Real gases
Phase transitions
Q = m · cl
In the isotherm process of phase transition a body absorbs
or give a quantity Q of heat where m is the mass and cL is
the latent heat.
Newton’s law of cooling
T (t) = Ti e−kt , k = gradient of the cooling.
Electric charge
ELECTROSTATIC
The electric charges are integer multiples of the elementary electric charge
e associated to an electron (−e) or to the proton (+e). The are expressed in
Coulomb [C].
e = 1.602 · 10−19 C ,
→ 1C = 6.242 · 1018 e
In an isolated system the electric charge is conserved.
q1 q2
Coulomb’s law
r2
F force acting on each electric charge q1 e q2 . r is the distance between the
charges.
N.B.: The electric charge are characterised by a sign. For charges of the
same sign (+, + o −, −) the force is repulsive (F > 0), for charge of opposite
sign (+, − o −, +) the force is attractive.
F =k
For electric charges in the vacuum:
2
N
·
m
k = 9 · 109
Costante di Coulomb
2
C
q q
Other notation F =
1
1 2
4π�0 r 2
where �0 =
−12 C 2
8.85·10
N m2
Compare with Newton’s gravitational force!
Electric field
F�
�
E=
q0
Intensity of the electric field in a point: force acting on
a unitary charge located at rest in that point. q0 is the
test charge (supposed to be infinitesimal in order to do not
modify the electric field in that point). F� coulomb force
acting of q0 .
� pt = k q2 campo elettrico di un insieme di una carica.
E
r
� �
�
Etot = i Ei campo elettrico di un insieme di cariche i
Potential electric energy
U = k · q1rq0 potential energy of the charge q0 with respect to q1 and vice versa.
Electric potential
[Volt = V → Joule / Coulomb]
U
V =
q0
The potential in a point of the electric field is the potential
energy of a unitary charge q0 located at rest in the point.
Vtot =
�
Vi = k
N
�
qi
i=1
i
ri
Potential associated to N electric charges, where ri is the
distance on the ith charge from that point.
In general:
Vtot =
�
dV = k
�
dq
r
Potential electric energy
U = k · q1rq0 potential energy of the charge q0 with respect to q1 and vice versa.
Force of a field and
motion of a charge
� = m · �a
F� = q · E
F� force acting on a charge q located in an electromagnetic
� m mass of the charge, �a acceleration
field of intensity E.
of the charge. q > 0 → F�
� and vice versa.
q > 0 ⇒ F� has the same direction of E
W = ∆U
W work done by an electric force when a particle of charge
q moves from A to B. ∆U = UB − UA variation of potential
energy.
Force of a field and motion of a charge
�
F� = q · E(=
m · �a
II Newton’s law)
F� force acting on a charge q located in an electromagnetic
� m mass of the charge, �a acceleration
field of intensity E.
of the charge. q > 0 → F�
� and vice versa.
q > 0 ⇒ F� has the same direction of E
W = ∆U = −q∆V = q · (VA − VB )
W work done by an electric force when a particle of charge
q moves from A to B. ∆U = UB − UA variation of kinetic
energy, ∆V = VB − VA variation of potentian between A
and B.
q > 0, ∆V > 0 → ∆K > 0
q < 0, ∆V < 0 → ∆K > 0
Electrovolt (eV ): work done to move an unitary charge
between two points whose variation of potential is 1V .
Cathodic tube
Electric current
q
[Ampere → C / sec]
t
i intensity of electric current (charge flow). q charge flowing
through a section of de conductor in a time t.
i=
N.B: By convection, the direction of the electric charge
is that corresponding to the flow of positive charges; the
electric current flows in the same direction of the electric
field (for points at high potential to points at low potential).
Ohm’s law
V = R · i [Ohm = Ω → Volt / Ampere]
V difference of potential between the extremes of a metallic
conductor; i intensity of current through the conductor; R
resistance of a conductor.
Resistance R = ρ · Al . l and A length and section of a
conductor; ρ [Ohm m] resistivity characterising a material
(it increases with the temperature).
�
Series of resistors
Resistors in parallel
V = V 1 + V2 + V3 · · · = i V i
i = i1 = i2 = i3 = . . .
�
R = R 1 + R2 + R3 · · · = i R i
V = V1 = V2 = V3 = . . .�
i = i1 + i2 + i3 + · · · = �
i ii
1
1
1
1
1
=
+
+
·
·
·
=
i Ri
R
R1
R2
R3
Ri , ii , Vi = resistance, current and difference of potential
of the ith conductor.
R, i, V = resistance, current and difference of potential
of the whole circuit of resistor in series and parallel.
Kirchhoff’s law
The current flowing into a generic node of the circuit
is equal to the current exiting that node. The difference
of potential between two nodes of the circuit is the same
independently on the path between these two points.
Energy and power
Energy required to maintain a current i for a time t in
conductor whose ends at at a difference of potential V ,
E = V · i · t = V · qtot ,
The energy is converted in heat by the resistor: E = Ri2 t =
V2
3
R t and it expressed Watt-hour (Wh) 1W h = 3.6 · 10 J.
The power is:
P =V ·i
Alternating Current f = frequency
V = Vp sin[2πf t] , i = ip sin[2πf t]
Vp e ip = Diff. of pot. and pick current
Pevr = Vrms irms
V
Vrms = √p2 e irms =
(root-mean-square).
i
√p
2
= Effective Diff. of pot. and corr.
Flux on a surface
θ
θ
The flow of an homogeneous a field E through a surface S whose normal direction forms an angle θ w.r.t. the
direction of the field is:
ΦE = E · S · cos θ
Gauss’s law: the flow of electric field trough a closed
surface encodes the total charge inside the surface
�
� · dA
� = qenc
�0 E
63.2\%
Charged isolated conductor
Gauss’s law: the flow of electric field trough a closed
surface encodes the total charge inside the surface
�
� · dA
� = qenc
�0 E
If an excess charge is placed on an isolated conductor,
that amount of charge will move entirely to the surface
of the conductor, in such a way that all the point of the
conductor are at equal potential.
63.2\%
Charged isolated conductor
Gauss’s law: the flow of electric field trough a closed
surface encodes the total charge inside the surface
�
� · dA
� = qenc
�0 E
If an excess charge is placed on an isolated conductor,
that amount of charge will move entirely to the surface
of the conductor, in such a way that all the point of the
conductor are at equal potential.
63.2\%
Capacitor
The capacitor is a system of two isolated conductors
with equal quantity of charge and of opposite charge
C=
q
∆V
The C capacity of the capacity depends only on the geometry and the isolating medium between the conductors. ∆V
difference of potential of the conductors. q absolute value
of the charge in each conductor.
I(t) = I0 e−t/τ
τ = RC time constant of the circuit. Time necessary
for the capacitor to charge of 63.2% Tempo necessari al
condensatore per caricarsi del 63.2% or to discharge 36.8%
(e−1 = 0, 368).