A \(500-\mathrm{kV}, 300-\mathrm{km}, 60-\mathrm{Hz}\), three-phase overhead transmission line, assumed to be lossless, has a series inductance of \(0.97 \mathrm{mH} / \mathrm{km}\) per phase and a shunt capacitance of \(0.0115 \mu \mathrm{F} / \mathrm{km}\) per phase.

(a) Determine the phase constant \(\beta\), the surge impedance \(Z_{c}\), velocity of propagation \(v\), and the wavelength \(\lambda\) of the line.

(b) Determine the voltage, current, real and reactive power at the sending end, and the percent voltage regulation of the line if the receiving-end load is \(800 \mathrm{MW}\) at 0.8 power factor lagging and at \(500 \mathrm{kV}\).