Question

In a dielectric medium of relative permittivity 5, the amplitudes of the displacement current and conduction current are equal for an applied sinusoidal voltage of frequency f = 1 MHz. The value of conductivity (in Ω^-1m^-1) of the medium at this frequency is

• A. 2.78 × 10^-4
• B. 2.44 × 10^-4
• C. 2.78 × 10^-3
• D. 2.44 × 10^-3

Answer 1

The Correct Option is A

To determine the conductivity of a dielectric medium where the displacement current and conduction current are equal for a given sinusoidal voltage, we use the relationship between the conduction current density, displacement current density, and frequency of the applied voltage. Let's break down the steps to solve this problem: 

1. According to Maxwell's equations, the displacement current density \(J_d\) is given by: \(J_d = \varepsilon_0 \varepsilon_r \frac{\partial E}{\partial t}\) where
   • \(\varepsilon_0\) is the permittivity of free space (approximately \(8.854 \times 10^{-12} \, \text{F/m}\)),
   • \(\varepsilon_r\) is the relative permittivity of the medium, given as 5,
   • \(\frac{\partial E}{\partial t}\) is the rate of change of the electric field.
2. For a sinusoidal voltage, \(\frac{\partial E}{\partial t} = \omega E_0 \sin(\omega t)\), where \(\omega = 2\pi f\) is the angular frequency. Given \(f = 1 \, \text{MHz} = 1 \times 10^6 \, \text{Hz}\), the angular frequency \(\omega = 2\pi \times 10^6 \, \text{rad/s}\).
3. Substitute the values into the formula for the displacement current density: \(J_d = \varepsilon_0 \varepsilon_r \omega E_0 \sin(\omega t)\).
4. The conduction current density \(J_c\) is given by Ohm's law: \(J_c = \sigma E\) where \(\sigma\) is the conductivity.
5. For the currents to be equal, we have: \(\sigma E = \varepsilon_0 \varepsilon_r \omega E_0 \sin(\omega t) \implies \sigma = \varepsilon_0 \varepsilon_r \omega\).
6. Substituting the values: \(\sigma = 8.854 \times 10^{-12} \times 5 \times 2\pi \times 10^6\) \(\sigma = 2.78 \times 10^{-4} \, \text{Ω}^{-1}\text{m}^{-1}\).

Thus, the conductivity of the medium for the given conditions is 2.78 × 10^-4 Ω^-1m^-1. This matches option

2.78 × 10^-4

, confirming it as the correct answer.