) = J/C.
It exists in the
absence of a test
charge
• potential and potential energy
• equipotential surfaces
• electric discharge
I. At this position, the electric
potential due to the positive
charges is
a)
b)
c)
d)
using the potential energy of a collection
of point charges qi.
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Quiz of concept 9.1
Physics 1E03 Lecture 9
+
+
_
_
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II.
The electric potential energy of the charges on the
metal plate at distance d, compared to when it is
resting on the insulator, is
a)
b)
c)
d)
-2πkdσ2
2πkdσL2
-2πkdσ2L2
2πkdσ2L2
_ _ _ _ _ _ _ _
σ C/m2
Charge of positive test charge, in C
s1
q
V (r ) = k ∑ i
i ri
Physics 1E03 Lecture 9
d
U
q0
From previous expressions for U, there are two equivalent
ways to find the electric potential:
s2
r r
r
r
V ( s2 ) − V ( s1 ) = − ∫ E • ds directly from the definition of work,
Serway and Beichner
Parts of sections 25.1-.4, 25.6
The Electrophorus has a charge
density of +σ on the insulating
plate. The metal plate is
charged by induction and raised
a distance d<<L.
V=
Electric potential energy of the test charge
in J. It exists only when the test charge is
present.
_ _ _ _ _ _ _ _
d
+
σ C/m2
+
_
_
+ + + + + + + + + +
L
L
4πkdσ below the potential on the insulator.
4πkdσ above the potential on the insulator.
2πkdσ below the potential on the insulator.
2πkdσ above the potential on the insulator.
Physics 1E03 Lecture 9
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Concept 9.2:
Lines of constant potential
run perpendicular to the
electric field lines. Field
lines point in the direction
of decreasing potential.
lines of constant
potential
electric
field lines
Quiz of concept 9.2
A conducting object in equilibrium has an excess charge
–Q. The potential at the surface of the conductor
a) is different than the potential inside the conductor,
since the excess charges are forced to the surface
b) is constant everywhere on the surface
c) is greater at sharply curved regions where there is a
greater local charge density σ
d) none of the above is always true, since the potential
can depend on other nearby charges.
+Q
s2
r r
r
r
V ( s2 ) − V ( s1 ) = − ∫ E • ds
s1
Q=0
If the step ds along a path is
perpendicular to E, E ds=0.
V(s2) = V(s1), the path is an
equipotential.
Physics 1E03 Lecture 9
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Concept 9.3: In order to maintain a constant
potential, the charge density on a conductor
is greater in regions of greater surface
L>>R +R
curvature.
1
The potential of a conducting
sphere with excess charge Q is
R1
2
R2
Q
V (r ) = k
r
When connected to form a single conductor,
Q
Q
Q
Q
V = k 1 = k 2 so | E ( R1 ) |= k 12 >| E ( R2 ) |= k 22
R1
R2
R1
R2
Physics 1E03 Lecture 9
Application of concept 9.3
Sparks and lightning travel through the air only once it
becomes conducting. This happens when E>3x106 N/C.
Then a free electron can be accelerated through a constant E
to give
enough kinetic energy to
K f = U i − U f = qEd ≈ knock another electron from
average spacing of molecules
conducting wire
Thus charge density Q/4πR2 is greater for sphere of smaller R.
+
+
+
+ For a single object with different
+
curvatures, more charge collects
+
+
++
at the sharper curves.
Physics 1E03 Lecture 9
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+ +
+
+
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an atom it hits
This creates more free electrons, which knock more free,
etc. until there is a cascade with a high enough electron
density to conduct the current.
This cascade will occur first near sharply pointed conductors
where the electric field is strongest.
Q: How does a lightning rod work?
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Summary
• the electric potential, is the electric potential
energy divided by the size of the test charge
V = U/q0.
• lines of constant potential run perpendicular to
electric field lines
• every point of a conductor in equilibrium is at the
same potential
• to maintain a constant potential, charge
accumulates near sharp points of a conductor
Practice problems: Chapter 25, #9, 21, 49, 70
Next lecture: read sections 25.4-.6
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