- #1

QuarkDecay

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## Homework Statement

Prove that in an MHD equilibrium state in a pinch with flux and B

_{θ}(r) and B

_{z}(r), the equation for pressure is;

d/dr(P + B

^{2}/8π) + B

_{θ}

^{2}/4πr - ρ( u

^{2}

_{θ}/r)= 0

B is the total Magnetic field, u

_{θ}(r) the velocity of plasma, ρ= density, and solving in a (r,θ,z) system

## Homework Equations

MHD equilibrium equations ( solution for z and θ pinches)

**(1)**∇P= (J x B)/c

**(2)**∇ x B= (4π/c)J

**Edit;**

The MHD equlibrium equation when there's flux is

ρ(u∇)u= (j x B)/c -∇P

That I apparently have to use instead of

**(1)**

3. The Attempt at a Solution

3. The Attempt at a Solution

I know how to solve the problem with the θ-pinch, that has a solution for Pressure; d/dr( P(r) + B

_{z}

^{2}(r)/8π) = 0

We calculate the J first, from equation (2), by calculating the ∇ x B, with ∇= ∂/∂r r

^{^}+ 1/r(∂/∂θ) θ

^{^}+ ∂/∂z z

^{^}and B only in the z

^{^}axis, B-> B(r) z

^{^}= B

_{z}(r)

Then we calculate the J x B ( J

_{θ}= -c/4π (dB

_{z}/dr ),and equate it from (1) like;

∇P = (JxB)/c, and here ∇P=dP

_{r}/dr

dP

_{r}/dr= (JxB)/c = -1/4π (dB

_{z}/dr =>

=> d/dr( P(r) + B

_{z}

^{2}(r)/8π) = 0

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