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

[applestrudle]

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 = B_r(r)rhat + B_θ(r)θhat + B_z(r)zhat

The Attempt at a Solution

Divergence of Ampere's Law = 0 gives

- ∂B/∂r θhat + 1/r ∂/∂r(rB_θ) zhat = 0

Not really sure where to go from there =/

 

[Simon Bridge]

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.