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Physics 7D:
Electricity and
Magnetism
Chapter #25
Current, Resistance
and EMF
1
Chapter 25
2
Electric Current
So far we have been studying charges at rest (electrostatics). Now we
will begin studying charges in movement
We begin with a key definition: electric current
An electrical current is motion of charge from one region to
another
note that it is not limited to the
motion of electrons, although that
is the most common
Mathematically, it is defined as:
dQ
I=
dt
If I take a cross-section in a conductor, the
current is the infinitesimal amount of
charge dQ that crosses in an infinitesimal
amount of time dt
3
Electric Current
Some key points about electric current:
1) Units:
Ampere (A), which is the same as Coulomb/second
(note that a current of 1A is enormous, and in general we talk about mA
or μΑ. A current of ~30 mA can almost always kill a human being)
2) Sign convention:
By convention, the direction of the current is that in which the
positive charges flow (from + to -)
+

+
4

Electric Current
3) Current density:
The current is the same
even though the flux
changes direction
Current is technically a scalar. It’s a measure of the
charge flux going through a conductor and/or circuit,
independently of direction
I
J=
A
It’s useful to define a charge density J, whose
magnitude is the current per unit of crosssectional area
This quantity is a
vector, and points in
the direction in which
the positive charge
carriers would move
+
+
A
!
J
+
To obtain the current from J we just multiply by the cross-sectional area
(all of this is valid as long as J is uniform, which we will assume in this course)
5
Experiment
Electric current through water with and without salt:
We get at least two lessons from this experiment: (i) the more charge
carriers available, the higher the current; and (ii) charge carriers need not
be electrons (in this case they are Na+ and Cl- atoms)
6
Basic Questions
Let’s answer a few common questions
1) What causes a current?
The application of a potential difference to a
conductor
A changing potential implies the presence of
an electric field, which moves the charges
2) But a conductor was supposed to be
equipotential. How can there be a
difference of potential?
+
+

A conductor is equipotential if it has reached
electrostatic equilibrium
This is because an external electric field makes the free charges of a conductor
accommodate in such a way that the field inside becomes zero. However, a
conductor in an electric circuit does not reach electrostatic equilibrium. The charge
never finishes accommodating!
7
Basic Questions
3) Does this mean that charges accelerate inside a conductor
with a voltage?
No… in reality charges move at a constant average speed. Why?
This picture is too
simplistic
In reality the free electrons of a conductor are moving and colliding constantly,
even if no external voltage is applied
This movement is completely at
random in all directions, at very high
speeds (~106 m/s for copper at room
temperature)
This means that electrons are constantly colliding amongst themselves and/or
with nuclei (the average time between collisions is in the order of 10-14 s)
8
Basic Question #3
When a voltage is applied, electrons
feel an acceleration, but only for a very
short amount of time. They collide
almost right away, transfer that energy
to the nuclei they collided with, and
start again.
This means that the energy imparted to
the electrons is almost immediately
transferred to the atoms making up the
conductor. This causes the
conductor’s temperature to rise!
Despite all the collisions, the application
of the voltage however does cause a
small net displacement in the direction
of the field
9
Analogy
The best analogy I know to explain this:
If I drop a ball in an inclined plane it will
accelerate and its speed will increase
constantly
But if we put obstacles, despite the fact that
a force is constantly acting on it, the ball can
only accelerate for a short amount of time
before colliding
In the collision the little amount of kinetic
energy gained is converted into thermal
energy (of the stick and the ball). Then
the process resumes
Therefore, the ball is not constantly accelerated. Rather, it falls at a speed that is
roughly constant (if measured in a time frame that is much wider than the average
time between collisions) and which is much lower than if there were no collisions
A similar analogy: a Porsche
during traffic hour in the 405
freeway
10
Relating Current and Drift Velocity
This small average speed at which charges move in a conductor is called
“the drift velocity” vd:
Consider a small volume cross-sectional area
and let n be the concentration of charge carriers
(free charges per unit volume)
The amount of charges crossing area A in an
infinitesimal time dt will be given by:
dQ = q (nAυ d dt) = n q υ d Adt
q=charge of
each charge
carrier
vd=drift velocity
n=density of
charge carriers
(per unit volume)
We conclude that:
dQ
I=
= n q υ d A and therefore J = n q υ d
dt
11
Example: some realistic numbers
(problem 25.5 in Y&F)
Copper has 8.5×1028 free electrons per cubic meter. A 71.0 cm length of 12gauge copper wire that is 2.05 mm in diameter carries 4.85 A of current. (a)
How much time does it take for an electron to travel the length of the wire?
(b) Generally speaking, how does changing the diameter of the wire affect
the drift velocity?
I
(a) υ d =
nqA
=
( 8.5 × 10
4.85 C/s
28
/m
3
)(1.6 × 10
−19
C ) (1.025 × 10
−3
m) π
=1.080 × 10 -4 m/s
(photo of actual 12-gauge
copper wire)
L
0.710 m
t=
=
≅ 110 min
-4
υ d 1.080 × 10 m/s
(b) Increasing the diameter by some factor decreases the
drift speed by the square of that factor
12
2
The drift speed of electrons
Based on these calculations, we conclude that:
Electrons are not so
much like
but more like
How come when I flip a light switch the light turns on instantaneously?
Because electrons are along the conductor, and they begin their
movement at practically the same time
The order to “march” gets
to all electrons at the
same time
13
Basic Question #4
There is one more basic question we
4) Why do you need to close a circuit
for current to flow?
current ✔
no current

It’s simple: otherwise charge starts
accumulating in the ends, causing an
electric field that cancels the applied one
Conclusion: only in a closed circuit
can a current flow stably
14
Resistance
For many conductors, the current that goes through it is directly
proportional to the voltage across the two points
R is called
“resistance” and is
measured in Ohms
(1Ω=1V/A)
V = IR
(this equation is commonly
called “Ohm’s Law”)
What is the resistance?
It is the voltage needed to circulate 1A of
current through a conductor
It is the voltage needed to circulate 1A of current through a conductor. It can be
seen as the opposition that a given conductor has to a current forming in it
A circuit element that does nothing else
besides having a specific resistance
value is called a “resistor”
15
Resistivity
The resistance depends on (i) the choice of material, and (ii) the
geometry of the conductor
It’s useful to define a property that depends only on the choice of materials
regardless of geometry: resistivity
ρ is called
“resistivity” and is
measured in
Ohms x m (Ωm)
E
ρ=
J
This equation is in fact a more general version of Ohm’s law:
!
!
E = ρJ
Sometimes it is useful to talk of conductivity,
which is the reciprocal of the resistivity:
16
1
σ=
ρ
Resistivity and Resistance
Knowing the geometry it is possible to derive a relationship
between resistance and resistivity
For example, consider a conductor
with uniform cross-section A:
We know that E = ρ J
V
I
But E = and J = , so we get
L
A
V
I
⎛ L⎞
= ρ →V = ⎜ρ ⎟ I
⎝ A⎠
L
A
A
I
L
L
We conclude that for this type of conductor: R = ρ
A
As expected, the resistance depends on the choice of material (its
resistivity) and the geometry (in this case L and A)
17
Against Ohm
Not all materials obey
Ohm’s law.
Case in point:
semiconductor diodes
18
Superconductors
In general the resistivity of materials depends on temperature: the
higher the temperature, the higher the resistivity
(this is actually easy to understand: the higher the temperature the more collisions)
There is a class of materials for which the resistivity goes to zero at
very low temperatures. They are called superconductors.
(the discovery was done in 1911 with mercury at Tc=4.2K)
The potential is enormous, but it is very expensive to cool these materials to the required
temperatures. The challenge remains to find a superconductor at higher temperatures
19
Experiment
This experiment is called Jacob’s ladder, and it was first popularized by
the classic Frankenstein movie where it can be seen in the background:
Lessons to remember: (i) if the electric field is high enough, even
insulators become conductors; (ii) the passage of a current through a
medium can drastically increase its temperature
20
Need for an EMF
Positive charges (and thus currents) move from + to – . Since V=IR,
this means that a resistor (and any passive element) produces a
drop in potential
In a resistor the current
goes through the side with
higher potential (+) and
comes out through the
one with lower potential (-)
+

But the change in potential for a charge that goes through the whole circuit
has to be zero. This means that there has to be an element in the circuit
that increases the potential
We call this device a source of “Electromotive Force” (EMF)
21
Why the fancy name?
E
In passive elements like
resistors the field goes in
the same direction of the
current (that’s actually
what makes the current
flow)
+

E
But in a battery it’s the
opposite! Charges
move against the
electric force
Something has to push the charges against the electric
force: that is the EMF
In a battery this is done by chemical reactions, in a Van der Graaff
by the motor and the belt… etc. Whatever its nature, it cannot be
electrostatic (since it works against it!), so we call it electromotive.
Despite the name, EMF does not have units of Newtons, but of Volts. It is
much easier to quantify the impact of this force in terms of how much it can
increase the potential energy of charges
22
Real sources of EMF
The EMF pushes charges until the build-up
causes a voltage too high
That is why sources of EMF (like batteries)
typically hold a constant voltage across
their terminals
In real life, the charges encounter a small
resistance as they climb in potential inside
the sources of EMF:
A real source of EMF can
be modeled as an ideal
source (EMF ε only) and
an internal resistance ri

+
(this means that the voltage you measure across the terminals of a
battery is not the EMF ε, but a little less: Vab=ε – Iri)
A simple circuit
We can use a simple circuit to illustrate this graphically:
The net change in potential going
around the circuit must be zero, since we
begin and end in the same point:
ε − Ir − IR = 0
Points to remember:

In passive elements (e.g. resistors) the
potential drops

In sources of EMF the potential
increases*

Cables are modeled as perfect
conductors (R=0). This means that it
requires no work to move charges
through them, and therefore that they
are equipotential. This is a good
approximation in most cases.
(* unless it’s a battery that is charging)
Power
It’s very useful to be able to calculate the energy absorbed or provided by
circuit elements
Consider any generic circuit element with a potential difference Vab = Va – Vb
between its terminals and through which a current I passes through:
The change in potential energy for a charge
that moves from a to be is given by:
ΔU = qVab
(for example, if Vab > 0 and q >0, the particle
loses potential energy, like in a resistor)
We’ve seen how the energy gained by the charges does not change their kinetic
energy. This energy ends up being transferred to the circuit element itself
So this is energy transferred to the circuit element (or from the
circuit element if the sign is negative)
25
Power
We can derive both sides to determine the rate at which this energy is being
given (or absorbed) by the circuit element:
E = qVab
Energy given or
absorbed by circuit
element
taking the
derivative
dE dq
= Vab
dt dt
We end up with:
dE
P=
= Vab I
dt
Reminder: power is
measured in Watts (W).
1W is the same as 1J/s
26
We assume the voltage is
constant
Works for *any*
circuit element
Power and Resistors
We have seen that in a resistor, all the
energy gained by the charges as they get
accelerated by the electric field is quickly
lost in the collisions with the atoms
This increases the temperature of the
conductor. The energy is eventually
dissipated into the environment in the
form of heat
For a resistor
2
ab
V
P = Vab I = I R =
R
2
(Note that the conversion of electric energy
into heat is called the “Joule effect”)
27
Example
(not in Y&F)
A 5.8 m length of 2.0 mm
diameter wire carries a current of
750 mA when 22.0 mV is applied
to its ends. If the drift velocity is
1.7⨉10-5 m/s, find
a)The resistance of the wire
b)The resistivity
c)The current density
d)Electric field inside wire
e)Number of free electrons per unit
volume
+
+
A
+
28
Review of Current, Resistance & EMF
(with iClickers)
Electrons in an electric circuit pass through a resistor. The
wire has the same diameter on each side of the
resistor. Compared to the potential energy of an electron
before entering the resistor, the potential energy of an
electron after leaving the resistor is
A)greater
B)less ✔
C)the same
D)exactly zero
E)infinity
29
Comment: for a
source of EMF it
would be the
opposite
Review of Current, Resistance & EMF
(with iClickers)
Two copper wires of different diameter are joined end to
end, and a current flows in the wire combination. When
electrons move from the larger-diameter wire into the
smaller-diameter wire, their drift speed
I
A)increases ✔
B)decreases
C)stays the same
D)is exactly zero
E)goes up to infinity
30
Review of Current, Resistance & EMF
(with iClickers)
Consider a beam with a constant square crosssectional area. The resistivity is measured by taking the
ratio of E and J when connecting a battery to the two
square faces (a and b) and to the two rectangular faces
(c and d). Select the true statement:
b
a
c
31
A)ρab = ρcd ✔
B)ρab < ρcd C)ρab > ρcd
D)ρab = 0
E)ρcd = 0
Review of Current, Resistance & EMF
(with iClickers)
Consider a beam with a constant square crosssectional area. The resistance is measured by taking
the ratio of V and I when connecting a battery to the two
square faces (a and b) and to the two rectangular faces
(c and d). Select the true statement:
b
a
c
32
A)Rab = Rcd
B)Rab < Rcd C)Rab > Rcd ✔
D)Rab = 0
E)Rcd = 0
Experiment
(with iClickers)
We have the following setup. When we close the circuit, what will
happen to the reading in the voltmeter?
You can treat the
voltmeter as a device
that measures the
voltage between the
terminals of a circuit
element without
disturbing it in any way
V
+

A) increase by a lot (tens of volts) D) decrease a little (few volts) ✔
B) decrease by a lot (tens of volts) E) stay the same
C) increase a little (few volts)
33
Why?
This is due to something we said during last class:
A real source of EMF can
be modeled as an ideal
source (EMF ε only) and
an internal resistance ri

+
There is a small internal resistance ri in the source of EMF. The voltage
seen by the voltmeter is:
Vab = ε − Iri
If I=0, then Vab=ε. But if I is different from zero, Vab < ε. 34 Next chapter: DC circuits 35 Physics 7D: Electricity and Magnetism Chapter #24 Capacitance and Dielectrics 1 Chapter 24 2 Basic Definition A capacitor consists of two conductors separated by an isolator (which could be vacuum) At all times the two conductors have charges of equal magnitude but opposite signs Symbols used to represent capacitors (of all types): What do capacitors do? If I connect a capacitor to a battery (whose function is to provide a voltage that will set electrons in motion), the electrons from one of the sides of the capacitor will move to the other and a charge of equal magnitude but opposite sign will start accumulating in the two sides of the capacitor battery (voltage V) + - + - capacitor electrons from the top conductor travel to the bottom one Analogy When the charge in the capacitor is such that its voltage is equal to the battery’s, the latter no longer has enough “strength” to keep moving more charges. At this moment the current ceases, but the charge remains in the capacitor plates. Enough potential energy to make it to the other side Not enough potential energy to make it to the other side That is why the capacitor’s voltage cannot be higher than the battery’s At this point we say the capacitor is charged What do capacitors do? After this you can take the charged capacitor and connect it to, for instance, a resistor or an LED: + LED - charged capacitor electrons from the lower conductor travel to the upper one (the resistance is there to slow down the capacitor’s discharge and regulate the current, as we will see later) Now the capacitor acts like a battery. The electrons at the lower conductor repel and want to travel to the other conductor where there is a deficit. The LED turns on and a current circulates until the capacitor is complete discharged So, going back to the original question: what do capacitors do? A capacitor is a device that stores charge (and therefore energy) (Please note that capacitors have other functions too, but this is the main one) 5 Practical Application One of the most common applications of capacitors (with a circuit that is very similar to the one showed in the last two slides): 6 Experiment Let’s see this in real life: lightbulb + resistance 7 How do you fabricate a capacitor? There are smart ways and not-so-smart ways of fabricating a capacitor
For instance, I could put two
conducting cubes separated by
vacuum
Is this a capacitor?
+
+
+ +
– – – — – – – – – – –
++
Yes!
Is it the best capacitor in the world?

No…
Charge wants to accumulate in the faces that look towards the other cube.
But once a certain amount of charge has accumulated there, it will not be
easy to bring more due to electrostatic repulsion
What is therefore commonly sought in a capacitor?
Maximize the surface area of the “faces” so it will be easier to store
charge
8
how do you fabricate a capacitor?
The easiest way to do this: two parallel plates face to face
Typically the
separation is much
smaller than the
area, so the plates
can be treated as
infinite plates
Typically an isolator
is put between them
(a dielectric) and
then the “sandwich”
is rolled up
You can also make a capacitor with concentric spheres or cylinders
9
Capacitance
In all capacitors the charge in the plates is proportional to the voltage
between them. The proportionality constant is called capacitance:
Q
C=
ΔV
In other words, if at any time I measure the amount of Coulombs in one of the plates and the
voltage between the two terminals, I will always get the same number
1) Units?
1 F = 1 farad=1 C/V=1coulomb/volt
2) In words, what is the capacitance?
It’s the capacity to store charge
If I have a 100μF capacitor and a 10μF, and I put a 1V potential difference on both, the
first one will store 100μC and the second one 10μC
3) What does capacitance depend on?
Only on the geometry of the capacitor and the choice of dielectric
10
Recipe for Calculating Capacitance
Steps to calculate the capacitance:

Assume that the plates have charge ±Q

Calculate the electric field between the plates using
Gauss’s Law

Int …