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Electric Field and Circuits
The Basics
In electric circuits, conductors are in non-equilibrium states, so
~ 6= 0 inside the conductor and free charges are moving,
E
generating current.
Electric Field and Circuits
The Basics
In electric circuits, conductors are in non-equilibrium states, so
~ 6= 0 inside the conductor and free charges are moving,
E
generating current.
When in a steady state, an electric circuit experiences no changes
in current and any deposits of excess charge change neither
magnitude nor position.
Electric Field and Circuits
The Basics
In electric circuits, conductors are in non-equilibrium states, so
~ 6= 0 inside the conductor and free charges are moving,
E
generating current.
When in a steady state, an electric circuit experiences no changes
in current and any deposits of excess charge change neither
magnitude nor position.
Current is the flow of charge and is conserved. In a steady state,
charge cannot accumulate anywhere in a circuit — e.g in a simple
loop circuit, current must be the same everywhere.
Electric Field and Circuits
The Basics
In electric circuits, conductors are in non-equilibrium states, so
~ 6= 0 inside the conductor and free charges are moving,
E
generating current.
When in a steady state, an electric circuit experiences no changes
in current and any deposits of excess charge change neither
magnitude nor position.
Current is the flow of charge and is conserved. In a steady state,
charge cannot accumulate anywhere in a circuit — e.g in a simple
loop circuit, current must be the same everywhere.
A node is a junction where two or more wires join. Since current is
conserved, the current entering a node is equal to the current
exiting the node.
Electric Field and Circuits
E Field and Current
Kinetic energy of moving electrons is converted to thermal energy
~ is
through collisions with the atomic lattice. An external E
required to keep charges in motion.
Electric Field and Circuits
E Field and Current
Kinetic energy of moving electrons is converted to thermal energy
~ is
through collisions with the atomic lattice. An external E
required to keep charges in motion.
Current carrying wires are neutral — moving electrons (on
average) experience no net electric force due to other sources of
charge inside the conductor.
Electric Field and Circuits
E Field and Current
Kinetic energy of moving electrons is converted to thermal energy
~ is
through collisions with the atomic lattice. An external E
required to keep charges in motion.
Current carrying wires are neutral — moving electrons (on
average) experience no net electric force due to other sources of
charge inside the conductor.
~
Ethin
~
Ethick
v̄thin
v̄thick
While current is uniform in a simple loop
~ | need
circuit, electron drift speed v̄ and |E
not be — both depend upon the
cross-sectional area A of elements in the
circuit.
Electric Field and Circuits
E Field and Current
Kinetic energy of moving electrons is converted to thermal energy
~ is
through collisions with the atomic lattice. An external E
required to keep charges in motion.
Current carrying wires are neutral — moving electrons (on
average) experience no net electric force due to other sources of
charge inside the conductor.
~
Ethin
~
Ethick
v̄thin
v̄thick
While current is uniform in a simple loop
~ | need
circuit, electron drift speed v̄ and |E
not be — both depend upon the
cross-sectional area A of elements in the
circuit.
Question: Explain why E and v̄ are larger in the thin wire.
Electric Field and Circuits
E Field, Current, and Surface Charge
Current is uniform and parallel to the wire in a steady-state simple
~ ∝ ~I , E
~ is also uniform and parallel within the loop.
loop. Since E
Electric Field and Circuits
E Field, Current, and Surface Charge
Current is uniform and parallel to the wire in a steady-state simple
~ ∝ ~I , E
~ is also uniform and parallel within the loop.
loop. Since E
−
+
1
2
3
4
5
~ and v̄ due to
A: Draw in E
source at numbered points
Electric Field and Circuits
E Field, Current, and Surface Charge
Current is uniform and parallel to the wire in a steady-state simple
~ ∝ ~I , E
~ is also uniform and parallel within the loop.
loop. Since E
−
+
1
−
+
1
2
2
v̄
~
E
v̄
~
E
3
3
~
E
v̄
4
4
~
E
v̄
5
5
v̄
~
E
~ and v̄ due to
A: Draw in E
B: Draw in accumulations of +
source at numbered points
and – surface charge
Electric Field and Circuits
E Field, Current, and Surface Charge
Current is uniform and parallel to the wire in a steady-state simple
~ ∝ ~I , E
~ is also uniform and parallel within the loop.
loop. Since E
−
+
1
−
+
+
1
2
−
2
v̄
~
E
v̄
~
E
3
3
~
E
v̄
4
4
~
E
v̄
5
5
v̄
~
E
~ and v̄ due to
A: Draw in E
B: Draw in accumulations of +
C: Draw in accumulations of
source at numbered points
and – surface charge
steady-state surface charges.
Electric Field and Circuits
E Field, Current, and Surface Charge
Current is uniform and parallel to the wire in a steady-state simple
~ ∝ ~I , E
~ is also uniform and parallel within the loop.
loop. Since E
−
+
1
−
+
+
1
2
−
2
v̄
~
E
v̄
~
E
3
3
~
E
v̄
4
4
~
E
v̄
5
5
v̄
~
E
~ and v̄ due to
A: Draw in E
B: Draw in accumulations of +
C: Draw in accumulations of
source at numbered points
and – surface charge
steady-state surface charges.
~ and v̄ within a circuit are produced by variations in
Uniform E
surface charge density throughout the circuit.
Electric Field and Circuits
Initial Transient
Before a circuit is closed, there is a buildup of
surface charge near the gap. As always,
~ net = 0 inside the conductor.
E
+ + + + +
+
+
+
+
~
Egap
+
+
+
+
+ + + + +
−−−−−
−
−
−
−
~
Eother
−
−
−
−
−−−−−
Electric Field and Circuits
Initial Transient
Before a circuit is closed, there is a buildup of
surface charge near the gap. As always,
~ net = 0 inside the conductor.
E
Immediately after the circuit is closed, the gap
~ other ; this
charges neutralize, leaving only E
causes charge to flow, reducing surface charge
densities in the region of the join.
+ + + + +
+
+
+
+
~
Egap
+
+
+
+
+ + + + +
+ +
+
+
−−−−−
−
−
−
−
~
Eother
−
−
−
−
−−−−−
− − −−
~
Eother
+ +
+
+
− − −−
Electric Field and Circuits
Initial Transient
Before a circuit is closed, there is a buildup of
surface charge near the gap. As always,
~ net = 0 inside the conductor.
E
Immediately after the circuit is closed, the gap
~ other ; this
charges neutralize, leaving only E
causes charge to flow, reducing surface charge
densities in the region of the join.
+ + + + +
+
+
+
+
~
Egap
+
+
+
+
+ + + + +
+ +
+
+
−−−−−
−
−
−
−
~
Eother
−
−
−
−
−−−−−
− − −−
~
Eother
+ +
+
+
− − −−
The electric field propagates at the speed of light, inducing
rearrangement of surface charge until a steady-state is reached.
Electric Field and Circuits
Feedback & Resistors
ii
−
−
−
−
−
−
−
−
i2 <i1
Feedback is the process whereby a current
gradient induces rearrangement of surface
charge until the gradient disappears and
current is uniform.
Electric Field and Circuits
Feedback & Resistors
ii
−
−
−
−
−
−
−
−
−
−
−
−
− − − −
−
−
−
−
i
−−
− − − − −
i2 <i1
Feedback is the process whereby a current
gradient induces rearrangement of surface
charge until the gradient disappears and
current is uniform.
Exercise: Consider a straight,
current-carrying wire that is bent. Explain
what happens at the bend, and how feedback
restores uniform current.
Electric Field and Circuits
Feedback & Resistors
ii
−
−
−
−
−
−
−
−
−
−
−
−
− − − −
−
−
−
−
i
−−
− − − − −
i2 <i1
Feedback is the process whereby a current
gradient induces rearrangement of surface
charge until the gradient disappears and
current is uniform.
Exercise: Consider a straight,
current-carrying wire that is bent. Explain
what happens at the bend, and how feedback
restores uniform current.
Exercise: Use feedback and the relation
~ in both wire and
i = nAuE to determine E
resisitor and an approximate distribution of
surface charge.
“resistor”
Electric Field and Circuits
Energy: Batteries
Given circuit elements 1, 2, · · · , energy conservation gives us the
following loop rule for the change in potential on a closed path
around the circuit:
∆V1 + ∆V2 + · · · = 0 .
Electric Field and Circuits
Energy: Batteries
Given circuit elements 1, 2, · · · , energy conservation gives us the
following loop rule for the change in potential on a closed path
around the circuit:
∆V1 + ∆V2 + · · · = 0 .
A battery is a circuit element that maintains a constant potential
difference across its terminals; this potential is equal in magnitude
to the emf of the battery, which is a measure of the battery’s
ability to separate charge.
Electric Field and Circuits
Energy: Batteries
Given circuit elements 1, 2, · · · , energy conservation gives us the
following loop rule for the change in potential on a closed path
around the circuit:
∆V1 + ∆V2 + · · · = 0 .
A battery is a circuit element that maintains a constant potential
difference across its terminals; this potential is equal in magnitude
to the emf of the battery, which is a measure of the battery’s
ability to separate charge.
~ | inside a battery and what is its direction
Question: What is |E
~
relative to E in an attached circuit?
Electric Field and Circuits
Energy: Loops
iin
i1
1
i2
2
∆V12 =∆V12
A parallel circuit is one in which current has
more than one path through which it can flow.
∆V across paths connected to the same nodes
must be the same, and charge conservation
requires that iin = iout at each node.
Electric Field and Circuits
Energy: Loops
iin
i1
1
i2
2
∆V12 =∆V12
A parallel circuit is one in which current has
more than one path through which it can flow.
∆V across paths connected to the same nodes
must be the same, and charge conservation
requires that iin = iout at each node.
This is an example of the loop rule. Consider the closed loop
1 → 2 → 1. The loop rule tells us
∆V12 − ∆V12 = 0 .
so ∆V12 = ∆V12 .
Electric Field and Circuits
Energy: Loops
iin
i1
1
i2
2
∆V12 =∆V12
A parallel circuit is one in which current has
more than one path through which it can flow.
∆V across paths connected to the same nodes
must be the same, and charge conservation
requires that iin = iout at each node.
This is an example of the loop rule. Consider the closed loop
1 → 2 → 1. The loop rule tells us
∆V12 − ∆V12 = 0 .
so ∆V12 = ∆V12 .
Typically, the connecting wires offer very little resistance when
compared to other circuit elements, so Ewire Lwire ≈ 0.
Electric Field and Circuits
Energy: Loops
iin
i1
1
i2
2
∆V12 =∆V12
A parallel circuit is one in which current has
more than one path through which it can flow.
∆V across paths connected to the same nodes
must be the same, and charge conservation
requires that iin = iout at each node.
This is an example of the loop rule. Consider the closed loop
1 → 2 → 1. The loop rule tells us
∆V12 − ∆V12 = 0 .
so ∆V12 = ∆V12 .
Typically, the connecting wires offer very little resistance when
compared to other circuit elements, so Ewire Lwire ≈ 0.
Exercise: Use the loop rule to determine ∆V of the battery.
Electric Field and Circuits
Applications
We have two rules for circuit analysis:
Electric Field and Circuits
Applications
We have two rules for circuit analysis:
I
Current Node Rule: Iin = Iout , with I = |q|nAuE .
Electric Field and Circuits
Applications
We have two rules for circuit analysis:
I
I
Current Node Rule: Iin = Iout , with I = |q|nAuE .
P
Loop Rule:
i ∆Vi = 0 about any closed loop.
Electric Field and Circuits
Applications
We have two rules for circuit analysis:
I
I
Current Node Rule: Iin = Iout , with I = |q|nAuE .
P
Loop Rule:
i ∆Vi = 0 about any closed loop.
Exercise: How does current I change when we double the wire
length in a circuit consisting only of a wire and a battery?
Electric Field and Circuits
Applications
We have two rules for circuit analysis:
I
I
Current Node Rule: Iin = Iout , with I = |q|nAuE .
P
Loop Rule:
i ∆Vi = 0 about any closed loop.
Exercise: How does current I change when we double the wire
length in a circuit consisting only of a wire and a battery?
Exercise: How does current I change if we double instead the
cross-sectional area of the wire in the same simple circuit.
Electric Field and Circuits
Applications
We have two rules for circuit analysis:
I
I
Current Node Rule: Iin = Iout , with I = |q|nAuE .
P
Loop Rule:
i ∆Vi = 0 about any closed loop.
Exercise: How does current I change when we double the wire
length in a circuit consisting only of a wire and a battery?
Exercise: How does current I change if we double instead the
cross-sectional area of the wire in the same simple circuit.
Doubling the length of the wire is analogous to placing two wires
in series, while doubling the cross-sectional area is analogous to
placing the wires in parallel.
Electric Field and Circuits
Applications: Bulbs in Series
Al ,Ll ,El ,v̄l
Ar ,Lr ,Er ,v̄r
A = cross-sectional area of filament, L =
filament length, E = electric field, v̄ = drift
velocity. Quantities refer to round bulb and
long bulb, with Ar > Al .
Electric Field and Circuits
Applications: Bulbs in Series
Al ,Ll ,El ,v̄l
Ar ,Lr ,Er ,v̄r
A = cross-sectional area of filament, L =
filament length, E = electric field, v̄ = drift
velocity. Quantities refer to round bulb and
long bulb, with Ar > Al .
Question: Using the node rule, write an equation relating v̄r to v̄l .
Electric Field and Circuits
Applications: Bulbs in Series
Al ,Ll ,El ,v̄l
Ar ,Lr ,Er ,v̄r
A = cross-sectional area of filament, L =
filament length, E = electric field, v̄ = drift
velocity. Quantities refer to round bulb and
long bulb, with Ar > Al .
Question: Using the node rule, write an equation relating v̄r to v̄l .
Question: Now relate Er to El and state which is larger.
Electric Field and Circuits
Applications: Bulbs in Series
Al ,Ll ,El ,v̄l
Ar ,Lr ,Er ,v̄r
A = cross-sectional area of filament, L =
filament length, E = electric field, v̄ = drift
velocity. Quantities refer to round bulb and
long bulb, with Ar > Al .
Question: Using the node rule, write an equation relating v̄r to v̄l .
Question: Now relate Er to El and state which is larger.
Question: Explain why the thick-filament (round) bulb doesn’t
glow.
Electric Field and Circuits
Applications: Bulbs in Series
Al ,Ll ,El ,v̄l
Ar ,Lr ,Er ,v̄r
A = cross-sectional area of filament, L =
filament length, E = electric field, v̄ = drift
velocity. Quantities refer to round bulb and
long bulb, with Ar > Al .
Question: Using the node rule, write an equation relating v̄r to v̄l .
Question: Now relate Er to El and state which is larger.
Question: Explain why the thick-filament (round) bulb doesn’t
glow.
Bulb achieves maximum brightness when energy radiated = energy
dissipated in filament. The rate of energy dissipation (power) is
proportional to E 2 .
Electric Field and Circuits
Atomic Level