Download Electric Field and Circuits

Electric Field and Circuits
Read: Chapter 19
The Basics
~ =
In electric circuits, conductors are in non-equilibrium states, so E
6 0 inside the conductor and free charges are moving,
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.
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Electric Field and Circuits
E Field and Current
~
Kinetic energy of moving electrons is converted to thermal energy through collisions with the atomic lattice. An external E
is 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.
~ thin
E
~ thick
E
v̄thin
~
While current is uniform in a simple loop circuit, electron drift speed v̄ and |E|
need not be — both depend upon the cross-sectional area A of elements in the
circuit.
v̄thick
Question: Explain why E and v̄ are larger in the thin wire.
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Electric Field and Circuits
E Field, Current, and Surface Charge
~ ∝ I,
~ E
~ is also uniform and parallel within
Current is uniform and parallel to the wire in a steady-state simple loop. Since E
the loop.
~ and v̄ due to source at numbered points.
A: Draw in E
−
+
1
2
3
4
5
−
+
B: Draw in accumulations of + and – surface charge.
1
2
v̄
~
E
v̄
~
E
3
v̄
~
E
4
v̄
~
E
5
v̄
+
~
E
−
C: Draw in accumulations of steady-state surface charges.
~ and v̄ within a circuit are produced by variations in surface charge density throughout the circuit.
Uniform E
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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
+
+
+
+
+
+
~ gap
E
~ other ;
Immediately after the circuit is closed, the gap charges neutralize, leaving only E
this causes charge to flow, reducing surface charge densities in the region of the join.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
− − − − −
−
−
−
−
~ other
E
−
−
−
−
− − − − −
−
−
− −
~ other
E
+
+
+
+
−
−
− −
The electric field propagates at the speed of light, inducing rearrangement of surface charge until a steady-state is reached.
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Electric Field and Circuits
Feedback & Resistors
ii
− − − − i <i
2
1
Feedback is the process whereby a current gradient induces rearrangement of surface
charge until the gradient disappears and current is uniform.
− − − −
−
−
−
−
− − − −
−
−
−
i
−
−
−
− − − − −
Exercise: Consider a straight, current-carrying wire that is bent. Explain what
happens at the bend, and how feedback restores uniform current.
~ in both
Exercise: Use feedback and the relation i = nAuE to determine E
wire and resisitor and an approximate distribution of surface charge.
“resistor”
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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 relative to E
~ in an attached circuit?
Question: What is |E|
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Electric Field and Circuits
Energy: Loops
iin
i1
1
i2
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.
2
∆V12 = ∆V12
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.
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Electric Field and Circuits
Applications
We have two rules for circuit analysis:
• 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.
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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 .
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