Question

An electromagnetic wave of frequency 100 MHz propagates through a medium of conductivity, $\sigma = 10$ mho/m. The ratio of maximum conduction current density to maximum displacement current density is _________.
$\left[ \text{Take } \frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \text{ Nm}^2/\text{C}^2 \right]$

Hint:

In a good conductor, the conduction current density is much larger than the displacement current density, which is consistent with the large ratio obtained here.

Answer 1

Correct Answer: 180

Step 1: Understanding the Concept:
The conduction current density ($J_c$) in a medium is determined by Ohm's law in its microscopic form, relating it to the electric field and conductivity.
The displacement current density ($J_d$) arises from the time-varying electric field, as described by Maxwell's equations.
For a sinusoidal electromagnetic wave, both current densities vary with time, and we seek the ratio of their peak values.

Step 2: Key Formula or Approach:
The maximum conduction current density is given by:
\[ J_{c, max} = \sigma E_0 \]
The displacement current density is defined as:
\[ J_d = \epsilon \frac{\partial E}{\partial t} \]
For a field $E = E_0 \sin(\omega t)$, the maximum displacement current density is:
\[ J_{d, max} = \epsilon_0 \omega E_0 \]
The required ratio is:
\[ \text{Ratio} = \frac{J_{c, max}}{J_{d, max}} = \frac{\sigma}{\epsilon_0 \omega} = \frac{\sigma}{\epsilon_0 (2\pi f)} \]

Step 3: Detailed Explanation:
Given values are:
$\sigma = 10$ mho/m
$f = 100$ MHz $= 10^8$ Hz
$\frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \Rightarrow \epsilon_0 = \frac{1}{36\pi \times 10^9} \text{ C}^2/(\text{Nm}^2)$
The angular frequency $\omega$ is:
$\omega = 2\pi f = 2\pi \times 10^8$ rad/s
Now, calculate the product $\epsilon_0 \omega$:
\[ \epsilon_0 \omega = \left( \frac{1}{36\pi \times 10^9} \right) \times (2\pi \times 10^8) \]
\[ \epsilon_0 \omega = \frac{2\pi}{36\pi \times 10} = \frac{1}{180} \]
Substituting this into the ratio formula:
\[ \text{Ratio} = \frac{10}{1/180} = 10 \times 180 = 1800 \]

Step 4: Final Answer:
The ratio of the maximum conduction current density to the maximum displacement current density is 1800.