18
2 Basic Electromagnetism
B =
μ 0 I
2π r
.
(2.24)
This result can also be derived from (2.22). Since the electric current is concentrated
on the line (2.22) leads to the following curvilinear integral:
B(r) =
μ 0 I
4π
C
dr
×
r − r
|r − r |
3
.
(2.25)
The coordinates are introduced as shown in Fig. 2.5, and the foot of a line perpendicular to the current from observation point P is set to be z = 0. dr
is a vector directed
along the z-axis with magnitude dz. Hence, dr
×
r − r
is along the azimuthal
direction (normal direction of this sheet), and its magnitude is
r
2
+ z
2
1/2 sinθ dz,
where θ is the angle of the observation point measured from the direction of the
current. Since all the magnetic flux density produced by the elementary current Idz
is directed along the same azimuthal direction, the integral is simply done and we
have
B =
μ 0 I
4π
∞
−∞
sinθ dz
r 2 + z 2 .
(2.26)
Using the relationship:
Fig. 2.5 Linear current I
and observation point P
2 Basic Electromagnetism
B =
μ 0 I
2π r
.
(2.24)
This result can also be derived from (2.22). Since the electric current is concentrated
on the line (2.22) leads to the following curvilinear integral:
B(r) =
μ 0 I
4π
C
dr
×
r − r
|r − r |
3
.
(2.25)
The coordinates are introduced as shown in Fig. 2.5, and the foot of a line perpendicular to the current from observation point P is set to be z = 0. dr
is a vector directed
along the z-axis with magnitude dz. Hence, dr
×
r − r
is along the azimuthal
direction (normal direction of this sheet), and its magnitude is
r
2
+ z
2
1/2 sinθ dz,
where θ is the angle of the observation point measured from the direction of the
current. Since all the magnetic flux density produced by the elementary current Idz
is directed along the same azimuthal direction, the integral is simply done and we
have
B =
μ 0 I
4π
∞
−∞
sinθ dz
r 2 + z 2 .
(2.26)
Using the relationship:
Fig. 2.5 Linear current I
and observation point P