Understanding the Electromagnetic Wave Equation

The provided magnetic field of the electromagnetic wave is given by the equation:

$B = 0.3\cos(ky-10^8t)i$ tesla

This equation represents a plane wave propagating along the positive y-axis. We can compare this to the general form of a magnetic field wave:

$B = B_0 \cos(k y - \omega t)\hat{i}$

From the given equation, we can identify the following parameters:

• Amplitude ($B_0$): 0.3 tesla
• Wave number ($k$): $k$
• Angular frequency ($\omega$): $10^8$ rad/s
• Direction of propagation: Positive y-axis
• Direction of magnetic field: x-axis ($\hat{i}$)

Relating Wave Properties in a Medium

The electromagnetic wave is propagating through a dielectric medium with a refractive index ($n$) of 2. The speed of the wave ($v$) in this medium is related to the speed of light in vacuum ($c$) and the refractive index ($n$) by the formula:

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

We know the standard value for the speed of light in vacuum is $c \approx 3 \times 10^8$ m/s.

Calculating the Wavelength

The relationship between angular frequency ($\omega$), wave speed ($v$), and wave number ($k$) is:

$\omega = v k$

The wavelength ($\lambda$) is related to the wave number ($k$) by:

$k = \frac{2\pi}{\lambda}$

Substituting the expression for $k$ into the angular frequency equation:

$\omega = v \left(\frac{2\pi}{\lambda}\right)$

Now, we can rearrange this formula to solve for the wavelength ($\lambda$):

$\lambda = \frac{2\pi v}{\omega}$

Substitute the relation $v = c/n$ into the wavelength formula:

$\lambda = \frac{2\pi (c/n)}{\omega} = \frac{2\pi c}{n\omega}$

Substituting Values and Finding the Result

Now, we substitute the known values into the derived formula:

• $c = 3 \times 10^8$ m/s
• $n = 2$
• $\omega = 10^8$ rad/s

Calculation:

$\lambda = \frac{2\pi \times (3 \times 10^8 \text{ m/s})}{2 \times (10^8 \text{ rad/s})}$

$\lambda = \frac{2\pi \times 3 \times 10^8}{2 \times 10^8}$ m

$\lambda = 3\pi$ m

Therefore, the wavelength of the electromagnetic wave in the dielectric medium is $3\pi$ meters.