Short Notes: Electromagnetic Fields & Theory

 Page 3


Magnetic dipoles:
Acurrentloopcreatesamagneticdipole~ µ =I
~
Awhere
I is the current in the loop and
~
A is a vector normal to
the plane of the loop and equal to the area of the loop.
The torque on a magnetic dipole in a magnetic ?eld is
~ t =~ µ×
~
B
Biot-Savart Law:
The magnetic ?eld d
~
B produced at point P by a dif-
ferential segment d
~
l carrying current I is
d
~
B =
µ
0
4p
Id
~
l׈ r
r
2
where ˆ r points from the segment d
~
l to the point P.
Magnetic ?eld produced by a moving charge:
Similarly, the magnetic ?eld produced at a point P by
a moving charge is
~
B =
µ
0
4p
q~ v׈ r
r
2
Amp` ere’s Law: (without displacement current)
I
~
B·d
~
l =µ
0
I
encl
Faraday’s Law:
The EMF produced in a closed loop depends on the
change of the magnetic ?ux through the loop
E =-
dF
B
dt
Self Inductance:
A changing current i in any circuit generates a chang-
ing magnetic ?eld that induces an EMF in the circuit:
E =-L
di
dt
where L is the self inductance of the circuit
L =N
F
B
i
For example, for a solenoid of N turns, length l, area A,
Amp` ere’s law gives B = µ
0
(N/l)i, so the ?ux is F
B
=
µ
0
(N/l)iA, and so
L =µ
0
N
2
l
A
When an EMF is produced by a changing magnetic ?ux
thereisaninduced, nonconservative, electric?eldE
~
such
that
I
~
E·dl
~
=-
d
dt
Z
A
~
B·d
~
A
Mutual Inductance:
Whenachangingcurrenti
1
incircuit1causesachang-
ingmagnetic?uxincircuit2, andvice-versa, theinduced
EMF in the circuits is
E
2
=-M
di
1
dt
and E
1
=-M
di
2
dt
where M is the mutual inductance of the two loops
M =
N
2
F
B2
i
1
=
N
1
F
B1
i
1
where N
i
is the number of loops in circuit i.
Capacitance:
A capacitor is any pair of conductors separated by an
insulatingmaterial. Whentheconductorshaveequaland
opposite charges Q and the potential di?erence between
the two conductors is V
ab
, then the de?nition of the ca-
pacitance of the two conductors is
C =
Q
V
ab
The energy stored in the electric ?eld is
U =
1
2
CV
2
If the capacitor is made from parallel plates of area A
separated by a distance d, where the size of the plates is
much greater than d, then the capacitance is given by
C =²
0
A/d
Capacitors in series:
1
C
eq
=
1
C
1
+
1
C
2
+...
Capacitors in parallel:
C
eq
=C
1
+C
2
+...
If a dielectric material is inserted, then the capacitance
increases by a factor of K where K is the dielectric con-
stant of the material
C =KC
0
Page 4


Magnetic dipoles:
Acurrentloopcreatesamagneticdipole~ µ =I
~
Awhere
I is the current in the loop and
~
A is a vector normal to
the plane of the loop and equal to the area of the loop.
The torque on a magnetic dipole in a magnetic ?eld is
~ t =~ µ×
~
B
Biot-Savart Law:
The magnetic ?eld d
~
B produced at point P by a dif-
ferential segment d
~
l carrying current I is
d
~
B =
µ
0
4p
Id
~
l׈ r
r
2
where ˆ r points from the segment d
~
l to the point P.
Magnetic ?eld produced by a moving charge:
Similarly, the magnetic ?eld produced at a point P by
a moving charge is
~
B =
µ
0
4p
q~ v׈ r
r
2
Amp` ere’s Law: (without displacement current)
I
~
B·d
~
l =µ
0
I
encl
Faraday’s Law:
The EMF produced in a closed loop depends on the
change of the magnetic ?ux through the loop
E =-
dF
B
dt
Self Inductance:
A changing current i in any circuit generates a chang-
ing magnetic ?eld that induces an EMF in the circuit:
E =-L
di
dt
where L is the self inductance of the circuit
L =N
F
B
i
For example, for a solenoid of N turns, length l, area A,
Amp` ere’s law gives B = µ
0
(N/l)i, so the ?ux is F
B
=
µ
0
(N/l)iA, and so
L =µ
0
N
2
l
A
When an EMF is produced by a changing magnetic ?ux
thereisaninduced, nonconservative, electric?eldE
~
such
that
I
~
E·dl
~
=-
d
dt
Z
A
~
B·d
~
A
Mutual Inductance:
Whenachangingcurrenti
1
incircuit1causesachang-
ingmagnetic?uxincircuit2, andvice-versa, theinduced
EMF in the circuits is
E
2
=-M
di
1
dt
and E
1
=-M
di
2
dt
where M is the mutual inductance of the two loops
M =
N
2
F
B2
i
1
=
N
1
F
B1
i
1
where N
i
is the number of loops in circuit i.
Capacitance:
A capacitor is any pair of conductors separated by an
insulatingmaterial. Whentheconductorshaveequaland
opposite charges Q and the potential di?erence between
the two conductors is V
ab
, then the de?nition of the ca-
pacitance of the two conductors is
C =
Q
V
ab
The energy stored in the electric ?eld is
U =
1
2
CV
2
If the capacitor is made from parallel plates of area A
separated by a distance d, where the size of the plates is
much greater than d, then the capacitance is given by
C =²
0
A/d
Capacitors in series:
1
C
eq
=
1
C
1
+
1
C
2
+...
Capacitors in parallel:
C
eq
=C
1
+C
2
+...
If a dielectric material is inserted, then the capacitance
increases by a factor of K where K is the dielectric con-
stant of the material
C =KC
0
Current:
When current ?ows in a conductor, we de?ne the cur-
rent as the rate at which charge passes:
I =
dQ
dt
Wede?nethecurrentdensityasthecurrentperunitarea,
and can relate it to the drift velocity of charge carriers
by
~
J =nq~ v
d
where n is the number density of charges and q is the
charge of one charge carrier.
Ohm’s Law and Resistance:
Ohm’sLawstatesthatacurrentdensityJ inamaterial
is proportional to the electric ?eldE. The ratio? =E/J
is called the resistivity of the material. For a conductor
with cylindrical cross section, with area A and length L,
the resistance R of the conductor is
R =
?L
A
A current I ?owing through the resistor R produces a
potential di?erence V given by
V =IR
Resistors in series:
R
eq
=R
1
+R
2
+...
Resistors in parallel:
1
R
eq
=
1
R
1
+
1
R
2
+...
Power:
The power transferred to a component in a circuit by
a current I is
P =VI
whereV isthepotentialdi?erenceacrossthecomponent.
Kirchho?’s rules:
The algebraic sum of the currents into any junction
must be zero:
X
I = 0
The algebraic sum of the potential di?erences around
any loop must be zero.
X
V = 0
Force on a charge:
An electric ?eldE
~
exerts a forceF
~
on a chargeq given
by:
F
~
=q
~
E
Coulomb’s law:
A point charge q located at the coordinate origin gives
rise to an electric ?eld E
~
given by
~
E =
q
4p²
0
r
2
r ˆ
where r is the distance from the origin (spherical coor-
dinate), r ˆ is the spherical unit vector, and ²
0
is the per-
mittivity of free space:
²
0
= 8.8542×10
-12
C
2
/(N·m
2
)
Superposition:
The principle of superposition of electric ?elds states
that the electric ?eld E
~
of any combination of charges
is the vector sum of the ?elds caused by the individual
charges
~
E =
X
i
~
E
i
To calculate the electric ?eld caused by a continuous dis-
tribution of charge, divide the distribution into small el-
ements and integrate all these elements:
~
E = dE
~
=
Z Z
q
dq
4p²
0
r
2
r ˆ
Electric ?ux:
Electric ?ux is a measure of the “?ow” of electric ?eld
through a surface. It is equal to the product of the
area element and the perpendicular component of E
~
in-
tegrated over a surface:
F
E
= EcosfdA =
~
E·n ˆdA =
Z Z Z
~
E·d
~
A
where f is the angle from the electric ?eld E
~
to the sur-
face normal ˆ n.
Gauss’ Law:
Gauss’ law states that the total electric ?ux through
any closed surface is determined by the charge enclosed
by that surface:
F
E
=
I
~
E·dA
~
=
Q
encl
²
0
Page 5


Magnetic dipoles:
Acurrentloopcreatesamagneticdipole~ µ =I
~
Awhere
I is the current in the loop and
~
A is a vector normal to
the plane of the loop and equal to the area of the loop.
The torque on a magnetic dipole in a magnetic ?eld is
~ t =~ µ×
~
B
Biot-Savart Law:
The magnetic ?eld d
~
B produced at point P by a dif-
ferential segment d
~
l carrying current I is
d
~
B =
µ
0
4p
Id
~
l׈ r
r
2
where ˆ r points from the segment d
~
l to the point P.
Magnetic ?eld produced by a moving charge:
Similarly, the magnetic ?eld produced at a point P by
a moving charge is
~
B =
µ
0
4p
q~ v׈ r
r
2
Amp` ere’s Law: (without displacement current)
I
~
B·d
~
l =µ
0
I
encl
Faraday’s Law:
The EMF produced in a closed loop depends on the
change of the magnetic ?ux through the loop
E =-
dF
B
dt
Self Inductance:
A changing current i in any circuit generates a chang-
ing magnetic ?eld that induces an EMF in the circuit:
E =-L
di
dt
where L is the self inductance of the circuit
L =N
F
B
i
For example, for a solenoid of N turns, length l, area A,
Amp` ere’s law gives B = µ
0
(N/l)i, so the ?ux is F
B
=
µ
0
(N/l)iA, and so
L =µ
0
N
2
l
A
When an EMF is produced by a changing magnetic ?ux
thereisaninduced, nonconservative, electric?eldE
~
such
that
I
~
E·dl
~
=-
d
dt
Z
A
~
B·d
~
A
Mutual Inductance:
Whenachangingcurrenti
1
incircuit1causesachang-
ingmagnetic?uxincircuit2, andvice-versa, theinduced
EMF in the circuits is
E
2
=-M
di
1
dt
and E
1
=-M
di
2
dt
where M is the mutual inductance of the two loops
M =
N
2
F
B2
i
1
=
N
1
F
B1
i
1
where N
i
is the number of loops in circuit i.
Capacitance:
A capacitor is any pair of conductors separated by an
insulatingmaterial. Whentheconductorshaveequaland
opposite charges Q and the potential di?erence between
the two conductors is V
ab
, then the de?nition of the ca-
pacitance of the two conductors is
C =
Q
V
ab
The energy stored in the electric ?eld is
U =
1
2
CV
2
If the capacitor is made from parallel plates of area A
separated by a distance d, where the size of the plates is
much greater than d, then the capacitance is given by
C =²
0
A/d
Capacitors in series:
1
C
eq
=
1
C
1
+
1
C
2
+...
Capacitors in parallel:
C
eq
=C
1
+C
2
+...
If a dielectric material is inserted, then the capacitance
increases by a factor of K where K is the dielectric con-
stant of the material
C =KC
0
Current:
When current ?ows in a conductor, we de?ne the cur-
rent as the rate at which charge passes:
I =
dQ
dt
Wede?nethecurrentdensityasthecurrentperunitarea,
and can relate it to the drift velocity of charge carriers
by
~
J =nq~ v
d
where n is the number density of charges and q is the
charge of one charge carrier.
Ohm’s Law and Resistance:
Ohm’sLawstatesthatacurrentdensityJ inamaterial
is proportional to the electric ?eldE. The ratio? =E/J
is called the resistivity of the material. For a conductor
with cylindrical cross section, with area A and length L,
the resistance R of the conductor is
R =
?L
A
A current I ?owing through the resistor R produces a
potential di?erence V given by
V =IR
Resistors in series:
R
eq
=R
1
+R
2
+...
Resistors in parallel:
1
R
eq
=
1
R
1
+
1
R
2
+...
Power:
The power transferred to a component in a circuit by
a current I is
P =VI
whereV isthepotentialdi?erenceacrossthecomponent.
Kirchho?’s rules:
The algebraic sum of the currents into any junction
must be zero:
X
I = 0
The algebraic sum of the potential di?erences around
any loop must be zero.
X
V = 0
Force on a charge:
An electric ?eldE
~
exerts a forceF
~
on a chargeq given
by:
F
~
=q
~
E
Coulomb’s law:
A point charge q located at the coordinate origin gives
rise to an electric ?eld E
~
given by
~
E =
q
4p²
0
r
2
r ˆ
where r is the distance from the origin (spherical coor-
dinate), r ˆ is the spherical unit vector, and ²
0
is the per-
mittivity of free space:
²
0
= 8.8542×10
-12
C
2
/(N·m
2
)
Superposition:
The principle of superposition of electric ?elds states
that the electric ?eld E
~
of any combination of charges
is the vector sum of the ?elds caused by the individual
charges
~
E =
X
i
~
E
i
To calculate the electric ?eld caused by a continuous dis-
tribution of charge, divide the distribution into small el-
ements and integrate all these elements:
~
E = dE
~
=
Z Z
q
dq
4p²
0
r
2
r ˆ
Electric ?ux:
Electric ?ux is a measure of the “?ow” of electric ?eld
through a surface. It is equal to the product of the
area element and the perpendicular component of E
~
in-
tegrated over a surface:
F
E
= EcosfdA =
~
E·n ˆdA =
Z Z Z
~
E·d
~
A
where f is the angle from the electric ?eld E
~
to the sur-
face normal ˆ n.
Gauss’ Law:
Gauss’ law states that the total electric ?ux through
any closed surface is determined by the charge enclosed
by that surface:
F
E
=
I
~
E·dA
~
=
Q
encl
²
0