# Using stokes theorem to find magnetic field

1. May 10, 2010

### indie452

1. The problem statement, all variables and given/known data

magnetic field is azimuthal B(r) = B(p,z) $$\phi$$
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

2. Relevant equations

stokes - {integral}dS.[curl A] = {closed integral}dl.A
curl B(r) = J(r)

3. 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) $$\phi$$

{integral}dS.J(r) = {integral}dS.p*exp[-p] p + (p-2)*z*exp[-p] z = {closed integral}dl.B(p,z) $$\phi$$

dl = pd$$\phi$$ $$\phi$$
dS = pd$$\phi$$dz p + pd$$\phi$$dp z

So:
{closed integral}pd$$\phi$$.B(p,z) $$\phi$$ - 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$$\phi$$dz.p*exp[-p] p - with limits 0-2pi and 0-R
= 2pi*p2*R*exp[-p]

{integral}pd$$\phi$$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]

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. May 10, 2010

### vela

Staff Emeritus
What's the difference between R, r, and p (by which I think you mean ρ)? That's the first thing you need to clear up.

You didn't say what surface S you're integrating over. In any case, I think your evaluation of

$$\oint_S (\nabla\times\textbf{B})\cdot d\textbf{S}$$

is incorrect.