Page 180 - Electromagnetics

The Biot–Savart law. We can obtain an expression for B in unbounded space by
performing the curl operation directly on the vector potential:

µ J(r ) µ J(r )

B(r) =∇ × dV = ∇× dV .
4π V |r − r | 4π V |r − r |

Using (B.43)and ∇× J(r ) = 0, we have

µ 1

B(r) =− J ×∇ dV .
4π V |r − r |

The Biot–Savart law
ˆ
µ R

B(r) = J(r ) × dV (3.164)
4π V R 2
follows from (3.57).

For the case of a line current we can replace J dV by Idl and obtain

ˆ
µ R

B(r) = I dl × . (3.165)
4π R 2
For an infinitely long line current on the z-axis we have
∞
µ ˆ z(z − z ) + ˆρρ µI

ˆ
B(r) = I ˆ z × dz = φ . (3.166)

2 3/2
2
4π [(z − z ) + ρ ] 2πρ
−∞
This same result follows from taking ∇× A after direct computation of A, or from direct
application of the large-scale form of Ampere’s law.
3.3.6 Force and energy
Ampere force on a system of currents. If a steady current J(r) occupying a region
V is exposed to a magnetic field, the force on the moving charge is given by the Lorentz
force law
dF(r) = J(r) × B(r). (3.167)
This can be integrated to give the total force on the current distribution:

F = J(r) × B(r) dV. (3.168)
V
It is apparent that the charge flow comprising a steady current must be constrained in
some way, or the Lorentz force will accelerate the charge and destroy the steady nature
of the current. This constraint is often provided by a conducting wire.
As an example, consider an infinitely long wire of circular cross-section centered on
the z-axis in free space. If the wire carries a total current I uniformly distributed over
2
the cross-section, then within the wire J = ˆ zI/(πa ) where a is the wire radius. The
resulting field can be found through direct integration using (3.164), or by the use of
ˆ
symmetry and either (3.118)or (3.120). Since B(r) = φB φ (ρ), equation (3.118)shows
that

2π µ 0 I 2
a 2 ρ ,ρ ≤ a
B φ (ρ)ρ dφ =
0 µ 0 I, ρ ≥ a.

© 2001 by CRC Press LLC