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CH25: Current, Resistance and Electromotive Force
•  Electric Current and Current Density
•  Drift Velocity
•  Resistivity
•  Ohm’s Law: resistance and resistors
•  Circuits Connection and emf
•  Energy and Power in Circuits
Introduction
•  How do we transfer/transform
electric potential energy?
–  We need circuits! (Or do we?)
•  Circuits allow the transportation
of energy without moving parts.
–  Does the electrons moving
inside the circuits?
•  Before we study circuits, we need to
understand “Electric Current”
Cause of Current low
•  An external field would causes current flow
–  Current: Motion of charge from one region to another.
•  Otherwise, electrons move randomly in a conductor. If a field
exists near the conductor, its force on the electron imposes a drift.
Random motion of electrons:
v avg ≈ 10 5−6 m /s ∝ T

v avg = 0
The external field destroys the
randomness of the motion

v avg ≠ 0
Current flowing in the presence of field
• 
Positive charges would move with the electric field, electrons move in
opposition.
• 
Current (thorugh the cross-sectional area A): net charge flowing the area per
unite time!
dQ
I=
dt
1ampere=1A=1C/s
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Is current a vector?
What does positive and negative current mean?
Current, Drift Velocity and Current Density
vd: drift velocity
dQ = nqv d Adt
I=
dQ
= nqv d A
dt
I
J=
= nqv d
A


J = nqv d
Example: A copper lamp wire has a cross
sectional area A=8.17x10-7m2, and carries a
Current of 1.67A. The free electron density is
n=8.5x1028/m3. What is the current density J
and drift velocity vd. How does it compare
with the random motion of the electrons?
n: concentration of particles
I is not a vector!
J is a vector: same direction as E (and vd)
If A changes, does I change? Does J change?
€
Resistivity is intrinsic to a metal sample (like density is)
Certain Diamond (CVD)
1018
Ohm’s law: (At a given temperature), the ratio of the magnitudes of E
and J is nearly constant (for some materials)
E
J
Ω⋅ m = (V /m) /(A /m 2 ) = (V / A)⋅ m
1Ω = 1V / A
ρ=
Ohmic conductor: r does not depent on E
Non-ohmic (non-linear) conductor:
r depends on E
Resistivity usually depends on temperature
•  Resistivity rises with increasing
temperature. The electronic motion is
analogous to shopping on quiet days
(lower T) or busy days (higher T). See
Figure 25.6.
ρ(T) = ρ 0 [1+ α (T − T0 )]
Semiconductor can be used to
measure temperature (thermistor)
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Superconductor: No resistance! Current can
continue without field!
A lot of energy is wasted (lost in heat) when transporting electric energy.
Superconducting circuits (at room temperature) is what we hope for!
Resistance and Ohm’s law (restated)


E = ρJ ⇒
V
I
=ρ ⇒
L
A
L
V = ( ρ )I = RI
A
L
(R ≡ ρ ) Resistance:
A
R=V/I
1ohm=1Ω=1V/A
€
: Intrinsic quality of a material
: Depends on the geometry of the material
Example:2 conducting copper wires
with different diameters (D0 and D1=3D0),
and different lengths (L0 and L1=16L0).
What’s the ratio of their resistors?
R1:R0=?
Just like fire hose needs enough water
Pressure at the upstream to produce the
water flow,
an electric potential difference is needed
to produce electric current.
Q25.3
Electrons in an electric circuit pass through a resistor. The wire has the same diameter
on each side of the resistor.
Compared to the potential energy of an electron before entering the resistor, the potential
energy of an electron after leaving the resistor is
A. greater.
B. less.
C. the same.
D. not enough information given to decide
Current–voltage relationships
•  Ohm’s Law is linear (and good only for certain type of devices) Current flow
through other devices may not be linear.
•  Example: From the reading of the voltmeter and ammeter, can you tell me what
is the resistance of the resistor?
Resistances usually depends on temperature
ρ(T) = ρ 0 [1+ α (T − T0 )]
L
R≡ρ ⇒
A
R(T) = R0 [1+ α (T − T0 )]
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Example: The resistance of €a wire is 0.97Ω at
0oC, and 1.38Ω at 100oC.
What is it’s resistance
At 20oC?
Example of a radially flowing current (like the
axon of a nerve cell)
The resistance of this hollow cylinder (inner
And outer radii a and b, resistivity ρ, length L),
is the sum of the resistances of a series of
cylindrical shells.
R=ρ
L
A
dr
⇒
A = 2πrL
b
dr
ρ
b
R = ∫ dR = ∫ a ρ
=
ln
2πrL 2πL a
dR = ρ
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Electromotive force and circuits
•  Steady current only exist in a
complete circuit that is not
isolated.
•  When charge moves through
resistors, the potential energy
decreases. When it comes back to
the origin, how come the electric
potential energy is the same as
before?
•  Need a pump!
Source of emf
(electromotive force):
Battery, generator,
solar cell, etc.
Ideal diagrams of “open” and “complete” circuits
E = Vab = IR
Internal resistance
•  Charges moving
inside a emf also
encounter resistance:
Internal resistance, r
•  Let’s make a
measurement of the
potential difference of
the battery before and
after (V0, and V1) it is
connected to a resistor
(R).
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•  Can you figure out
what is the value of r?
Vab = E − Ir
V0 = E
V1 = E − Ir = IR ⇒
IR
V1 /V0 =
⇒
I(R + r)
r = ?R
A function of V0 and V1
Symbols for circuit diagrams
•  Shorthand symbols are in use for all wiring components. See
below.
Resistance infinitely large
Resistance negligible
Q25.5
Electrons in an electric circuit pass through a source of emf. The wire has the same
diameter on each side of the source of emf.
Compared to the potential energy of an electron before entering the source of emf, the
potential energy of an electron after leaving the source of emf is
A. greater.
B. less.
C. the same.
D. not enough information given to decide
Source in an open circuit I
Voltmeter:
• Resistance infinitely large (so it won’t divert any current);
• Connected parallel to measure potential difference
Ammeter:
• Resistance negligible (so it doesn’t change
• the voltage difference across the resistor);
• Connected in series with the resistor
•  What are the readings of
the voltmeter and ammeter
in this circuit?
Source in an open circuit II
•  What are the readings of the voltmeter and ammeter?
Voltmeters and ammeters
•  a) from previous example, what if the voltmeter is setup to
measure the potential difference between a’, b’, instead of a,b?
•  b)What if the voltmeter is connected in the circuit in series?
A source with a short circuit
•  What if we have a short circuit (R=0)? Should the following
voltmeter measure 12V or 0V?
Potential changes around a circuit
•  The net change in potential energy must be zero for the entire
circuit.
E − Ir − IR = 0
•  Local differences in potential and emf do occur. See Figure 25.21
below.
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Energy and Power in Electric Circuits: Power into
a pure resistance
In a time dt, charge (dQ=Idt) experiences potential
Change of Vab,
Therefore the time rate of energy transfer
(output/input) is P, power of the circuit element:
P = dE /dt
dE = Vab dQ = Vab Idt
⇒ P = Vab I
Unit: (1 J/C) (1C/s) = 1J/s=1W
Power input to a pure resistance:
Vab = IR ⇒
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Vab2
P = (IR)I = I R =
R
2
Va>Vb for a resistor, the current enters it.
Where does this energy transferred to the resistor become:
How doe T (temperature) change?
€
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Power output of a source/input to a source
Power output of a source:
Vab = ε − Ir
Power input to a source:
I leaves the source
⇒ P = Vab I = εI − I 2 r
Vab = ε + Ir
I enters the source
⇒ P = Vab I = εI + I 2 r
Internal resistance
dissipates energy
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Energy conversion
Where did this go?
(Think about your rechargeable battery)
Power and energy in Electric Circuits
•  See circuit below:
•  What is the rate of energy
conversion?
•  The rate of dissipation of
energy in the battery? The net
power output of the battery?
See short circuit below:
What is the dissipation of
energy in the battery?
Q25.6
In the circuit shown, the two bulbs A and B are
identical. Compared to bulb A,
A. bulb B glows more brightly.
B. bulb B glows less brightly.
C. bulb B glows just as brightly.
D. answer depends on whether the mobile charges in the wires are positively or
negatively charged
Q25.7
In the circuit shown in (a), the two bulbs A and B are identical. Bulb B is removed and the
circuit is completed as shown in (b). Compared to the brightness of bulb A in (a), bulb A in
(b) is
A. brighter.
B. less bright.
C. just as bright.
D. any of the above, depending on the rated wattage of the bulb.
Q25.8
An ideal voltmeter
A. has zero resistance and should be connected in parallel with the circuit
element being measured.
B. has zero resistance and should be connected in series with the circuit
element being measured.
C. has infinite resistance and should be connected in parallel with the circuit
element being measured.
D. has infinite resistance and should be connected in series with the circuit
element being measured.
Q25.9
An ideal ammeter
A. has zero resistance and should be connected in parallel with the circuit
element being measured.
B. has zero resistance and should be connected in series with the circuit
element being measured.
C. has infinite resistance and should be connected in parallel with the circuit
element being measured.
D. has infinite resistance and should be connected in series with the circuit
element being measured.