Electric field of plane electromagnetic wave propagating through a non-magnetic medium is given by E

Q: Why is the speed of the wave $\omega/k$?

The phase of the wave is constant for a point moving with the wave, so $\omega t - kx = \text{constant}$. Differentiating with respect to time gives $\omega - k \frac{dx}{dt} = 0$, which means the wave speed $v = \frac{dx}{dt} = \frac{\omega}{k}$.

Q: What does a non-magnetic medium imply?

A non-magnetic medium does not respond strongly to magnetic fields, which means its relative permeability $\mu_r$ is approximately equal to 1, just like a vacuum.

Q: Is the refractive index always equal to $\sqrt{\epsilon_r \mu_r}$?

Yes, from Maxwell's equations, the speed of light in a medium is $v = \frac{1}{\sqrt{\epsilon \mu}}$. Since $c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}$, the refractive index $n = \frac{c}{v} = \sqrt{\frac{\epsilon \mu}{\epsilon_0 \mu_0}} = \sqrt{\epsilon_r \mu_r}$.

Q: Why did we ignore the amplitude $20$ V/m?

The amplitude $E_0 = 20$ V/m tells us the maximum strength of the electric field, but it doesn't affect the speed of propagation or the properties of the medium like the dielectric constant.

Q: Can the dielectric constant be less than 1?

No, the dielectric constant (relative permittivity) of a vacuum is exactly 1. For all other materials, it is greater than 1 because the material's polarization always opposes the applied electric field, increasing the overall permittivity.

Question

Electric field of plane electromagnetic wave propagating through a non-magnetic medium is given by $E = 20\cos(2 	imes 10^{10} t - 200x)$ V/m. The dielectric constant of the medium is equal to
(Take, $\mu_r = 1$)

Options:

(A) 9

(B) 2

(C) $\frac{1}{3}$

(D) 3

Sigma Percentile · Accepted Answer

### Decoding the Electromagnetic Wave

Imagine an electromagnetic wave surfing through space. When it enters a medium, its speed changes, and this change is governed by the electrical and magnetic properties of that medium. In this problem, we are given the electric field equation of a plane electromagnetic wave propagating through a non-magnetic medium.

Our mission? To uncover the dielectric constant of this mysterious medium. Let's break down the given equation and see what secrets it holds.

### Extracting the Wave Parameters

The electric field is given by:

$E = 20\cos(2 	imes 10^{10} t - 200x)$

To make sense of this, we compare it with the standard equation of a traveling wave:

$E = E_0 \cos(\omega t - kx)$

By simply matching the terms, we can extract two vital pieces of information. First, the angular frequency $\omega$ is the coefficient of $t$:

$\omega = 2 	imes 10^{10} 	ext{ rad/s}$

Second, the wave number $k$ is the coefficient of $x$:

$k = 200 	ext{ rad/m}$

These two parameters are the keys to finding how fast the wave is moving.

### Finding the Speed of the Wave

The speed of any wave, $v$, is the ratio of its angular frequency to its wave number.

$v = \frac{\omega}{k}$

Let's plug in our extracted values:

$v = \frac{2 	imes 10^{10}}{200}$

Calculating this gives us the speed of the wave in the medium:

$v = 10^8 	ext{ m/s}$

Notice that this speed is significantly less than the speed of light in a vacuum ($3 	imes 10^8 	ext{ m/s}$). This slowing down is exactly what happens when light enters a denser medium!

### The Refractive Index Connection

Now that we have the speed of the wave, we can easily find the refractive index, $\mu$, of the medium. The refractive index is defined as the ratio of the speed of light in a vacuum, $c$, to the speed of light in the medium, $v$.

$\mu = \frac{c}{v}$

Substituting the known values:

$\mu = \frac{3 	imes 10^8}{10^8}$

This simplifies beautifully to:

$\mu = 3$

### Unveiling the Dielectric Constant

We are almost at the finish line. The refractive index $\mu$ is intimately connected to the medium's relative permittivity (dielectric constant), $\epsilon_r$, and relative permeability, $\mu_r$, through Maxwell's relation:

$\mu = \sqrt{\epsilon_r \mu_r}$

The problem explicitly states that the medium is **non-magnetic**, which is a crucial hint. It means the relative permeability $\mu_r$ is equal to $1$.

Let's substitute $\mu = 3$ and $\mu_r = 1$ into our relation:

$3 = \sqrt{\epsilon_r 	imes 1}$

To isolate the dielectric constant, we simply square both sides of the equation:

$3^2 = \epsilon_r$

$\epsilon_r = 9$

And there we have it! The dielectric constant of the medium is **$9$**. By systematically decoding the wave equation and applying fundamental electromagnetic principles, we've successfully solved the problem.

Animated Solution for Physics - Electromagnetic Waves: Electric field of plane electromagnetic wave propagating through a non-magnetic medium is given by $E = 20 cos (2 \times 1 0^{10} t - 200 x)$ V/m. The dielectric constant of the medium is equal to (Take, $μ_{r} = 1$ )

Select Answer:

Visualized Solution

Standard Equation of EM Wave

Comparing with Standard Form

Speed of the Wave

Calculating Speed

Refractive Index

Relation with Dielectric Constant

Substituting Values

Final Answer

The Sigma Insight: Characteristics of Electromagnetic Waves

Decoding the Electromagnetic Wave

Extracting the Wave Parameters

Finding the Speed of the Wave

The Refractive Index Connection

Unveiling the Dielectric Constant

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