Transient response in an imperfect dielectric

This paper gives the transient electric field response to an electric current element in an infinite linear homogeneous isotropic medium for all values of the parameter
b = r\sigma \sqrt{\mu/\epsilon}
upon which its shape depends. It is shown that the response cannot be separated into that resulting from 1) the charge, 2) the current, and 3) its derivative when
b
is appreciably different from zero. The initial response occurs at the time
t = q = r \sqrt{\epsilon \mu}
. The radial component of the field is a monotonically increasing function of time approaching a constant asymptote. Its initial value has a maximum of 0.7358 times its final value at
b = 2
. The shape of the transient changes radically at
b = 2
. For values of
b > 20
the initial value is negligible and the response is closely approximated by the asymptotic expression for
b
large. The tangential component approaches its constant asymptote from larger values. For small
b
the maximum occurs when
t
is large. The tangential component is approximately the same as the radial component for
b < \frac{1}{2}
. For larger values of
b
the maximum occurs at earlier times, occurring when
t = q
for
2.243 < b < 6.600
. It has its maximum initial value of 1.692 times its final value for
b = 5.043
. For values of
b>6.6
the maximum occurs at increasingly later times. For
b>24
the initial value is negligible and the response is approximated by the asymptotic expression for
b
large. Curves are given for the response not only as a function of time for various values of
b
but also as a function of
b
for various times. Comparison of experiments with these curves will allow the determination of
b
and, hence, the conductivity. The response is also given as a function of distance for various times. The Bessel function integral involved in this problem has been evaluated and presented in the form of curves for all values of
b
and all values of
t
for which it makes an appreciable contribution to the result.