2.2 Static Magnetic Phenomena
17
represented by the magnetic flux density, B. When a current I is placed in this space,
a force
F
= I × B
(2.21)
is exerted on a unit length of the current. This force is called the Lorentz force. When
an electric current of density i
r
flows at position r
in space V, the magnetic flux
density at position r is given by
B(r) =
μ 0
4π
V
i
r
×
r − r
|r − r |
3
dV
.
(2.22)
This is called the Biot-Savart law. In the above, μ 0 is the magnetic permeability
of vacuum given by (1.1), and the volume integral is with respect to r
. When the
magnetic flux density is integrated on a closed line C, the following relationship
holds:
C
B · ds = μ 0
S
i · dS.
(2.23)
This is called Ampere’s law. In the above, S is a plane surrounded by C, and the
elementary surface vector dS points along the motion of a screw when a screw driver
is rotated along the direction of the curvilinear integral ds (see Fig. 2.4). This is the
right-hand rule. While the Biot-Savart law describes the local magnetic flux density
produced by an electric current, Ampere’s law gives the global relationship between
magnetic flux density and electric current. This is similar to the relationship between
Coulomb’s law and Gauss’ law.
We consider the case where electric current I flows along the z-axis. Here we
estimate the magnetic flux density at point P separated by distance r from the current.
Ampere’s law is applied on a closed circle of radius r with its center on the z-axis. It
is derived that the magnetic flux density is parallel to this circle from (2.22), and its
value is constant on the circle from symmetry. Hence, the left side of (2.23) is given
by 2π rB. The right side is equal to μ 0 I . Hence, we have
Fig. 2.4 The directions of
the elementary surface
vector dS and the curvilinear
integral ds based on the
right-hand rule
dS
S
C
ds
17
represented by the magnetic flux density, B. When a current I is placed in this space,
a force
F
= I × B
(2.21)
is exerted on a unit length of the current. This force is called the Lorentz force. When
an electric current of density i
r
flows at position r
in space V, the magnetic flux
density at position r is given by
B(r) =
μ 0
4π
V
i
r
×
r − r
|r − r |
3
dV
.
(2.22)
This is called the Biot-Savart law. In the above, μ 0 is the magnetic permeability
of vacuum given by (1.1), and the volume integral is with respect to r
. When the
magnetic flux density is integrated on a closed line C, the following relationship
holds:
C
B · ds = μ 0
S
i · dS.
(2.23)
This is called Ampere’s law. In the above, S is a plane surrounded by C, and the
elementary surface vector dS points along the motion of a screw when a screw driver
is rotated along the direction of the curvilinear integral ds (see Fig. 2.4). This is the
right-hand rule. While the Biot-Savart law describes the local magnetic flux density
produced by an electric current, Ampere’s law gives the global relationship between
magnetic flux density and electric current. This is similar to the relationship between
Coulomb’s law and Gauss’ law.
We consider the case where electric current I flows along the z-axis. Here we
estimate the magnetic flux density at point P separated by distance r from the current.
Ampere’s law is applied on a closed circle of radius r with its center on the z-axis. It
is derived that the magnetic flux density is parallel to this circle from (2.22), and its
value is constant on the circle from symmetry. Hence, the left side of (2.23) is given
by 2π rB. The right side is equal to μ 0 I . Hence, we have
Fig. 2.4 The directions of
the elementary surface
vector dS and the curvilinear
integral ds based on the
right-hand rule
dS
S
C
ds