Question

A sinusoidal electromagnetic wave is given by \( \vec{E} = 20 \sin \left( \frac{2}{300} x - 10^6 t \right) \) propagating in a non-magnetic material. Dielectric constant of the material is.

• A. \( 9 \times 10^4 \)
• B. \( 3 \times 10^2 \)
• C. 4
• D. 2

Hint:

In non-magnetic materials, the dielectric constant can be found by comparing the wave number and angular frequency in the wave equation.

Answer 1

The Correct Option is C

Step 1: Wave equation and relation to dielectric constant.
The wave equation for an electromagnetic wave is given as \( \vec{E} = E_0 \sin(kx - \omega t) \), where \( k = \frac{2\pi}{\lambda} \) is the wave number, and \( \omega = 2\pi f \) is the angular frequency. The relation between the speed of light \( c \), the dielectric constant \( \epsilon_r \), and the speed of the wave \( v \) in the material is: \[ v = \frac{c}{\sqrt{\epsilon_r}} \] where \( v = \frac{\omega}{k} \). Step 2: Compare with given values.
By comparing the given wave equation with the standard form, we can find the speed of the wave and use the above relation to solve for the dielectric constant \( \epsilon_r \). After solving, we find that \( \epsilon_r = 4 \). Final Answer: \[ \boxed{4} \]