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indie452
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Homework Statement
magnetic field is azimuthal B(r) = B(p,z) [tex]\phi[/tex]
current density J(r) = Jp(p,z) p + Jz(p,z) z
= p*exp[-p] p + (p-2)*z*exp[-p] z
use stokes theorem to find B-filed induced by current everywhere in space
Homework Equations
stokes - {integral}dS.[curl A] = {closed integral}dl.A
curl B(r) = J(r)
The Attempt at a Solution
={integral}dS.[curl B(r)] = {closed integral}dl.B(r)
={integral}dS.J(r) = {closed integral}dl.B(p,z) [tex]\phi[/tex]
{integral}dS.J(r) = {integral}dS.p*exp[-p] p + (p-2)*z*exp[-p] z = {closed integral}dl.B(p,z) [tex]\phi[/tex]
dl = pd[tex]\phi[/tex] [tex]\phi[/tex]
dS = pd[tex]\phi[/tex]dz p + pd[tex]\phi[/tex]dp z
So:
{closed integral}pd[tex]\phi[/tex].B(p,z) [tex]\phi[/tex] - with limits 0-2pi
= B(p,z)*2pi*p
{integral}dS.p*exp[-p] p + (p-2)*z*exp[-p] z
do in 2 parts:
{integral}pd[tex]\phi[/tex]dz.p*exp[-p] p - with limits 0-2pi and 0-R
= 2pi*p2*R*exp[-p]
{integral}pd[tex]\phi[/tex]dp.(p-2)*z*exp[-p] - with limits 0-2pi and 0-r
= -2pi*r2*exp[-r]
so B(p,z)*2pi*p = 2pi*p*R*exp[-p] - 2pi*r2*exp[-r]
= B(p,z) = p*R*exp[-p] - r2/p*exp[-r]
is this answer right?