Magnetostatics
1
Introduction
Charges at rest produce electric fields. When charges move with constant velocity, a
static magnetic field or a magnetostatic field is produced. Thus, a magnetostatic field is
produced by a constant current flow, which could be due to the magnetization currents in
a permanent magnet, electron beam current in vacuum tubes or conduction currents in
a current carrying conductor. The strength of the magnetic field intensity is represented
→
−
→
−
→
−
as H (analogous to E ) and the magnetic flux density is represented as B (analogous to
→
−
D ).
2
Biot-Savart’s Law and Magnetic Fields
Just like Coulomb’s law that deals with the force exerted by a point charge, Biot-Savart’s
law deals with the force exerted by a differential current element. The law states that
−→
the differential magnetic field intensity, dH, produced at a point P due to a differential
→
−
current element I dl at a distance r is given by
→
− −
−→ I dl × →
r
dH =
3
4πr
where × represents the cross product.
Figure 1: Magnetic field at point P due to a current carrying conductor
Thus, the total magnetic field due to a line current distribution is given by
− −
Z →
→
−
I dl × →
r
H =
3
4πr
L
1
3
Ampere’s Law
→
−
Ampere’s circuit law states that the line integral of H around a closed path is the same
as the net current Ienclosed enclosed by the path. Thus,
I
−
→
− →
H . dl = Ienclosed
L
Ampere’s law in magnetostatics is very similar to Gauss’ law in electrostatics. The above
equation can be modified as follows,
I
Z
−
→
→
− →
→
− −
H . dl =
J .dS
L
S
H →
R
−
→
−
− →
where J is the current density. Now if we apply stokes law so that L H . dl = S (∇ ×
→
→
− −
H ).dS,
Z
Z
→
→
→
− −
→
− −
(∇ × H ).dS =
J .dS
S
4
S
→
− →
−
→
−
⇒ ∇×H = J
Magnetic Flux Density
→
−
→
−
Similar to D in electrostatics, there exist an analogous magnetic flux density B in mag→
−
→
−
netostatics. B is related to H according to
→
−
→
−
B = µ0 H
where µ0 is known as the permeability of free space. Its value is 4π × 10−7 H/m (Henry
per metre). The magnetic flux through an surface S is given by
Z
→
→
− −
B .dS
ψ=
S
The unit of magnetic flux is Weber (Wb) and that of magnetic flux density is Wb/m2 or
Tesla (T). We know that magnetic monopoles don’t exist. Thus the total magnetic flux
through any closed surface S will always be zero. That is
I
→
→
− −
B .dS = 0
S
Figure 2: Net flux through the blue surface is 0
2
Now if we apply divergence theorem, we have
I
→
− →
−
∇. B dV = 0
V
→
− →
−
∇. B = 0
→
− →
−
Alternatively, if ∇. B = 0 is accepted as a fundamental postulate of magnetostatics then
it can be proved that magnetic monopoles do not exist.
5
Magnetic Force
As in case of electric fields, the force experience by a moving charge in a magnetic field
is given by
→
−
→
−
−
F = q(→
v × B)
→
−
−
where →
v is the velocity of the charge particle and B is the magnetic flux density. We
can rewrite the equation as
I
→
− →
→
−
−
F =
I dl × B
L
where I is the current in a current carrying inductor whose path is given by L. Note that
the magnetic field produced by the element does not exert a force on itself (similar to the
→
−
fact that a point charge does not exert a force on itself). The B in the above equation
is due to another current carrying element.
3