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 v avg
≈ 10 5 − 6 m / s ∝ T
= 0
The external field destroys the 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!
I = dQ dt
1ampere=1A=1C/s
Is current a vector?
What does positive and negative current mean?

Current, Drift Velocity and Current Density v d
: drift velocity
Example: A copper lamp wire has a cross sectional area A=8.17x10
-7 m 2 , and carries a
Current of 1.67A. The free electron density is n=8.5x10
28 /m 3 . What is the current density J and drift velocity v d
. How does it compare with the random motion of the electrons ?
€ dQ = nqv d
Adt
I = dQ dt
= nqv d
A
J =
I

A
J = nq
= nqv d
 d n: concentration of particles
I is not a vector!
J is a vector : same direction as E (and v d
)
If A changes, does I change? Does J change?

Resistivity is intrinsic to a metal sample (like density is)
€
Certain Diamond (CVD) 10 18
Ohm’s law: (At a given temperature), the ratio of the magnitudes of E and J is nearly constant (for some materials)
E
ρ =
J
Ω⋅ m = ( V / m ) /( A / m 2 ) = ( V / A ) ⋅ m
1 Ω = 1 V / A
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.
( T ) =
0
[1 +
( T
T
0
)]
Semiconductor can be used to
measure temperature (thermistor)
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 = ρ

J ⇒
V
L
= ρ
V = ( ρ
I
⇒
A
L
A
) I = RI
( R ≡ ρ
L
A
) Resistance: R=V/I
1ohm=1 Ω =1V/A
: Intrinsic quality of a material
: Depends on the geometry of the material
Example:2 conducting copper wires with different diameters (D
0
and D
1
=3D
0 and different lengths (L
0
and L
1
=16L
0
What’s the ratio of their resistors?
).
),
R
1
:R
0
=?
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 − T
0
)]
L
R ≡ ρ ⇒
A
R ( T ) = R
0
[1 + α ( T − T
0
)]
€
Example: The resistance of a wire is 0.97
Ω at
0 o C, and 1.38
Ω at 100 o
What is it’s resistance
C.
At 20 o C?

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.
L
R = ρ
A dr dR = ρ ⇒
A = 2 π rL
R = ∫ dR = ∫ a b
ρ dr
2 π rL
=
ρ
2 π L ln b a
€

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 = V ab
= 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).
€
•   Can you figure out what is the value of r?
V ab
= E − Ir
V
0
= E
V
1
= E − Ir = IR ⇒
V
1
/ V
0
=
IR
I ( R + r )
⇒ r = ?
R
A function of V
0
and V
1

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.

€
Energy and Power in Electric Circuits: Power into a pure resistance
In a time dt, charge (dQ=Idt) experiences potential
Change of V ab
,
Therefore the time rate of energy transfer
(output/input) is P, power of the circuit element:
P = dE / dt dE = V ab dQ = V ab
Idt
⇒ P = V ab
I
Unit: (1 J/C) (1C/s) = 1J/s=1W
Power input to a pure resistance:
V ab
= IR ⇒ €
P = ( IR ) I = I 2 R =
V 2 ab
R
V a
>V b
for a resistor, the current enters it.
Where does this energy transferred to the resistor become:
How doe T (temperature) change?

€
Power output of a source/input to a source
Power output of a source:
V ab
= ε − Ir I leaves the source
⇒ P = V ab
I = ε I − I 2 r
Internal resistance dissipates energy
Power input to a source:
V ab
= ε + Ir I enters the source
⇒ P = V ab
I = ε I + I 2 r
Where did this go?
(Think about your rechargeable battery)
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Energy conversion

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.