Download CHAPTER 25: CURRENT, RESISTANCE, AND EMF • So far we

CHAPTER 25: CURRENT, RESISTANCE, AND EMF
• So far we have studied electric interactions where the charges involved are
at rest. In this chapter, we will now consider charges that are in motion.
• We will do so first by studying circuits and the motion of charge through
them – current
Current
• A current is the motion of charge from one region to another. More
specifically, it is the rate at which charge crosses a given area.
• For a conductor in electrostatic equilibrium, the electric field is zero
everywhere inside the conductor, the charges are at rest, and there is no
current
• When we apply an electric field to a metal, the field exerts a force (F = eE)
on the electrons and they begin to accelerate
• Collisions between the electrons and the atoms of the metal slow them
down, transforming the kinetic energy of the electron into thermal energy,
making the metal warmer
• The motion of the electrons will cease unless you continue pushing by
maintaining an electric field
• In a constant field, an electron’s average motion will be opposite the field –
this motion is called the electron’s drift velocity vd, which corresponds to a
net current
• We learned that the electric field is zero inside a conductor in electrostatic
equilibrium but a conductor with charges moving through it is NOT in
electrostatic equilibrium
• In different materials, the charges of moving particles can be positive or
negative but in conductors, the moving charges are always electrons
(negative)
• Figure (a) shows positive charges drifting to the right because the electric
force is in the same direction as the field
• Figure (b) shows negative charges drifting to the left because the electric
force is opposite to the direction of the field
• In both cases there is a net flow of positive charge to the right OR you could
also say there is a net flow of negative charge to the left – these are
equivalent statements. We can either view current as negative charges
moving in one direction OR view current as positive charges moving in the
other direction – it makes no difference.
• So, like most textbooks, we adopt the convention that current is the flow of
positive charge – this is commonly referred to as “conventional current” (as
opposed to “electron current”, which is in the opposite direction)
• We will use the symbol I to denote conventional current
• I is defined as the rate at which net charge crosses a given area (units: C/s,
defined to be an ampere, A)
• For a steady current (flow of charge is constant in time), I = ΔQ/Δt. For
current that varies in time, the instantaneous current is I = dQ/dt.
• Like charge, current has an algebraic sign, but neither are vectors – both
charge and current are scalars
• We can relate the current to the drift velocity of the moving charges
• Consider the situation shown in the figure below, and let A be the crosssectional area of the cylinder, perpendicular to the field E
• Let n denote the concentration of charge carriers – the number of moving
charged particles per unit volume (units: 1/m3) and assume that all particles
have the same drift velocity
• In a small time interval Δt, each particle moves a distance Δx = vdΔt
• The volume of the slice shown above (with thickness Δx) is AΔx or AvdΔt. If
each particle has charge q, the amount of charge that flows out of the right
end of this slice is given by ΔQ = nqAvdΔt
• So, the current is I = ΔQ/Δt = n|q|Avd
• The current density J is defined as the current per unit cross-sectional area
and is given by J = I/A = n|q|vd (units: A/m2)
• If the moving charges are positive, the drift velocity is in the same direction
as the electric field; if the moving charges are negative, the drift velocity is
in the opposite direction as the electric field
• We can define the current density as a vector that takes into account the
�⃗d
direction of the drift velocity: 𝑱𝑱⃗ = 𝑛𝑛𝑛𝑛𝒗𝒗
**SEE EXAMPLE #1**
Resistivity
• The current density in a conductor depends on the electric field and the
material properties of the conductor – for many conductors at a given
temperature, 𝑱𝑱⃗ ∝ ��⃗
𝑬𝑬, and the ratio E/J is constant
• This relationship is called Ohm’s “law”. Like Hooke’s “law,” it is an
idealized model that is only valid under certain circumstances
• The ratio of the magnitudes of electric field to current density is called the
resistivity of the material, ρ = E/J. For a given current density, the larger
the resistivity of a metal is, the larger the field must be.
• For ohmic materials, ρ is constant; in contrast, the resistivity of nonohmic
materials depends on the field strength: ρ = ρ(E)
• The units of resistivity are: (V/m)/(A/m2) = V*m/A. We will define another
unit that we will soon discuss – the ohm: 1 Ω = 1 V/A. In terms of Ω,
resistivity has units: Ω*m
• An ideal conductor has zero resistivity and an ideal insulator has infinite
resistivity
• The reciprocal of resistivity is called conductivity: σ = 1/ρ (units: (Ω*m)-1)
��⃗ or
• The microscopic version of Ohm’s law is generally written as 𝑱𝑱⃗ = 𝜎𝜎𝑬𝑬
��⃗
𝑬𝑬 = 𝜌𝜌𝑱𝑱⃗. This describes the behavior at a single point in a material in terms
of the electric field and the current density at that point. We will discuss the
macroscopic version of Ohm’s law, but first let’s discuss the dependence of
resistivity on changes in the temperature of the material.
Dependence on Temperature
• In almost all cases, the resistivity of metallic conductors increases with
increasing temperature
• Physically, the ion cores of a conductor vibrate with larger amplitude as
temperature increases, making it more probable that a collision with a
moving electron occurs. These collisions cause the drift velocity to decrease,
which leads to a reduction in the current.
• Over a certain range of temperatures, the resistivity of a metal can be
modeled as having a linear dependence on temperature T: 𝜌𝜌(𝑇𝑇) = 𝜌𝜌0 [1 +
𝛼𝛼(𝑇𝑇 − 𝑇𝑇0 )]. 𝜌𝜌0 is the resistivity at some reference temperature 𝑇𝑇0 and 𝛼𝛼 is
called the temperature coefficient of resistivity.
Resistance
• ��⃗
𝑬𝑬 and 𝑱𝑱⃗ are microscopic quantities and difficult to measure. The
��⃗ is the potential difference V and
macroscopic quantity corresponding to 𝑬𝑬
the current I is the macroscopic quantity corresponding to 𝑱𝑱⃗.
• Consider the case of a section of conducting wire with length L and crosssectional area A shown below. The potential difference between the ends is
V and the field always points “downhill” (from higher potential to lower
potential). The force on the positive charges is in the same direction as the
field, so the (conventional) current I is also in the direction of decreasing
potential.
• If the current density and the electric field are uniform then the current is I =
JA and the potential difference is V = EL. ρ = E/J = (V/L)/(I/A). Thus,
𝑉𝑉 =
𝜌𝜌𝜌𝜌
𝐴𝐴
𝐼𝐼
𝜌𝜌𝜌𝜌
• The quantity is defined as the resistance R of the conductor and is the
𝐴𝐴
macroscopic version of resistivity
• The macroscopic version of Ohm’s law is then V = IR, where 𝑅𝑅 =
𝜌𝜌𝜌𝜌
𝐴𝐴
(units:
Ω = V/A) and is usually taken to be constant (only true for ohmic materials)
• Resistance increases with the length of the wire and decreases with its crosssectional area. As long as Ohm’s law is obeyed, there is a linear relationship
between I and V, and the slope is 1/R (see figure below).
• In addition to understanding how to use equations quantitatively, it is a good
idea to also have a qualitative understanding of how a particular physical
quantity changes when you vary some other quantity (see figure above)
**SEE EXAMPLE #2**
Electromotive Force and Circuits
• If a conductor is not part of a closed loop (a complete circuit), then a steady
current through it is not possible.
• Imagine establishing an electric field ��⃗
𝑬𝑬1 in a conductor such as the wire
segment shown in the figure below. Current begins to flow in the direction
of the applied field and positive charge builds up on the right end which
causes a negative charge buildup on the left end (conservation of charge).
• This separation of charge creates a field ��⃗
𝑬𝑬2 in the direction opposite to the
applied field. As the charge density on the ends increases so does the field
strength of E2 until the net field is zero and the current stops flowing.
• Let’s now consider a complete circuit and how to establish a steady current
• Because the electric force is conservative, there can be no change in energy
for a charge q making one complete circuit around a closed loop (ΔUloop = 0)
• As discussed earlier, current flows thru a conductor in the direction of
decreasing potential. So, if ΔVloop = 0, then there must be some part of the
circuit where the potential also increases.
Electromotive Force
• The interaction that causes current to flow “uphill” – from lower to higher
potential is called the electromotive “force” (abbreviated emf). This is not a
force but rather a source of voltage that does work on a charge, increasing its
potential (or potential energy). It has units of energy per unit charge (J/C =
V) and the physical source of emf is often a battery. We will use the symbol
ℰ to denote emf.
• We will often treat a source of emf in a circuit as ideal, meaning that it
maintains a constant potential difference across its terminals, independent of
the amount of current through it
• So inside of an emf source, some type of energy (chemical, mechanical, etc)
is converted into electric potential energy, causing the positive charges to
move “uphill” or “against the field”. Once the charge reaches the positive
terminal and enters the wire, it flows through the wire towards the negative
terminal of the source and its potential (energy) decreases. It then enters the
emf source and moves “uphill” again and the process repeats.
• For an ideal emf source connected to a resistor R, the potential increase due
to the emf ℰ is exactly equal to the potential drop IR as the current flows
thru the remainder of the circuit: ℰ = 𝐼𝐼𝐼𝐼
Internal resistance
• Real sources of emf do not behave like this – the potential difference
between the terminals is not equal to its emf. This is because there is always
some amount of resistance in any material and we call this the internal
resistance r of the source.
• Assuming this resistance obeys Ohm’s law, the potential difference between
the terminals of an emf source is ∆𝑉𝑉 = ℰ − 𝐼𝐼𝐼𝐼
• For a real emf source, the potential increase is now ℰ − 𝐼𝐼𝐼𝐼, which is exactly
equal to the potential drop IR as the current flows thru the remainder of the
circuit: ℰ − 𝐼𝐼𝐼𝐼 = 𝐼𝐼𝐼𝐼 so 𝐼𝐼 =
ℰ
𝑅𝑅+𝑟𝑟
• We will use the concept of an ideal wire  R = 0. So, the potential
difference everywhere in the wire is also zero, even if there is current in it.
• An ideal voltmeter has infinite resistance so that it can measure the
potential difference between two points without changing it, which it would
if some of the current flowed thru it
• The resistance of an ideal ammeter is zero so that it can measure the
current without changing it. If its resistance were not zero, then the current
thru it would be smaller than it is in the circuit.
**SEE EXAMPLE #3**
Potential Changes in a Circuit
• Because the electric force is conservative, the net change in potential energy
for a charge making one complete trip around the circuit (a closed loop)
must be zero. This means that the change in potential around the circuit is
also zero. There is a potential gain across a source of emf and a potential
drop associated with the resistance due to other circuit elements.
Energy and Power in a Circuit
ℰ − 𝐼𝐼𝐼𝐼 − 𝐼𝐼𝐼𝐼 = 0
• So we have talked about how electric potential increases or decreases as one
goes around a circuit. This leads to changes in the potential energy
associated with the electric potential of the circuit and the charge Q moving
thru it.
• A circuit involves various conversions of one form of energy to another as
charges traverse different components of the circuit. Often we are interested
in the rate at which this energy transfer takes place.
• Consider some component in a circuit with a steady current I in it. Then in a
time interval ∆𝑡𝑡, an amount of charge ∆𝑄𝑄 = 𝐼𝐼∆𝑡𝑡 has passed thru it
• Denote the potential difference between the endpoints associated with the
time interval ∆𝑡𝑡 as ∆𝑉𝑉. Then the change in potential energy is ∆𝑈𝑈 =
∆𝑄𝑄∆𝑉𝑉 = 𝐼𝐼∆𝑡𝑡∆𝑉𝑉. Dividing thru by ∆𝑡𝑡 gives an expression for the rate of
energy transfer between the moving charges and a circuit element. This is
usually referred to as power, P.
•
𝑃𝑃 =
∆𝑈𝑈
∆𝑡𝑡
= 𝐼𝐼∆𝑉𝑉, here ∆𝑉𝑉 is the potential difference across the entire circuit
element (as usual, the SI unit for power is the watt: 1 W = 1 J/s)
Resistor
• Circuit elements called resistors have a fixed resistance R and the potential
drop ∆𝑉𝑉 across a resistor depends on the current I flowing thru it, ∆𝑉𝑉 = 𝐼𝐼𝐼𝐼.
Using Ohm’s law, the rate at which electric potential energy is converted to
heat is given by 𝑃𝑃 = 𝐼𝐼∆𝑉𝑉 = 𝐼𝐼2 𝑅𝑅 =
∆𝑉𝑉 2
𝑅𝑅
. The current always enters a resistor
at a higher potential and exits at a lower potential.
• At an atomic level, the moving charges collide with the atoms in the resistor
and some of their potential energy gets transferred to (or dissipated in) the
resistor mostly in the form of heat or thermal energy
Emf Source
• A circuit element that supplies an emf to a circuit has a potential difference
across its terminals. The current enters at the lower potential terminal
(negative) and exits at the higher potential terminal (positive).
• If the emf source also has an internal resistance then the potential difference
between the terminals ∆𝑉𝑉 = ℰ − 𝐼𝐼𝐼𝐼, which is positive in the direction of the
current
• So instead of the moving charges delivering energy to a circuit element, as
with a resistor; an emf source delivers energy to the free charges in the
conductor. In the case of a battery, chemical energy is converted to electric
potential energy.
• The rate at which this energy transfer takes place is 𝑃𝑃 = 𝐼𝐼∆𝑉𝑉 = 𝐼𝐼ℰ − 𝐼𝐼2 𝑟𝑟
The term 𝐼𝐼ℰ represents the rate at which nonelectrical energy is converted to
electrical energy. 𝐼𝐼 2 𝑟𝑟 represents the rate at which electrical energy is
dissipated due to the internal resistance of the source. 𝐼𝐼ℰ − 𝐼𝐼 2 𝑟𝑟 is the net
power output of the source.
**SEE EXAMPLE #4**
Microscopic Theory of Metallic Conduction
• In the figure above, the ball represents a free electron colliding with the
stationary ion cores of the conductor. The incline of the ramp indicates that
the charge is moving in the direction of decreasing potential (in the direction
of the electric field).
• The average time between collisions is called the mean free time, τ
• Earlier we defined the resistivity as ρ = E/J and the current density as J =
nqvd
• Metals make good conductors because they have a large number of free
electrons that are readily accelerated by an applied electric field. The
positive ion cores assemble in a crystal lattice and in the absence of an
applied field, the free electrons move randomly thru this lattice at high
speeds (~106 m/s) and, as a result, collide frequently with the ions and
rebound in random directions. With no applied field, this random motion
results in a zero average drift velocity (i.e., no net current).
• Applying an electric field to the conductor gives the free electrons an
acceleration in a direction opposite the field. They already have large kinetic
energies, so an electron does not gain much energy from the field before
colliding with another ion core. The result is that they acquire a small
average velocity in a direction opposite the field – the drift velocity, vd. This
constitutes a net current that is proportional to vd.
• The drift velocity depends on 2 things: 1) the acceleration of the electrons
and 2) the rate at which they undergo collisions
• The acceleration is due to the electric force and opposite to the field,
�𝒂𝒂⃗ =
�⃗
𝑭𝑭
𝑚𝑚
=−
�⃗
𝑒𝑒𝑬𝑬
𝑚𝑚
. This causes a change in the electron’s velocity during the
time interval between collisions which, on average, is the mean free time.
Thus, the electron’s average velocity due to the presence of the electric field
(the drift velocity) is �𝒗𝒗⃗𝑑𝑑 = −
�⃗
𝑒𝑒𝑬𝑬
𝑚𝑚
𝜏𝜏 (~10-3 m/s). This is in addition to the
random, high-speed motion that averages out to zero.
�⃗𝑑𝑑 =
• We defined the current density in terms of the drift velocity as 𝑱𝑱⃗ = 𝑛𝑛𝑛𝑛𝒗𝒗
�⃗𝑑𝑑 =
−𝑛𝑛𝑛𝑛𝒗𝒗
𝑛𝑛𝑒𝑒 2 �𝑬𝑬⃗
𝑚𝑚
𝜏𝜏. Comparing this with the microscopic version of Ohm’s
2
��⃗� shows that the conductivity can be expressed as 𝜎𝜎 = 𝑛𝑛𝑒𝑒 𝜏𝜏 or
law �𝑱𝑱⃗ = 𝜎𝜎𝑬𝑬
the 𝜌𝜌 =
𝑚𝑚
𝑛𝑛𝑒𝑒 2 𝜏𝜏
.
𝑚𝑚
• The time between collisions depends on several factors: the spacing of the
ions in the crystal lattice, the presence of impurities, and how fast the
electrons are moving
• There are two kinds of motion of the electrons – the high-speed, random
thermal motion and the slow, drift velocity acquired due to the presence of
the applied field. vd is essentially negligible in determining the speed of the
electrons and is not a factor in determining the time between collisions.
• This means that the time between collisions and the conductivity are
essentially independent of the applied electric field. If, also, the electron
number density n does not depend on the applied electric field (which is
��⃗) is a linear relationship for
usually the case) then Ohm’s law (𝑱𝑱⃗ = 𝜎𝜎𝑬𝑬
metallic conductors.
• Almost all metals are ohmic (they obey Ohm’s law) provided the
temperature of the metal is not too high and the applied electric field
strength is not too large