Classical Electrodynamics · Problem 8.8

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · schumann-resonances-q-values

Problem

For the Schumann resonances of Section 8.9 calculate the

Q

values on the assumption that the earth has a conductivity

\sigma_e

and the ionosphere has a conductivity

\sigma_i

, with corresponding skin depths

\delta_e

and

\delta_i

. (a) Show that to lowest order in

h/a

the

Q

value is given by

Q = Nh/(\delta_e + \delta_i)

and determine the numerical factor

N

for all

l

. (b) For the lowest Schumann resonance evaluate the

Q

value assuming

\sigma_e = 0.1\;(\Omega\text{m})^{-1}

\sigma_i = 10^{-5}\;(\Omega\text{m})^{-1}

h = 10^2

km. (c) Discuss the validity of the approximations used in part a for the range of parameters used in part b.