### charges in motion produce magnetic fields

```Andres La Rosa
Portland State University
Lecture Notes
PH-212
Magnetic field produced by a moving
point charge
Magnetic field at point P,
Z
P
q
Y
q
electrical charge (Coulomb)
velocity (m/s)
vector position (m)
It aims from the charge q to the point P
where we want to evaluate the magnetic field
Magnetic field
The unit is Tesla (T)
P
Z
S
q
Y
L
Exercise
Draw the vector MAGNETIC
FIELD at the different points P,
S, and L, respectively.
Magnetic field produced by a current
P
Strategy:
Divide the wire into small sections of length Δl
P
But notice,
Example: Magnetic field produce by a current
that flows along a straight wire
Z
P
0
We want to evaluate the magnetic
field at a point "P' located at a
distance "R" from the wire
I
Z
I
Δz
P
Contribution to the
MAGNETIC FIELD at P
from just one current
segment Δ z
Δz
Δz
For the case of
an infinitely long
wire:
Q
I
P
Infinitely long
wire:
Lateral View
BP = BQ
I
Lateral View
P
Magnetic field
line
I
Question:
Question: What is the contribution to the magnetic field at P from
the finite wire segment AB of length L ?
T
L/2
P
L/2
A
Z
Y
X
Question: What is the magnetic filed at the point P caused by the
segment AB that carries a current I ?
T
b
L
0
a
A
Z
X
Y
Exercise: Calculate the magnetic field at the points "G", "H", and "K"
produced by the 2-meter wire that carries a current I = 0.5 Amps.
T
G
H
L=2m
A
K
Z
X
Y
Force between two parallel currents
.
.
.
.
I1
.
.
.
.
. B.
. .
. I.
. .
2
I1
- The magnetic field B1 produced
by the current I1 at the site 2 is,
B1
Magnetic field affecting
the current I2
- A segment L of wire-2 is wire-2
immersed
in a magnetic field B1 . So the wire
will experience a force,
Y
Y
1
X
We are assuming the wires
are infinitely long
?
The magnetic dipole moment
The torque τ makes the dipole μ to have a tendency to be
aligned along the external magnetic field.
Magnetic potential energy of a dipole immersed in an
external magnetic field
As usual, the potential energy of the dipole will be given
with respect to a configuration reference
How much external work is needed to take the loop from the
configuration-1 to the configuration-2?
Example
μ = NiA
= 20 (0.1 Amp) (10-1m ) ( 5x10-2m )
Y
= - 10-2 A-m2
X
B = 0.5 T
Z
=
=
=
Definition of the unit current: The Ampere
Example
. . . .
. . . .
x x x x
x x x
Amp
3030
Amp
. . . . . . . .
. . . . . . . .
C
B
A
20 Amp
D
x x x x
x x x
B
Calculate the net force acting on the loop that carries a 20
Amp currrent
Along the segment BC the magnetic field produced by the 30 Amp
wire is constant.
Also, along the segment DA the magnetic field produced by the
30 Amp wire is constant.
For these two cases, it is convenient to use the expression,
Along the segment AB the magnetic field produced by the 30
Amp wire varies with position. Hence, in order to calculate te
force on that segment of the loop, it is better to use,
Example
Y
K
P
X
L
KL is a semi-circle of radius "R"
What is the magnetic field at point "P"?
Solution
Contribution to the magnetic field at P from the
section (- to K)?
Contribution to the magnetic field at P from the
section (L to )?
Contribution to the magnetic field at P from
the semicircle KL?
90 degrees.
Inside the plane of the
figure.
Magnitude of the
magnetic field at
point P
Exercise: Sketch the magnetic field of a circular
coil of radius R carrying a current I.
..
Z
.
. . .
x x
.B
. .
.. .
..
x
x
x x
B
x x x
x x xx
..
Z
X
Z
B
B
Y
X
Y
Notation:
= # of turns
per unit length
Question: In the figure above, where is the magnitude of
the magnetic field higher, at point P or at point Q?
```