Question

Consider a long straight wire of radius a and electrical conductivity (\sigma) carrying a uniform current density J long it’s length. Find the magnitude and direction of the Poynting vector at the surface of the wire.

For this do we find J using amperes Law, then since magnetic field is constant have that \[\nabla \times \mathbf{E} = 0 \implies \mathbf{E}=\nabla U\] for some scalar U? then find U with seperation of variables? Then calculate Poynting Vector? If so I'm stuck with what the boundary conditions are. If not how is this done?

Answer 1

\[\vec{J}=J\vec{i}_z\]\[\vec{E}=\frac{\vec{J}}{\sigma}=\frac{J}{\sigma}\vec{i}_z\]\[\]\[\vec{H}=\frac{I}{2\pi a}\vec{i}_\phi=\frac{J\cdot\pi a^2}{2\pi a}\vec{i}_\phi=\frac{Ja}{2}\vec{i}_\phi\]\[\vec{P}=\vec{E}\times\vec{H}=\frac{J^2a}{2\sigma}(\vec{i}_z\times\vec{i}_\phi)=-\frac{J^2a}{2\sigma}\vec{i}_r\]

Answer 2

Thanks for helping me study! I have a few more that I've been stuck with. Exam is tomorrow morning !! lol