Question

A plane wave propagating in the dielectric medium has an electric field given as Ex = E0 cos (3 × 1010 t - 100 z). The dielectric constant of the medium is:
1. 1.33
2. 1.00
3. 1.01
4. 2.15

Answer 1

Correct Answer - Option 2 : 1.00

Concept:

The phase velocity is defined as the rate at which the phase of the wave propagates in space.

Mathematically this is calculated as:

\({V_p} = \frac{ω }{β }\)   ----(1)

Where,

ω = angular frequency

β = phase propagation constant

Phase velocity is also given by:

\({V_p} = \frac{1}{{\sqrt {μ \varepsilon } }}\)

Here, μ = μoμr

ε = εo εr

\({V_p} = \frac{1}{{\sqrt {{μ _o}{μ _r}{\varepsilon _o}{\varepsilon _r}} }}\)

\({V_p} = \frac{c}{{\sqrt {{μ _r}{\varepsilon _r}} }}\)  ---(2)

\(c = \frac{1}{{\sqrt {{μ _o}{\varepsilon _o}} }} = 3 × {10^8}\;m/sec\)

μ0 = permeability in free space 4π × 10-7 H/m

εo = permittivity in free space 8.854 × 10-12 C2/Nm2

Calculation:

Given:

Ex = E0 cos (3 × 1010 t - 100 z)

μ_r = 1 

From the given equation we can calculate V_p:

ω = 3 × 10¹⁰

β = 100

\(V_p \ = \ \frac{3 \ \times \ 10^{10} }{100}=3 \ \times \ 10^{8}\)

Comparing with equation (2) we get:

\(\frac{3 \ \times \ 10^{8}}{{\sqrt {{}{\varepsilon _r}} }} = 3 \times {10^8}\)

ϵ_r = 1

Hence the solution is option (2).