Potential Difference: Path Independence
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Transcript of Potential Difference: Path Independence
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Path independence principle:V between two points does not depend on integration path
Potential Difference: Path Independence
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What is E inside metal? E = 0
What is the potential difference (Vf – Vi)?
Is V zero everywhere inside a metal?No! But it is constant
In static equilibrium
Potential Difference in Metal
In static equilibrium the electric field is zero at all locations along any path through a metal.
The potential difference is zero between any two locations inside the metal, and the potential at any location must be the same as the potential at any other location.
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E
d =3 mm
+Q-Q
V = 6 Volt
Potential in MetalIn static equilibrium A Capacitor with large plates and a small gap
of 3 mm has a potential difference of 6 Volts from one plate to the other.
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d =3 mm
+Q1-Q1 1 mm
Charges +Q2 and –Q2
What are the charges Q1 and Q2?
Now we have 2 capacitors instead of one
V = 4 V
There is no “conservation of potential”!
Potential in MetalIn static equilibrium Insert a 1 mm thick metal slab into the
center of the capacitor.Metal slab polarizes and has charges +Q2 and -Q2 on its surfaces.
Q2=Q1E=0 inside metal
V inside metal slab is zero!
XAt X
𝐸1=𝐸2
Ignoring the fringe fields, E = 2000V/m in each capacitor (from previous slide).
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Metal is not in static equilibrium:• When it is in the process of being polarized• When there is an external source of mobile charges (battery)
If a metal is not in static equilibrium, the potential isn’t constant in the metal.
Potential in MetalThere can be a potential in metal if is NOT in static equilibrium
Nonzero electric field of uniform magnitude E throughout the interior of a wire of length L.Direction of the field follows the direction of the wire.
For each step , the potential difference is: V = -EL
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Question
0.02m 0.03m 0.04m
A B
300 V/m
0 V/m
300 V/m
What is VB-VA?A) 270 VB) -270 VC) -18 VD) 6 VE) -6 V
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Question
0.02m 0.03m 0.04m
A B
0 V/m
0x
𝑉 𝐵−𝑉 𝐴=−(300 Vm ) (0.02 m −0 )
=
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The electric field is uniform in the region below left. The point B is at <0, 0, 0> m. The point C is at <0,-2, 0>m. The potential difference along the path from B to C is +500V.
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Potential difference due to a stationary point charge is independent of the path
Potential difference along a closed loop is zero
Round Trip Potential Difference
+
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Is the following “curly” pattern of electric field possible?
dl
dl
dl
Predicting Possible Field Configuration
is always parallel to
This “curly” pattern of electric field is impossible to produce by arranging any number of stationary point charges!
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Method 1: Divide into point charges and add up contributions due to each charge
Q
Potential of a Uniformly Charged Ring
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Method 2: Integrate electric field along a path
Q
Potential of a Uniformly Charged Ring
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Q
What is V for z>>R ?
The same as for a point charge!
Potential of a Uniformly Charged Ring
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one ring:
integrate:
Potential of a Uniformly Charged Disk
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Potential of a Uniformly Charged Disk
Can find :
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Electric field in capacitor filled with insulator: Enet=Eplates+Edipoles
Eplates=const (in capacitor)
1 2 3 4 5
Edipoles,A
A B
Edipoles,B
Edipoles complicated f(x,y,z)
Travel from B to A:Edipoles is sometimes parallel to dl, and sometimes antiparallel to dl
Potential Difference in an Insulator
∆𝑉=−∫𝐴
𝐵
𝐸 ∙𝑑 �⃗�
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Instead of traveling through inside – travel outside from B to A:
BAEdipoles, average
Potential Difference in an Insulator
Effect of dielectric is to reduce the potential difference.
Enet< Eplates
Enet=Eplates+Edipoles,average ∆𝑉 𝐴 𝐵=−∫𝐵
𝐴
𝐸𝑑𝑖𝑝𝑜𝑙𝑒𝑠 ∙𝑑�⃗�¿0
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Electric field in capacitor filled with insulator: Enet=Eplates-Edipoles
K – dielectric constant
Dielectric Constant
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Dielectric constant for various insulators:
vacuum 1 (by definition)air 1.0006typical plastic 5NaCl 6.1water 80strontium titanate 310
Inside an insulator:
Dielectric Constant
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s
Potential Difference in a Capacitor with Insulator
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+Q-Q
s
d
K
Potential Difference in Partially Filled Capacitor
BAx
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Energy can be stored in electric fields
(for small s)
Evolume
Field energy density: (J/m3)
Energy expended by us was converted into energy stored in the electric field
Energy Density of Electric Field
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Energy Density of Electric Field
In the previous slide, the “system” is the set of two plates. Work, Wexternal > 0, is done on the system by you – part of the “surroundings.”
If the force exerted by you just offsets the attractive force, Fby-plates, so that the plate moves with no gain in KE,
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1 mm
90V 100Vx
Ex
Electric Field and Potential
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Suppose in some area of space V(x,y,z)=x2+yz. What is E(x,y,z)?
Exercise
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=0
In general, integration path may be complex
Potential Inside a Uniformly Charged Hollow Sphere
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If an electron moves from rest through a potential difference of 35 V, what would be its kinetic energy?
What if we had a proton?
0V 35 V
E
0 V 35 V
E
Which particle will move faster?
Example
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In most cases we are interested in V, not the absolute values of V
Shifting the Zero Potential