Chapter 25
Current Resistance, and Electromotive Force
1
Current
In previous chapters we investigated the properties of “charges at rest.” In this
chapter we want to investigate the properties of “charges in motion.” An electric
current consists of charges in motion from one region to another. If the charges
follow a conducting path that forms a closed loop, the path is called an electric
circuit.
1.1
The direction of current flow
~ is established inside a conductor, the “free”
When a constant, steady electric field E
~
electrons collectively move in a direction due to a steady force F~ = q E.
~ field. The
Figure 1: This figure shows the random motion of electrons in a conductor without an E
~ field is present.
second figure shows the collective motion of electrons in a conductor when an E
1
The collective motion of electrons in a conductor is characterized by their drift
velocity v~d . While the random motion of electrons has a very fast average speed
of ∼ 106 m/s, the drift speed is very slow, often on the order of 10−4 m/s. The
electric field is set up throughout the conductor at velocities approaching the speed
of light, v ∼ c.
Figure 2: The first figure (a) shows the direction of the conventional current, the flow of positive
charges. The second figure (b) shows the flow of electrons in a metallic conducting moving in the
opposite direction with respect to the current.
2
1.2
Current, Drift Velocity, and Current Density
I =
dq
∆Q
=
∆T
dt
(the definition of current)
(1)
Figure 3: The current I is the time rate of charge transfer through the cross-sectional area A. The
~
current is in the same direction as E.
dQ = q(nAvd dt) = nqvd A dt
I =
dQ
= nqvd A
dt
The current per unit cross-sectional area is called the current density J:
J =
J~ = nq ~vd
I
= nqvd
A
(the current density)
(the current density in vector form)
3
(2)
(3)
2
Resistivity
~ and on the
The current density J~ in a conductor depends on the electric field E
properties of the material. The dependence can be quite complex, but for some
materials, especially metals at a given temperature, J~ is nearly directly proportional
~ and the ratio of the magnitudes of E and J is constant. When materials
to E.
exhibit this kind of behavior, it is said that they follow Ohm’s Law.
~ = 1E
~
J~ = σ E
ρ
where σ is the conductivity and ρ is the resistivity shown in the table.
(4)
Figure 4: This table show the resistivities of various materials at room temperature (20o C)
4
2.1
Resistivity and Temperature
Figure 5: These figures show how the resistivity depends with the absolute temperature T for (a)
a normal metal, (b) a semiconductor, and (c) a superconductor.
The temperature dependence of the resistivity ρ can be written as:
ρ(T ) = ρo [1 + α (T − To )]
where α is the temperature coefficient of resistivity.
5
(5)
Figure 6: This table shows the coefficient of resistivities for various materials near room
temperature.
3
Resistance
~
For a conductor with resistivity ρ, the current density J~ and the electric field is E
are related by the following equation:
~ = ρJ~
E
(6)
Figure 7: This figure shows a conductor with uniform cross sectional area A. The current density
is uniform over any cross section, and the electric field is constant along the length.
Let V bet the potential difference between higher-potential and the lower-potential
ends of the conductor, so that V is positive. V is the voltage across the con6
ductor. The direction of the current is always from the higher-potential to the
lower-potential end.
We can also relate the value of the current I to the potential difference between
the ends of the conductor. Using the definition of current density J, we can write
the current as I = JA. Likewise, the potential difference V between the ends is
V = EL. Rewriting Eq. 6 above, we find:
V
ρI
=
L
A
or
V =
ρL
A
I
The ratio of V to I for a particular conductor is called its resistance R:
R =
V
I
or
V = IR
We identify the resistance R to be R = ρL/A.
7
(Ohm’s Law)
(7)
3.1
Interpreting Resistance
Figure 8: This resistor has a resistance of 57 kΩ with an accuracy (tolerance) of ±10%. This is
assuming that the color of the 3rd band is “orange.”
Figure 9: This table shows the color codes used for labeling the resistances of resistors.
The Resistance is Dependent Upon Temperature
R(t) = Ro [1 + α (T − To )]
where T0 is often taken to be 0o C or 20o C. The temperature coefficient of resistance
is represented by α, and the change in resistance is R0 α(T − T0 ).
8
Figure 10: This figure shows the current-voltage relationships for two devices. In figure (a) the
resistor obeys Ohm’s law and the current I is proportional to the voltage V . In figure (b), a
semiconductor diode, the current rises steeply with a modest amount of voltage, and the current
flows only in the positive direction.
4
Electromotive Force and Circuits
In order for a conductor to have a steady current, it must be part of a path that
forms a closed loop or complete circuit. Otherwise, opposite charges would even~ field that produced the initial
tually build up on the two ends and negate the E
separation of charges.
9
Figure 11: This figure demonstrates why a steady current cannot persist in a section of conductor
~
with a constant external E-field
unless it is part of a circuit. Charges eventually accumulate on the
~
~ field to E ≈ 0. J~ = 1 E.
two ends of the wire ultimately reducing the E
ρ
10
4.1
Electromotive Force
In order to produce a steady current in a circuit, there has to be a device somewhere that acts to raise the potential (measured in volts) from low potential to
high potential. Although “in the circuit” the electrostatic force is trying to push
positive charge from high potential to low potential, the device that raises the
potential pushes positive charges in the opposite sense (i.e., from low potential to
high potential ). Thus, the device must provide a force not derived from electrostatic means, buy by some other process. The non-electrostatic force that raises
the voltage in a circuit is called an electromotive force, or emf, and is many times
written as E.
Figure 12: This is a schematic diagram of a source of emf (E) in an “open-circuit” situation. The
~ and the nonelectrostatic force F~n are shown acting on a positive charge q.
electric-field force F~e = q E
The voltage across the battery terminals–not connected.
The most common source of emf is called a battery which uses an electrochemical
process to move charge from low potential to high potential. When the battery is
not connected, the voltage measured across the terminals is Vab = E.
Vab = E
Voltage measured across the battery–not connected
11
Figure 13: This is schematic diagram of an ideal emf source in a complete circuit. The electric-field
~ and the nonelectrostatic force F~n are shown for a positive charge q. The current is
force F~e = q E
in the direction from a to b in the external circuit and from b to a within the source.
12
4.2
Internal Resistance
When the battery is connected to a simple circuit as shown in Fig. 13, current
begins to flow through the whole circuit, including the battery. Because there
is a small internal resistance r inside the battery, there is a slight voltage drop
measured at the battery terminal Vab = E − Ir.
Vab = E − Ir
4.3
Voltage measured across the battery–when connected
Symbols for Circuit Diagrams
Figure 14: This table shows the common symbols found in a a circuit diagram.
13
4.4
Potential Changes around a Circuit
Figure 15: This figure shows the potential rises and drops in a circuit.
14
5
Energy and Power in Electric Circuits
When a charge q passes through a circuit element, there is a change in potential
energy equal to qVab . The potential energy decreases as the charge “falls” from potential Va to a lower potential Vb . However, the moving charge does not gain kinetic
energy (because there is conservation of charge and flux –current is constant).
In electric circuits we are more interested in the rate at which energy is either
delivered to, or extracted from, a circuit element. If the current through the element
is I, then a charge dQ = I dt passes through the element in a time interval dt.
The change in potential energy for this amount of charge crossing a potential
difference Vab is dU = Vab dQ = Vab I dt. The power delivered to the circuit
element must be
Power =
dU
= Vab I
dt
(power delivered to a circuit element)
(8)
Figure 16: This figure shows the power input to the circuit element between a and b .
P = (Va − Vb )I = Vab I.
5.1
Power Input to a Pure Resistance
Power = V I
V2
R
=
15
=
I2 R
(9)
5.2
Power Output of a Source
Figure 17: This figure shows the energy conversion in a simple circuit.
Power output of a source–a battery
Power = Vab I = (E − Ir)I = EI − I 2 r
16
5.3
Power Input to a Source
Figure 18: This figure shows two sources connected in a simple loop circuit. The source with the
large emf delivers energy to the other source.
Power input to a source–a battery
Power = Vab I = (E + Ir)I = EI + I 2 r
17