8.2 Surface Integrals, Stokes 127
8.3 Exercise: Verify the Stokes Law for a Vorticity Field
Compute the integrals on both sides of the Stokes law (8.36) for the vorticity vector
fieldv=w×r, with a constant angular velocityw. The surface integral should be
evaluated for a circular disc with radiusR. The disc is perpendicular tow.
Hint: choose a coordinate system with itsz-axis parallel tow.
8.2.7 Application: Magnetic Field Around an Electric Wire
The Stokes law can be used to evaluate the strength of the magnetic fieldHoutside a
straight wire. The electric current densityj, inside the wire, is assumed to be steady,
i.e. it does not change with time. In this stationary situation, one of the Maxwell
equations reduces to
∇×H=j. (8.38)This equation underlies the findings of Oersted and Ampere on the coupling between
electricity and magnetism.
Next (8.38) is integrated over a circular surface, perpendicular to the wire. The
radiusRof the circle is larger than the diameter of the wire, cf. Fig.8.12. The resulting
“integral form” of (8.38)is
∫
(∇×H)·ds=∫
j·ds≡I. (8.39)HereIis the electric current, notice thatj=0, outside the wire. On the other hand,
the Stokes law implies
∫
(∇×H)·ds=
∮
H·dr. (8.40)Fig. 8.12 The magnetic field
Haround a long straight wire
carrying the electric current
I, due to the electric fluxj