Lecture 12
Electric Current
Suppose that charges are moving perpendicular to a surface of area A, as shown in
Figure 1. (This area could be the cross-sectional area of a wire, for example.) The
current is the rate at which charge flows through this surface.
Figure 1. Charges in motion
through an area A. The time rate at
which charge flows through the
area is defined as the current I.
The direction of the current is the
direction in which positive charges
flow when free to do so.
If ΔQ is the amount of charge that passes through this area in a time interval Δt, the
average current Iav is equal to the charge that passes through A per unit time:
(1)
If the rate at which charge flows varies in time, then the current varies in time; we
define the instantaneous current I as the differential limit of average current:
(2)
The SI unit of current is the ampere (A):
(3)
That is, 1 A of current is equivalent to 1 C of charge passing through the surface
area in 1 s.
The charges passing through the surface in Figure 1 can be positive or negative,
or both. It is conventional to assign to the current the same direction as the
flow of positive charge. In electrical conductors, such as copper or aluminum, the
current is due to the motion of negatively charged electrons. Therefore, when we
speak of current in an ordinary conductor, the direction of the current is opposite
the direction of flow of electrons.
Resistance
The electric field inside a conductor is zero. However, this statement is true only if
the conductor is in static equilibrium. What happens when the charges in the
conductor are not in equilibrium, in which case there is an electric field in the
conductor.
Consider a conductor of cross-sectional area A carrying a current I. The current
density J in the conductor is defined as the current per unit area. Because the
current I = nqvdA, the current density is
(4)
where J has SI units of A/m2. This expression is valid only if the current density is
uniform and only if the surface of cross-sectional area A is perpendicular to the
direction of the current. In general, current density is a vector quantity:
(5)
From this equation, we see that current density is in the direction of charge motion
for positive charge carriers and opposite the direction of motion for negative
charge carriers.
A current density J and an electric field E are established in a conductor
whenever a potential difference is maintained across the conductor. In some
materials, the current density is proportional to the electric field:
(6)
where the constant of proportionality & is called the conductivity of the conductor.
Materials that obey Equation 27.7 are said to follow Ohm’s law, named after
Georg Simon Ohm (1789–1854). More specifically, Ohm’s law states that
for many materials (including most metals), the ratio of the current density to the
electric field is a constant & that is independent of the electric field producing the
current.
Materials that obey Ohm’s law and hence demonstrate this simple relationship
between E and J are said to be ohmic. Experimentally, however, it is found that not
all materials have this property. Materials and devices that do not obey Ohm’s law
are said to be nonohmic. Ohm’s law is not a fundamental law of nature but rather
an empirical relationship valid only for certain materials.
Figure 27.5 A uniform conductor of length ! and cross-sectional area A. A potential
difference ΔV = Vb - Va maintained across the conductor sets up an electric field E,
and this field produces a current I that is proportional to the potential difference.
A potential difference ΔV = Vb -Va is maintained across the wire, creating
in the wire an electric field and a current. If the field is assumed to be uniform, the
potential difference is related to the field through the relationship
Therefore, we can express the magnitude of the current density in the wire as
Because J = I/A, we can write the potential difference as
The quantity
is called the resistance of the conductor. We can define
the resistance as the ratio of the potential difference across a conductor to the
current in the conductor:
(7)
We will use this equation over and over again when studying electric circuits.
From this result we see that resistance has SI units of volts per ampere. One volt
per ampere is defined to be one ohm Ω:
(9)
This expression shows that if a potential difference of 1 V across a conductor
causes a current of 1 A, the resistance of the conductor is 1Ω. For example, if an
electrical appliance connected to a 120-V source of potential difference carries a
current of 6 A, its resistance is 20 Ω.
The inverse of conductivity is resistivity3 ρ:
(10)
where ρ has the units ohm-meters ( Ω.m). Because
the resistance of a uniform block of material along the length
, we can express
as
(11)
Resistance and Temperature
Over a limited temperature range, the resistivity of a conductor varies approximately
linearly with temperature according to the expression
(12)
where ρ is the resistivity at some temperature T (in degrees Celsius), ρ0 is the resistivity
at some reference temperature T0 (usually taken to be 20°C), and - is the temperature
coefficient of resistivity. From Equation 12, we see that the temperature coefficient of
resistivity can be expressed as
(13)
where
is the change in resistivity in the temperature interval
Because resistance is proportional to resistivity (Eq. 27.11), we can write the variation of
resistance as
(14)
Superconductors
There is a class of metals and compounds whose resistance decreases to zero when
they are below a certain temperature Tc , known as the critical temperature. These
materials are known as superconductors.
Electrical Power
Let us consider the rate at which the system loses electric potential energy as the
charge Q passes through the resistor:
where I is the current in the circuit.
The power , representing the rate at which energy is delivered to the resistor, is
(15)
the fact that !V " IR for a resistor, we can express the power delivered to the
resistor in the alternative forms
(16)
When I is expressed in amperes, ΔV in volts, and R in ohms, the SI unit of power is the watt.
The process by which power is lost as internal energy in a conductor of resistance R is often
called joule heating ; this transformation is also often referred to as an I 2R loss.