Answer to Question #153980 in Classical Mechanics for Afraid

Question

A material has conductivity of 10-2 mho/m and a relative permittivity of 4. The frequency at which conduction current in the medium is equal to displacement current is

Answer 1

Let's first write the formula for the conduction current density:

$$J=\sigma E,$$

here, $J$ is the conduction current density, $\sigma=10^{-2}\ \dfrac{S}{m}$ is the conductivity of the material.

Then, we can write the formula for the displacement current density:

$$J=\omega D=2\pi fD,$$

here, $E$ is the electric field, $\omega$ is the angular frequency, $D=\epsilon\epsilon_0 E$ is electric displacement field, $\epsilon=4$ is a relative permittivity, $\epsilon_0=8.85\cdot10^{-12}\ \dfrac{F}{m}$ is the permittivity of the free space, $f$ is the frequency.

Since we searching for frequency at which conduction current in the medium is equal to displacement current, we must equate these two formulas:

$$\sigma E=2\pi fE\epsilon\epsilon_0.$$

Finally, from this formula we can find the frequency at which conduction current in the medium is equal to displacement current:

$$f=\dfrac{\sigma}{2\pi\epsilon},$$ $$f=\dfrac{10^{-2}\ \dfrac{S}{m}}{2\pi\cdot4\cdot8.85\cdot10^{-12}\ \dfrac{F}{m}}=45\ MHz.$$

Answer:

$f=45\ MHz.$