Question

A plane electromagnetic wave in a non-magnetic dielectric medium is given by $\vec{E}=\vec{E}_{0}\left(4\times10^{-7}\,x-50t\right)$ with distance being in meter and time in seconds. The dielectric constant of the medium is:

• A. $2.4$
• B. $5.8$
• C. $8.2$
• D. $4.8$

Answer 1

The Correct Option is B

To find the dielectric constant of the medium, we need to analyze the properties of the given electromagnetic wave. The electric field of the wave is given as:

\(\vec{E} = \vec{E}_{0}\left(4\times10^{-7}\,x - 50t\right)\)

The general expression for a plane electromagnetic wave is:

\(\vec{E} = \vec{E}_{0} \sin(kx - \omega t)\)

where \(k\) is the wave number, and \(\omega\) is the angular frequency. Comparing this with the given wave equation, we can identify:

• Wave number, \(k = 4 \times 10^{-7}\) m^-1
• Angular frequency, \(\omega = 50\) s^-1

The relationship in a dielectric medium is given by the formula:

\(\omega = v \cdot k\)

where \(v\) is the speed of the wave in the medium. This can also be expressed as:

\(v = \frac{\omega}{k}\)

Substituting the known values:

\(v = \frac{50}{4 \times 10^{-7}} = 1.25 \times 10^{8}\) m/s

The speed of light in a vacuum is \(c = 3 \times 10^{8}\) m/s.

The speed of light in a medium is related to the dielectric constant \(\varepsilon_r\) and the speed of light in vacuum as:

\(v = \frac{c}{\sqrt{\varepsilon_r}}\)

Rearranging gives:

\(\varepsilon_r = \left(\frac{c}{v}\right)^2\)

Substituting for \(c\) and \(v\):

\(\varepsilon_r = \left(\frac{3 \times 10^8}{1.25 \times 10^8}\right)^2\)

\(\varepsilon_r = \left(2.4\right)^2 = 5.76\)

Rounding to one decimal place, we have the dielectric constant:

\(\varepsilon_r \approx 5.8\)

Thus, the correct answer is \(5.8\).