Energy Transport and the Poynting Vector — Question 9

Question 9

The Poynting flux in a coaxial cable reproduces electrical power

An ideal vacuum-filled coaxial line has inner radius a and outer radius b, voltage V, and current I in +z on the inner conductor. In the dielectric region use $E_r=V/[r\ln(b/a)]$ and $B_\phi=\mu_0I/(2\pi r)$.

1. Find the Poynting vector between the conductors.

2. Integrate its axial flux over the annular cross section and compare with circuit power.

Question 9 - Solution

The Poynting flux in a coaxial cable reproduces electrical power

(a) Method and calculation

$$\hat r\times\hat\phi=\hat z\Rightarrow\mathbf S=\frac{E_rB_\phi}{\mu_0}\hat z.$$$$\boxed{\mathbf S=\frac{VI}{2\pi r^2\ln(b/a)}\hat z}.$$

(b) Interpretation and further calculation

$$P=\int_a^b S_z2\pi r\,dr=\frac{VI}{\ln(b/a)}\int_a^b\frac{dr}{r}.$$$$\boxed{P=VI}.$$

Physical check

The energy flows through the field-filled annulus; the Poynting vector is not limited to freely propagating radiation in empty space.