What will we learn in this chapter?
Chapter 19: Current, resistance, circuits
What are currents?
Resistance and Ohm’s law (no, there are no 3 laws).
Circuits and electric power.
Resistors in series and parallel.
Kirchhoff’s rules.
Resistor-Capacitor circuits.
G. Kirchhoff
(1824 - 1887)
Current
So far:
Focus on static fields (“electrostatics”) due to charges at rest.
In an electrostatic situation there is no field inside a conductor.
Now:
We study systems where the fields inside materials cause charge
motion.
Simple definition:
An electric current is any motion of charge from one region of a
conductor to another.
To keep the charges moving, we need a field E inside the
conductor which produces a force F = qE.
Definition: Current
Think of a liquid: What is the
amount of particles flowing trough
an area A at a given time t?
Here: What is the net charge ∆Q
flowing trough an area in time ∆t.
A
Definition of current: When a net charge ∆Q passes through a cross
section of a conductor during a time ∆t, the current I is
I=
∆Q
∆t
Unit: 1 coulomb/second = 1 C/s = 1 A (Ampere).
Note: The current is always a number (not a vector).
Currents typically range from pA (computer) to 100’s A (car starter).
Current contd.
A conventional current is
treated as a flow of positive
charges, regardless if the free
charges in the conductor are
positive or negative, or both.
In a metallic conductor, the
moving charges are
electrons. Note that the
current still points in the
direction where positive
charges would flow.
Current contd.
Electrons move around randomly
in a metal. When you apply a
potential (i.e., a field), a net drift is
induced. Note that the E-field
travels at the speed of light in that
medium.
Analogy: Liquid
In absence of gravity, molecules
wiggle around.
Adding a slope, the liquid flows.
Current contd.
Closed loops:
Due to conservation of charge, you can imagine a tube filled with
peas.You push one in at one end, another one comes out on the
other end.
The current at any
instant is the same
+
at all cross sections. +
Notes:
In circuit analysis we will always assume
that currents consist of positive charges even though we know
that for conductors electrons are negatively charged.
Resistance and Ohm’s law
Idea behind Ohm’s law:
The E-field causing a current in a conductor is related to a
potential difference.
The current is proportional to the average drift speed of the
moving charges, which is proportional to the E-field and thus to
the potential.
It follows, that the current I should be proportional to the
potential V.
I
Mathematically:
If I is proportional to V, then
I = const · V
V
Resitance
Resistors
Definition of resistance: When the potential difference V between the
ends of a conductor is proportional to the current I, the ratio V/I is
called the resistance of the conductor:
V
R=
I
Unit: The SI unit is 1 ohm = 1 volt/ampere
1Ω = 1V/A
Ohm’s law: The potential difference V between the ends of a conductor
is proportional to the current I trough the conductor; the
proportionality factor is the resistance R.
Note:
Ohm’s law is only valid when V and I are proportional!
This implies that R is a constant independent of V and I.
Clearly, this is an idealized scenario.
Resistor color codes
Resistors are common parts in electric circuits.
You will encounter them sooner or later in labs.
Tables with the different color codings can be obtained on the
internet.
Example:
Green: !
Violet: !!
Red: ! !
Silver: !!
5
7
102
10% tolerance
Result: 57x102 Ohm with
10% tolerance.
Resistivity
What happens with R if we double the diameter of a conductor?
What happens with R if we double the length of a conductor?
Can we define a geometry-independent quantity? Yes!
Definition of resistivity: The resistance R is proportional to the
length L and inversely proportional to the cross-section area A, with a
proportionality factor ρ called the resistivity of the material:
R=ρ
L
A
Note: The resistivity is material dependent and thus characterizes the
conductance properties of given materials.
Units: 1 ohm meter = 1Ωm
Typical resistivities at room temperature
Temperature dependence of resistance
In general, the resistance of a metallic conductor increases with
increasing temperature. Up to approximately 373K:
RT = R0 [1 + α(T − T0 )]
Conductors have resistivities 18 - 22 orders of magnitude smaller
than insulators.
These numbers depend on temperature…
Non-ohmic conductors
R0 is a reference resistance at a given temperature T0 (usually 273
or 293K).
For common metals α is between 0.003 and 0.005 K-1.
Superconductors:
The resistance drops to zero for
temperatures less than Tc.
Highest Tc known: 160K. Usually
only a few K.
Discovered 1911 by KammerlinghOnnes in Leiden.
Electromotive force (emf)
What is needed such that a conductor has a steady current?
A complete (closed) circuit.
A potential difference across the circuit.
A potential difference which “overcomes” the resistance of the
path.
I
“Ohmic” conductor:
Diode:
The I–V characteristic is
linear with slope 1/R.
Strongly nonlinear characteristic.
Diodes act like one-way valves.
How can we move charge from lower to higher
potential such that we can later use it?
emf: ElectroMotive Force E .
Think of a water tower: We move
water up there to later have pressure
in the pipes.
Note: It is not a force (units are 1 V = 1 J/C)
but an energy.
Sources of emf
ideal emf source not connected to a circuit
Any device that converts energy of some form into electrical
potential for a circuit.
Examples:
Batteries
Solar cells
Fuel cells
Generators
Nuclear fission
…
The field associated with
the potential difference Vab
produces a force FE on the
charges q.
There is an intrinsic
“push” (e.g., chemical
reaction) in the device Fn
moving charges from a to
b that maintains the
potential difference.
When the emf source is
not part of the circuit
Fn = FE
and no charge flows.
ideal emf source connected to a circuit
potential causes fields making charges move
When an emf source is
connected to a circuit with
resistance R
Vab = IR
An electric field is set up in
the wire and a current flows.
Note: The current is the same
at every point in the circuit
(charge conservation…).
Internal resitance in an emf source
Ideal emf source:
Charge inside the source does not loose energy when in motion.
Real emf source:
The movement of charge encounters resistance inside the source.
If the internal resistance r follows Ohm’s law, r is a constant. The
current trough the emf source has an associated potential drop
equal to Ir.
The terminal potential difference is thus:
Vab = E − Ir = V − Ir
The terminal voltage Vab is thus smaller than the emf.
It follows for a real emf source: I = E/(R + r)
The internal resistance can be taken into account as a resistance
in series.
Common circuit symbols
Example: Source in an open circuit
E = 12V
r = 2.0Ω
What are the readings on
the volt and ammeter?
Voltmeter: There is no closed circuit. Hence no current flows. Since
there is no current trough the battery, there is no potential
difference across the internal resistance r. Thus Vab = E = 12V .
Ammeter: There is no complete circuit. Hence no current flows and
the ammeter reads 0A.
Example: Source in a closed circuit
E = 12V
R = 4.0Ω
r = 2.0Ω
What are the readings on
the volt and ammeter?
We have a closed circuit,
thus a current flows:
E
12V
I=
=
= 2A
r+R
4Ω + 2Ω
Voltmeter: Vab = IR = 2A · 4Ω = 8V
Vab = E − Ir = 12V − 2A · 2Ω = 8V
Example contd.
The potential difference
starts and ends at the
same point.
Note that the potential
on the VM can be taken
across the resistor or
the battery itself.
Note: The potential
between an old and
new battery is the
same. The internal
resistance increases
drastically with age.
Energy and power in electric circuits
When a charge moves in a circuit, the electric field performs work
on it.
For a current I = ∆Q/∆t the associated charge does a work
∆W = Vab ∆Q = Vab I∆t
The rate of work performed (energy transferred into the circuit) on
a certain time is called power P. We obtain
∆W
P =
= Vab I
∆t
Unit: 1 Watt = 1 W = (1 J/C)(1 C/s) = 1 J/s.
Pure resistance
When a current flows trough a resistor, electric energy is
transformed into thermal energy. Using Vab = IR it follows:
P = Vab I = I 2 R =
2
Vab
R
What happens to this energy?
Charges collide with atoms and increase
the internal energy of the material.
The energy is dissipated into the resistor.
Overheating might cause damage.
Conversely, we can use the heat (oven, toaster, hair dryer, …).
If the potential at b is higher than a then Vab is negative and there is a
net transfer of energy out of the circuit element.
Power output of a (real) source
The emf source converts energy into electrical energy at a rate EI .
The internal resistance dissipates energy at a rate: Pinternal = I 2 r.
Net power output: P = EI − Pinternal = EI − I 2 r
Resistors in series and in parallel
Why should we care?
Vendors make a finite set of resistors, thus, if we want a given
resistance for an appliance, we need to combine different ones.
Two possible options:
Series connection (“one after the other”)
Parallel connection (“next to each other”)
Quiz:
Why are LED lamps more
efficient than incandescent
bulbs?
Think: which ones are hotter?
What is the equivalent resistance for a given circuit?
Resistors in series
Resistors in parallel
Characteristics:
There is only one path for the current to flow and it has to go
trough all resistors.
It follows that the current I
is the same in all resistors.
Potentials across resistors:
Vax = IR1 Vxy = IR2
Vyb = IR3
Vab = Vax + Vxy + Vyb = I(R1 + R2 + R3 )
Req = R1 + R2 + R3
Equivalent resistance for resistors in series:
Req = R1 + R2 + R3 + R4 + . . .
it is always the sum and greater than any individual resistance.
Warning: Resistors vs Capacitors.
Characteristics:
There are different paths the current
can take, but the potential across all
resistors is the same.
Currents across resistors:
Vab
Vab
Vab
I2 =
I3 =
R1
R2
R3
�
�
1
1
1
Current conservation: I = I1 + I2 + I3 = Vab
+
+
R1
R2
R3
I1 =
Equivalent resistance for resistors in parallel:
1
1
1
1
1
=
+
+
+
+ ...
Req
R1
R2
R3
R4
it is always the reciprocal sum and smaller than any resistor.
Kirchhoff’s rules
Some networks cannot be reduced easily to parallel–series
combinations.
Some examples:
The rules “series vs parallel” are inverted!
Battery-Charger-Lamp
Rectifyer-like circuit
Gustav Kirchhoff figured out simple rules to treat these difficult
cases.
Kirchhoff’s rules: some definitions
Kirchhoff’s rules
Kirchhoff’s junction (or point) rule:
!
The algebraic sum of the currents into any junction is zero; that is
�
I=0
×
Kirchhoff’s loop rule:
!
!
!
Junction: a point where 3 or more conductors meet.
Loop: a closed conducting path.
The algebraic sum of the potential differences in any loop, including
those associated with emf’s and those of resistive elements, must
�
equal zero; that is
V =0
�
Kirchhoff’s rules contd.
Example
Notes:
The junction rule is based on the
conservation of electric charge (no
charge can accumulate at a junction).
Think again of a water pipe: If 2L/min
flow in, then 2L/min flow out.
What are the potential
differences across A, B, C
when the switch is open
and when it is closed?
Note: I is conserved.
1L
The loop rule is based on the
conservation of energy: If we walk
along a path measuring potential
differences, then the sum of all these
when we reach the starting point has
2L
to be zero.
If this were not the case, the force would not be conservative!
1L
Open: Only one loop.
Since A, B, C are identical, it follows that the voltage is the same
across all bulbs.
Closed: Two loops.
Since the switch is an ideal conductor, it follows from the small loop
that VC = 0. The emf of the source is now split only between 2 bulbs
and so the potential across A and B increases by 50%.
General problem-solving strategy
Draw a large circuit diagram so there is plenty of space to write
within it.
Often you will not know the actual current flow direction. No
problem: Your result might simply need a sign switch.
Whenever possible, apply the junction rule to remove as many
unknown quantities as possible:
General problem solving strategy contd.
Choose any closed loop in the network, and designate a direction to
go around it.
Go around that loop in the designated direction adding potential
differences as you cross them. Special considerations:
An emf is counted positive if you cross it from – to + and negative
when crossed from + to –.
An IR product is negative if your path passes the resistor in the
same direction as the assumed current and positive otherwise.
“bar gets bigger”
“bar gets smaller”
Relative signs are essential!
General problem solving strategy contd.
Example: Charging a car battery
Apply Kirchhoff’s loop rule to the previously-obtained potential
differences.
If necessary, choose a different loop to obtain another relation
between the different unknowns until you have enough equations to
determine them all.
Solve the equations!
Special case: calculation of a potential drop between points a and b
Start at b and add the potential changes on the way to a.
The algebraic sum of the potential changes is Vab = Va - Vb.
What is the current I in the circuit?
What is the potential difference Vab between points a and b?
Note: there is only one loop, thus we do not need to use the
junction rule.
Example contd.
Example contd.
Set up:
Assume the current flows from the +
terminal of the charged battery. If this is
not right, then the current will come out
negative. Big deal…
Apply the loop rule:
Start at a and use V = IR.
−I(4Ω) − 4V − I(7Ω) + 12V − I(2Ω) − I(3Ω) = 0
The – in the resistor terms come from the fact that we have a
potential drop. We find I = 0.5A.
Resistance–Capacitance (RC) circuits
So far we have assumed that resistances and emfs are constant.
A simple example where this is not the case is an RC circuit.
Potential drop between a and b:
Use the upper path from b to a.
Remember that Vab = Va - Vb.
Follow that loop!
Vb + 12V − (0.5A)(2.0Ω) − (0.5A)(3.0Ω) = Va
Result Vab = 9.5V.
Taking the lower path we obtain the same result.
RC circuit contd.
Initial setup:
The capacitor is uncharged, i.e., the potential difference is zero
and its initial charge Q0 = 0.
Following Kirchhoff’s rules, the voltage across R is equal to the
terminal voltage E .
The initial current I0 trough the resistor is I0 = E/R (Ohm’s).
Switch closed:
From Kirchhoff’s loop rule it follows:
capacitor initially uncharged
when the switch is closed the capacitor is
charged, while the current decreases
E = iR +
q
C
The solution of the differential equation is
i = I0 e−t/RC
q = Qfinal (1 − et/RC )
the relaxation time when i or q are 1/e is τ = RC .
i = ∂q/∂t
RC circuit contd.
Graphical representation of i and q:
Strictly speaking, it takes an infinite amount of time for the current to
be zero and the capacitor to be fully charged.