Department of Industrial Engineering
Lectures of
ELECTRICAL ENGINEERING
General part
Elettrotecnica generale
M.Fauri, F.Gnesotto, G.Marchesi, A.Maschio
Esculapio Editore, Bologna
Presentation edited by:
Prof. Alvise Maschio
Department of Industrial Engineering
University of Padova
4/3/13
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Department of Industrial Engineering
Index of lecture 1

Physics of current, electric field and voltage


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1.1
1.2
Electric charge and current intensity
Electric field, voltage, potential
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1.1

In the electromagnetic physics the matter is characterized by
the electric charge.



proton charge (q+ positive)
electron charge (q- negative)
In the SI, the Coulomb [C] is the measuring unit of electric
charge:

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The electric charge may be either positive or negative (+ or -)
The electric charges are multiple of the elementary charges:


Electric charge and current intensity - 1
the elementary charge value corresponds to 1.6021.10-19 C
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1.1

Electric charge and current intensity - 2
When in a given space region some elementary charges
are grouped, the total charge Q in the volume τ:
Qτ = ∑ q + + ∑ q − = Q + + Q − = Q + − Q −
τ
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1.1
Electric charge and current intensity - 3
P


The charge may be considered:
  concentrated (or punctiform)
  distributed (in a macroscopic sense)
Δτ
τ
By definition, the charge density δc in the point P inside the volume
Δτ where the charge ΔQ is present corresponds to:
ΔQ
ΔQ + + ΔQ −
δ c = lim
= lim
Δτ → 0 Δ τ
Δτ → 0
Δτ
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1.1

Electric charge and current intensity - 4
Surface charge density σc in the point P of a surface ΔS where a
charge ΔQ is present:
P
ΔQ
ΔQ + + ΔQ −
σ c = lim
= lim
ΔS → 0 ΔS
ΔS → 0
ΔS
ΔS
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1.1

A charge Q may be standing or moving



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Electric charge and current intensity - 5
electrons in a metal vthermic ~ 105 m/s.
Casual movement: for every surface, the total charge passing through the
surface in any time interval is zero.
The orderly movement of charges through a surface S is called electric
current
Intensity of electric current I:
ΔQ
= lim
Δ t→ 0 Δ t
Δ t→ 0
I = lim

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+
ΔQ − ΔQ
−
S
n̂
Δt
ΔQ is the total charge flowing through the surface S in the time interval Δt
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1.1


To determine the sign of I the surface S must be oriented
I>0: the positive charges move in the same direction of

I<0: the positive charges move in the opposite direction with respect to
n
Metals: the electrons are the free electric charges, the current is positive
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n

if
€
Electric charge and current intensity - 6
n
is opposite to the electron movement direction:
Drift velocity of the order of
(10‑3
÷
10‑2)
€
m/s
€
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1.1


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Electric charge and current intensity - 7
The electric current intensity is measured in
Amp [A]
Definition: 1 Amp corresponds to the current
intensity that, flowing through two parallel
conductors, rectilinear, of infinite length, with
negligible section, at 1 m distance in vacuum,
produces a force of 2.10-7 [N/m] between them
1 Coulomb = 1 Amp * 1 Second -> [C] = [A s]
I
F
I
1m
F
1m
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1.1

The electric current is measured
instrument called ammeter:


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Electric charge and current intensity - 8
with
an
It is inserted, in the circuit, along the current path where
the current has to be measured
If the current enters from the positive, the instrument
will show a positive value and vice-versa.
ideal ammeter: it does not modify the value of
the current intensity flowing without the
instrument
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I
+
A
A
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1.2

Forces of different type acts on the electric charge:
  Electrochemical
  Inertial
  Photoelectric
  Piezoelectric
  ……..
  Electromagnetic:

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Electric field, voltage, potential - 1
Electrostatic: interaction among charges in static conditions
Electro-dynamic: interaction among moving charges or charges
moving in a magnetic field
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1.2
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Electric field, voltage, potential - 2
Consider a positive charge ΔQ, on which an electric force ΔF is acting; the
electric field E is defined as:
ΔF
E = lim
Δ Q→ 0 Δ Q
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It is the force acting on the unitary positive charge.
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1.2

Electric field, voltage, potential - 3
Electric voltage: VAB
tˆ
dl

E
B
l
P
B
VAB =
∫
+
E ⋅ t d
A
B
VAB =
∫
A

€
ΔF
lim
⋅ t d
Δ Q→ 0 Δ Q
the charge ΔQ is the same along the
integration path:
€
A
B
∫
VAB = lim
Δ Q→ 0
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Δ F ⋅ t d
A
ΔQ
Δ LAB( )
Δ Q→ 0
ΔQ
= lim
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1.2
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Electric field, voltage, potential - 4
VAB is:



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a scalar quantity
the value may be positive or negative (in the first case the work is done by the
forces, in the second one “against” them)
it depends on the integration path between A and B
the starting point for the evaluation of the electric voltage is denoted with + in
the electric schemes
it corresponds to the work done by the electric forces to move an unitary electric
charge from A to B along the path shown
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1.2

Electric field, voltage, potential - 5
Following the Coulomb law, between two charges Q1 and Q2:

d = distance between the two charges
⎡ N m ⎤
.
9
α = in vacuum= 9 10 ⎢⎣ C ⎥⎦
Q1 Q2
F =α
d2
2


2
Considering the space around the charge Q, the force acting on a small
“exploratory charge” ΔQ positive placed at a distance r from Q is given by:
€
Q
€
ΔQ
Q ΔQ
ΔF = α
ur
2
r
ûr
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1.2
Electric field, voltage, potential - 6
Q
Q ΔQ
ΔF = α
ur
2
r
ΔQ
ûr

The corresponding electric field is given by:
€
Q
E e = α 2 ur
r
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1.2

The electric field Ee is called coulomb-type or
electrostatic:
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Electric field, voltage, potential - 7
conservative
the work to move a charge from one point to the
other does not depend on the path followed
the direction of the electric field corresponds to
the one of the decreasing potentials
E e = −gradU
Therefore it is possible to define in every point a
scalar function U called electric potential
€
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1.2
Electric field, voltage, potential - 8
In a conservative field the electric voltage between two points A and B is equal to

the difference of the A and B potentials:
B
B
VAB = ∫ E e ⋅ t d = ∫ −gradU ⋅ t d = U A − U B
A

€
A

Ee
dl
tˆ
P
l
B
+
A
If the potential of a point at infinite distance is taken equal to zero, the potential of a
generic point P may be defined as the work done to move a positive unitary charge
from P to the infinite:
Δ LP∞
Δ Q→ 0 Δ Q
U P = lim
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1.2
Electric field, voltage, potential - 9
  The measurement unit of voltage and potential is:


⎡ kg m 2 ⎤
⎡ J ⎤
Volt [V] = ⎢ ⎥ = ⎢ 3 ⎥
⎣ A s ⎦
⎣ C ⎦
A
A
+
V
B
⎡ kg m ⎤
⎡ V ⎤
The measurement
unit of the electric field is: ⎢ ⎥ = ⎢⎣ A s3 ⎥⎦
€
⎣ m ⎦
  The electric voltage is measured with the voltmeter, with two connections
l
laid down along the integration line :
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€

with the voltmeter connections as in the figure, VAB is measured

the instrument which does not perturb the electric field is called ideal voltmeter

if a voltmeter is used in a conservative field, the measurement does not depend
on the connection path, but only on the terminals A and B.
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