Proving the B field in a wire only has theta component?

  • #1

Homework Statement



Prove in a current carrying wire the magnetic field only has a theta component.

Homework Equations



∇ ⋅ B = 0 (dive of magnetic field zero, 2nd Maxwell Eq)

∇ x B = μ J (Ampere's Law, 4th Maxwell Eq)

Cylindrical symmetry means B field only dependent on r (distance from z axis) so that

B = Br(r)rhat + Bθ(r)θhat + Bz(r)zhat

The Attempt at a Solution



Divergence of Ampere's Law = 0 gives

- ∂B/∂r θhat + 1/r ∂/∂r(rBθ) zhat = 0

Not really sure where to go from there =/
 

Answers and Replies

  • #2
Simon Bridge
Science Advisor
Homework Helper
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You'll end up having to solve a system of DEs.

Note: you can write the equations better in LaTeX... i.e. $$\vec B = B_r\hat r + B_\theta\hat\theta + B_\phi\hat\phi \\ -\frac{\partial B}{\partial r}\hat\theta + \frac{1}{r}\frac{\partial}{\partial r}\left(rB_\theta\right) \hat z = 0$$ ... If I understand you correctly that the last equation is supposed to be ##\nabla\cdot(\nabla\times\vec B) = 0## then that does not look right to me.
Please show your working.
 
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