How to calculate phase constant for a coaxial cable?

Question

inner and outer conductor diameters of 1 mm and 5 mm respectively, is filled with a non-magnetic PTFE dielectric having εr = 2.55. frequency = 1 GHz.

Answer 1

The velocity of propagation for the TEM mode $V_p = \frac{c}{\sqrt{\epsilon_r}}$, i.e. in your case about $0.626\,c$, or $1.877 \times 10^8\,\frac{\mathrm{m}}{\mathrm{s}}$. (This is a bit different from the numbers usually used for PTFE, which are $\epsilon_r = 2.07$ and $V_p = 0.695\,c$, but we'll roll with it.)

The wavelength at 1 GHz is then $\lambda = \frac{V_p}{1 \mathrm{GHz}} \approx 18.77\,\mathrm{cm}$.

The phase constant (not commonly used by hams) is $\beta = \frac{2\pi}{\lambda} \approx 33.47\,\mathrm{m}^{-1}$.

The dimensions of the coax don't matter so long as the frequency falls below the cutoff frequency for non-TEM waveguide modes. 1GHz is nowhere near the cutoff for 5mm, so that's fine.