Electric Currents, Resistors, and DC-Circuits
(Chapters 30, 31)
• Electric current I
• Resistance R and resistors
• Relation between current and resistance: Ohm’s Law
• Resistivity ρ
• Energy dissipated by current. Electric power
• Electromotive force: emf, ε
• Simple resistive circuits:
• Series and parallel circuits. Circuits reducible to simple combinations
• Circuits non-reducible to series and parallel: Kirchhoff Rules
• Resistors and capacitors combined: dc RC circuit
Electric Current – Charge Carrier Motion in a Conductor
• In an electrically insulated wire, the electrons undergo thermal random motion
• When an potential difference is set up between the ends of a wire, the surface charge
gets nonuniformly distributed, producing an electric field inside the wire, such that
electrons tend to drift against the average field producing an electric current, I
more positive
V+
a
++
+
+
+
+
+
Eb
+
Ea
Enet
b+
Eb
–
Ec
Enet
+
c
more negative
V–
–
–
• When a circuit is completed, the electric field sets in the wires with the speed of
light (of the order of 108 m/s), and all free electrons throughout the material move
very fast; however, their motion is impeded by collisions in the material, so their
drift speed, vd, is much smaller, (of the order of 10 –4 m/s)
Ex: The zigzag lines represent the drift against
the electric field of an electron in a conductor.
The drift is analog with the parabolic
trajectories of a ball drifting due to gravity
down a wall with pegs.
-e –
+
m
+
+
ion cores
+
+
+
+ E
+
+
pegs
vd
g
• The nature of the charges carrying the current depends on the nature of the material
Ex: electrons in metals, electrons and ions in plasma, holes in semiconductors, etc.
Electric Current – Definition
• So, whenever electric charges of like signs drift, an electric current is said to exist
• In order to define it quantitatively, consider at the charge flowing through a wire
perpendicular to a cross-sectional surface of area A
Def: The rate at which the electric charge flows through this
surface, that is, the amount of charge dQ flowing per unit of
time dt through the surface, is called an electric current
dQ
I
dt
Q
A
I
SI

1C
 Ampere (A)
1s
Comments:
• Conventional current direction: the direction of the current is the direction of the
drift of positive charge: that is, in the direction of the average field, or from high
potential to low potential
• Albeit its directionality, the current is not a vector
• The current across a potential difference is the same through any cross-section, that
is, the carriers drift identically as long as the conductive substance is the same
• The colloquial short for “Amperes” is “Amps”
• Order of magnitude: flashlight bulb ~1A, sensitive electronics ~μA, high-power
devices (such as large electro-magnets) ~kA
Electric Current – Relation to drift speed. Current density
• To find the relationship between the macroscopic
electric current and the microscopic details of the
electric-carrier drift, consider elementary charge
carriers of charge e drifting with constant drift speed
vd through a current carrying conductor of crosssectional area A, with the charge concentration given
by:
Number of carriers
n
Volume
• Then, as shown on the figure,
Q e
n
 Q  neAx
Ax
• Therefore, combining with the definition of current, we find that the current is related
to the drift speed vd as following:
Q
x
I
 neA
 I  neAvd
t
t
• So, if the drift speed is small (many collisions), the
current is small and vice versa.
• If the carriers are electrons (charge –e), the current
opposes the drift velocity.
Def: The current per unit cross-sectional area of a conductor
is a vector called current density:
I
J   nqvd
A
Quiz 1: We learned that the electron drift speed is relatively small: of the order of 10 –4 m/s.
How come, when the circuit in the figure is closed, the bulb light up almost instantaneously?
a) Actually it doesn’t: in my house we wait minutes
until bulbs light up…
b) Since the energy of the electric field is delivered to
the bulb before electrons from the battery reach to it
c) Because the “sea of electrons” fill all the circuit.
When the circuit is closed, the field is set with the
speed of light and electrons start to drift through
each cross-section of the circuit including the bulb.
Quiz 2: Both segments of the shown wire are made of the same metal. Current I1 flows into
segment 1 from the left. How does current I1 in segment 1 compare to current I2 in segment 2?
a) I1 > I2
b) I1 = I2
c) I1 < I2
d) There’s not enough information to compare them
Ohm’s Law – Statement
• Experiments show that in most metals the current density J depends on:
• the electric field E
• the tabulated properties of the material: resistivity ρ or conductivity σ = ρ–1
• The dependency is given by
Ohm’s Law: In an ohmic (or linear) material at constant temperature, the
resistivity and conductivity remain constant for any electric field and the current
density is proportional to the field:
J   E  E  J
constant
Comments:
• Ohm’s Law does not state the proportionality E ~ J, but the constancy of ρ
• For most metals, resistivity increases Ex: Only some metals follow this behavior
through the whole range of temperatures. The
approximately linearly with
resistivity of superconductors drops sharply to
temperature T, over limited T-ranges:
zero under a critical T, while the resistivity of
semiconductors decreases with increasing T
ρ
ρ
ρ
  0 1   T  T0 
ρ0
ρ is the resistivity at temperature T
ρ0 is the resistivity at a reference
temperature T0 (usually taken to be 20° C)
 is the temperature coefficient of resistivity
Slope = αρ0
T0
conductor
T
Tc
superconductor
T
T
seminconductor
Resistance and Resistors – A new element of circuit
• Ohm’s law can be reformulated in terms of the potential difference Vab responsible
for the electric field driving a current through a wire:
length parallel with current
Vab
I
L
E  J 
   Vab  
I
L
A
A
Vab  RI
Va
E
J
I
A
area perpendicular on current
• The quantity R is termed the resistance of the conductor: R
SI

Vb
L
1V
 Ohm (Ω)
1A
• The elements of circuit with resistance are called resistors. Symbol:
• In a resistor in circuit with a battery, the electric potential decreases in the direction
of the current – the resistor determines a voltage drop – while the potential increases
across a battery – it determines a voltage raise – since a battery does work on the
Constricted pipe ↔ resistance
carriers
Vab  Va  Vb  IR  Va  Vb
R
I
a
+
I
I
I
b
Analogy:
circuit of
pipes with
water in
laminar
flow
Flow rate ↔ current
High
Low
Pump: pressure difference ↔ potential
Quiz 3: An ohmic material is probed for drawn
current I by applying an increasing potential
difference V across the sample. Which of the
shown I vs. V graphs (called IV-characteristics)
represents the likely dependency of I on V?
I
I
a)
b)
V
V
Quiz 4: What geometrical aspect of the IV-characteristics is a measure of the resistance?
a) The intercept of V axis divided by the respective current
b) The slope of the graph
c) The inverse of the slope of the graph
Problems:
1. Temperature dependence of current: A resistor with uniform cross-section and
temperature coefficient α = 410-3 C-1 is heated from T0 = 20oC to T = 170oC. All this time a
constant potential difference is applied across the device. If the current through the resistor at
T0 is I0 = 2.0 A what is the current at temperature T?
2. Resistance and directionality: A metallic solid parallelepiped with resistivity ρ has length
L, width a and height b. Find the resistance of the object if
a) a potential difference is applied on the ab cross-sections
b) the potential difference is applied perpendicular on L
Meters in a Circuit – Measuring current and voltage
• An ammeter is used to measure current. Symbol:
A
• It must be mounted in line with the element along which the current is measured: all
the charge passing through the element must also pass through the meter
• In order to measure current without modifying it, the ammeter must have a very
small resistance, so no potential difference across it
• A voltmeter is used to measure potential difference. Symbol:
V
• It must be connected to the two ends of the element
• In order to measure the voltage without affecting it, the voltmeter must carry no
current so its resistance is very large
Energy Dissipated in a Resistor – Qualitative approach
• We’ve seen that the capacitor is an element of circuit
that stores electric energy. What about a resistor? What
does a resistor do from an energy point of view?
• Consider an element of positive charge, dq, moving
around a closed circuit from point a back to point a
• As the charge moves through the battery from a to b, the
potential energy of the charge increases by Vdq (on behalf
of the chemical energy of the battery)
• As the charge moves through the resistor R, from c to d,
ground: taken as
it looses energy in collisions with the atoms of the resistor:
having zero potential
the energy is transferred to the internal energy
• When the charge returns to a, the net result is that some chemical energy of the
battery has been delivered to the resistor and caused its temperature to rise: we say
that the energy was dissipated across the resistor
• Notice that as long the charge flowing around the circuit (that is, the current) looses
energy in the resistor, the battery must resupply it in order to maintain the current
• Resistance acts like friction in a mechanics: without resistance (friction), the current
would flow (charges would move) around the circuit without the need of a battery
Energy Dissipated in a Resistor – Electric power
• Say that a circuit carries a charge dq through a difference of potential V across a
resistor R that dissipates energy dU = Vdq
• The rate at which the energy U is dissipated is the electric power of the resistor:
dU dq
P
 V  P  IV
dt
dt
• Using Ohm’s law, we can find two useful alternative forms:
• The unit of energy used by electric companies is the kilowatthour: defined in terms of the unit of power and the amount of
time it is supplied: 1 kWh = 3.60106 J
V2
PI R
R
Quiz:
5. Which of the resistors on the right will draw a larger current?
a) A
b) B
c) Both draw the same current
6. Which of the two resistors will get hotter?
a) A
b) B
c) Both will get equally hot
d) They don’t get hot
2
Types of Circuits
• Direct-Current or dc circuits are traveled by currents in only one direction: the
magnitude of the currents along the circuit branches may vary, but not their direction
• Alternating-Current circuits or ac circuits are traveled by currents in directions
alternating with a certain periodicity
• Different elements of circuit behave differently in dc and ac circuits
Electromotive Force – Sources and internal resistance
Def: An device – such as a battery or generator – that maintains the current in a
closed circuit is said to provide an electromotive force or emf, ε
 SI  Volts  V 
• A real battery has some internal resistance r, so some
of its energy is dissipated internally
• Therefore, the voltage Vab across a real battery when
it drives a current is not equal to the emf
Vab    Vr    Ir
• The voltage Vab is applied to the external circuit of
resistance R (called load), so
  IR  Ir
The internal resistance can
be represented as a resistor
in series with the battery
Electromotive Force – Comments
• Notice that the emf ε is equal to the terminal
voltage when the current is zero (also called the
open-circuit voltage)
• If the emf source is mounted across a load, a
current I is driven and the difference of
potential across the battery decreases by Ir
• The current provided by the battery depends
both on the load R and the internal resistance r
ε
r
R
I   R  r
• When R  r, the source is considered as ideal
• Ordinary batteries have a small internal
resistance, such that the voltage delivered is less
than the nominal voltage which is the emf
• The battery power is distributed both to the
load and internal resistance:
I   I 2 R  I 2r
• So, when R  r, most of the power delivered
by the battery is transferred to the load
  IR  Ir
Ex: Voltage diagram
The voltage V raised by the battery
by ε drops both across the internal
resistance r and the load R
Electric Circuits – Resistors in series
• When two or more resistors are connected end-to-end, they are said to be in series
• The current is the same in all resistors because
Vab
Vbc
any charge that flows through one resistor flows
through the other:
I1  I 2  I
• As a consequence of energy conservation around
the circuit, the sum of the potential differences
across the resistors is equal to the total potential
difference across the combination
Rs
V  Vab  Vbc
IRs  IR1  IR2
Rs  R1  R2
• In general, for n resistors in series:
Rs  R1  R2  R3  ...  Rn
• The series equivalent resistance is always greater than any of the individual resistors
Electric Circuits – Resistors in parallel
• When two or more resistors are connected to the same two points in the circuit, they
are said to be in parallel
• The potential difference across each resistor is
Rp
the same
V1  V2  V
• As a consequence of charge conservation, the
V1  Vab
current, that enters into a junction must be equal
to the total current leaving that junction, so,
using for instance junction a,
V2  Vab
I  I1  I 2
V
V V
 
Rp R1 R2
R1 R2
1
1 1
 
 Rp 
Rp R1 R2
R1  R2
• In general, for n resistors in parallel:
1
1
1
1
 
 .. 
Rp R1 R2
Rn
• So, the parallel equivalent resistance is always less than the smallest resistor
Problems:
3. Current with and without internal resistance: A battery with emf ε and internal
resistance r = 2.0 Ω delivers a current I0 = 100 mA when connected to a load R = 70 Ω.
Calculate the current I through the circuit if the internal resistance of the battery were zero.
4. Analysis of combined circuit: An electric circuit contains an ideal battery with ε and four
given resistors, R1, R2, R3, R4, arranged as in the figure.
a) Calculate the equivalent resistance of the circuit.
b) Calculate the current through resistor R4 and the potential difference Vbc across it
c) Calculate the potential difference Vab across points a and b on the circuit.
d) Calculate the currents I1 and I2 through R1 and R2,3 respectively.
e) Say that point c is “grounded”. What are the potentials in points a, b, and d with respect to
this ground? What changes if the ground changes?
a
R2
ε
R1
d
R3
c
R4
b
Kirchhoff’s Rules – Statements
• Resistors can be connected so that the circuits formed cannot be reduced to a single
equivalent resistor as a combination of parallel and series arrangements
• These generic circuits can be analyzed – that is, can be described in terms of
currents carried along various branches, and potential differences between different
points in the circuit – using Kirchhoff’s Rules:
1. Junction Rule
A statement of Charge Conservation
The sum of the currents entering any junction must equal the sum of the currents
leaving that junction
I  0
2. Loop Rule
A statement of Energy Conservation
The sum of the potential differences across all the elements around any closed
circuit loop must be zero
V  0
Kirchhoff’s Rules – Junction rule
I
How to use the rule:
• Assign symbols and directions to the currents in all
branches of the circuit
• The directions can be arbitrary: if a direction is
chosen incorrectly, the current resulting after solving
the equations will be negative, with a correct
magnitude
• Write the junction rule for as many junctions as
needed as needed, as long as each time you write an
equation you include in it a current that has not been
used in a previous junction rule equation
• In general, the number of times the junction rule
can be used is one fewer than the number of junction
points in the circuit
in
  I out
Ex: Hydrodynamic analog:
the current entering into a
junction splits into partial
currents (a) as water flowing
into bifurcating pipes (b)
Kirchhoff’s Rules – Loop rule
V
i
0
around
a loop
How to use the rule:
• Circuits contain loops of electric elements such as resistors and batteries
• Choose an arbitrary direction to travel around a circuit loop Ex: clockwise
Resistors:
loop
• If the resistor is traveled in the direction of the current: V = –IR (voltage drop)
• If the resistor is traveled opposite to the current: V = +IR (voltage raise)
Batteries:
• If the source of emf is traveled in the direction of the emf: V = +ε (voltage raise)
• If the source of emf is traveled opposite to the emf: V = –ε (voltage drop)
Kirchhoff’s Rules – Problem solving strategy
1. Draw the circuit diagram and assign labels and symbols to all quantities
2. Assign directions to the currents – no need for the directions to be all correct; a current with
an incorrectly chosen direction will eventually come out negative
3. Apply the junction rule to any junction in the circuit
4. Apply the loop rule to as many loops as are needed to solve for the unknowns
5. Solve the equations simultaneously for the unknown quantities. Check your answers
a
Problems:
5. Applying Kirchhoff’s Rules – one battery: An
electric circuit contains an ideal battery with ε = 6 V and
four given resistors, R1 = 10 Ω, R2 = 12 Ω, R3 = 2 Ω,
arranged as in the figure. Use Kirchhoff Rules to calculate
the currents I, I1, I2 through the branches of the circuit.
ε
b
a
6 . Applying Kirchhoff’s Rules – two batteries:
An electric circuit contains 4 resistors and two ideal
batteries, as in the figure. Write out Kirchhoff rules
for this circuit.
R2
R1
R4
ε1
R1
R2
c
d
ε2
b
R3
RC Circuit – Functionality
• A more complex behavior is expected when resistors are connected in the same
circuit with capacitors forming an RC dc-circuit: in these circuits, currents will be
unidirectional (dc), but their magnitudes will vary with time
• When the RC circuit is completed is series with a battery
ε, the capacitor C starts to charge and the circuit is called
in charging regime
• Due to the presence of resistance R, the flow of charge is
slowed down, such that the capacitor will take time to
approach its maximum charge Q = Cε
• The capacitor builds a potential opposing the battery, so
the current i – which is initially ε/R when there is no
charge on the capacitor – decreases until the capacitor is
fully charged and the current in the circuit tends to zero
• Subsequently, if the battery is removed, the capacitor
will enter in a discharging regime gradually releasing its
charge through the resistor like a finite-charge reservoir
• Initially the capacitor drives the current ε/R opposite to
the current in the charging regime, but the current
decreases since the charge on the capacitor depletes
ready for charging
ready for discharging
RC Circuit – Charging regime
• Using Kirchhoff rules around the circuit, one ca obtain an
equation for how the charge on the capacitor increases with
time exponentially, tending to Qfinal= Cε:

q  Q final 1  e
t RC

time constant
τ = RC
charging…
• Consequently, the current through resistor decreases with
time from I0 = ε/R to zero:
i  I 0et RC
• The constant  = RC is called the time constant and represents the time required
for the charge to increase from zero to 63.2% of its maximum
• In a circuit with a large time constant, the capacitor charges and discharges slowly
RC Circuit – Discharging regime
• When the battery is removed, the capacitor discharges and
it can be shown that the charge decreases exponentially with
time, from Q0 asymptotically to zero:
q  Q0e
t 
time
constant τ
• The current decreases exponentially from
I0 (but in opposite direction than when
charging)
i  I 0e
t 
discharging…
maximum voltage across the capacitor
V0 Q0
I0  
R RC
Problems:
7. Charging an RC circuit: Demonstrate that the time dependency of the charge on the plates
of a capacitor in an RC-circuit charged by a battery with emf ε is, indeed, given by



q   C 1  et RC  Q f 1  et RC

8. RC circuit: An RC circuit is connected to a battery with emf ε = 10 V. The capacitance is C
= 0.50 μF and the resistance is R = 4.0 MΩ.
a) Calculate the current and the amount of charge accumulated
in the capacitor at t = 0.20 s in the charging regime
b) How long will it take until the charge reaches half of its
maximum value
c) If the battery is removed (discharging regime), how long will
it take until the charge in the capacitor is half the maximum
charge
d) What is the current through the resistor at that moment?