Page 346 - Intro to Tensor Calculus

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For this problem the Biot-Savart law is

Z
µ 0 I ˆ e
~
B(P)= ds.
4π r 2
In terms of θ we find from the geometry of figure 2.6-6
s 2 h
tan θ = with ds = h sec θdθ and cos θ = .
h r
Therefore,
Z θ 2 Iˆ e sin αh sec θ
2
~
B(P)= µ 0 dθ.
2
2
π θ 1 h / cos θ
But, α = π/2+ θ so that sin α =cos θ and consequently
Z θ 2
µ 0 Iˆ e µ 0 Iˆ e
~
B(P)= cos θdθ = (sin θ 2 − sin θ 1 ).
4πh 4πh
θ 1
µ 0 Iˆ e
~
For a long straight wire θ 1 →−π/2and θ 2 → π/2 to give the magnetic field B(P)= .
2πh
For volume currents the Biot-Savart law is
ZZZ ~
~
B(P)= µ 0 J × b e r dτ (2.6.60)
4π V r 2
and consequently (see exercises)
~
∇· B =0. (2.6.61)
~
Recall the divergence of an electric field is ∇· E = ρ ∗ is known as the Gauss’s law for electric fields and so
0
~
~
in analogy the divergence ∇· B = 0 is sometimes referred to as Gauss’s law for magnetic fields. If ∇· B =0,
~
~
~
~
then there exists a vector field A such that B = ∇× A. The vector field A is called the vector potential of
~
~
~
~
~
B. Note that ∇· B = ∇· (∇× A)=0. Also the vector potential A is not unique since B is also derivable
~
from the vector potential A + ∇φ where φ is an arbitrary continuous and differentiable scalar.
Ampere’s Law
Ampere’s law is associated with the work done in moving around a simple closed path. For example,
~
consider the previous example 2.6-5. In this example the integral of B around a circular path of radius h
which is centered at some point on the wire can be associated with the work done in moving around this
path. The summation of force times distance is
Z Z µ 0 I Z
~
~

B · d~ =
B · ˆ e ds =
ds = µ 0 I (2.6.62)
r
2πh
C C C
Z
r
where now d~ = ˆ e ds is a tangent vector to the circle encircling the wire and
ds =2πh is the distance
C
around this circle. The equation (2.6.62) holds not only for circles, but for any simple closed curve around
the wire. Using the Stoke’s theorem we have
Z ZZ ZZ
~
~
~
r

B · d~ = (∇× B) · b e n dσ = µ 0 I = µ 0 J · b e n dσ (2.6.63)
C S S