An electromagnetic wave travelling in the $$x-$$ direction has frequency of $$2 \times 10^{14}$$ Hz and electric field amplitude of 27 V m$$^{-1}$$ oscillates in $$Y-$$direction. From the options given below, which one describes the magnetic field for this wave?

We are given an electromagnetic wave traveling in the $$x$$-direction with a frequency $$f = 2 \times 10^{14}$$ Hz and an electric field amplitude $$E_0 = 27$$ V/m oscillating in the $$Y$$-direction. We need to find the correct expression for the magnetic field $$\vec{B}(x, t)$$.

First, recall that in an electromagnetic wave, the electric and magnetic fields are perpendicular to each other and to the direction of propagation. Since the wave travels in the $$x$$-direction and the electric field oscillates in the $$Y$$-direction (using $$\hat{j}$$ for the unit vector), the magnetic field must oscillate in the $$Z$$-direction (using $$\hat{k}$$ for the unit vector). This is determined by the right-hand rule, where the direction of propagation ($$\hat{i}$$) is given by $$\vec{E} \times \vec{B}$$.

Next, the amplitude of the magnetic field $$B_0$$ is related to the electric field amplitude $$E_0$$ by the formula $$B_0 = \frac{E_0}{c}$$, where $$c$$ is the speed of light in vacuum. The speed of light $$c = 3 \times 10^8$$ m/s. Substituting the values:

$$ B_0 = \frac{27}{3 \times 10^8} = 9 \times 10^{-8} \text{ T} $$

So, the magnetic field amplitude is $$9 \times 10^{-8}$$ T, and it must be in the $$\hat{k}$$ direction.

Now, the general wave equation for a magnetic field traveling in the positive $$x$$-direction is $$\vec{B}(x, t) = B_0 \hat{k} \sin(kx - \omega t)$$, where $$k$$ is the wave number and $$\omega$$ is the angular frequency. We know $$\omega = 2\pi f$$, so:

$$ \omega = 2\pi \times (2 \times 10^{14}) = 4\pi \times 10^{14} \text{ rad/s} $$

The wave number $$k$$ is related to the wavelength $$\lambda$$ by $$k = \frac{2\pi}{\lambda}$$. The wavelength can be found using $$c = f \lambda$$, so:

$$ \lambda = \frac{c}{f} = \frac{3 \times 10^8}{2 \times 10^{14}} = 1.5 \times 10^{-6} \text{ m} $$

Thus, the wave number is:

$$ k = \frac{2\pi}{\lambda} = \frac{2\pi}{1.5 \times 10^{-6}} = \frac{4\pi}{3} \times 10^6 \text{ m}^{-1} $$

The argument of the sine function is $$kx - \omega t = \left(\frac{4\pi}{3} \times 10^6\right)x - (4\pi \times 10^{14})t$$. We can factor out $$2\pi$$ to match the form in some options:

$$ kx - \omega t = \frac{4\pi}{3} \times 10^6 \cdot x - 4\pi \times 10^{14} t = 2\pi \left( \frac{2}{3} \times 10^6 \cdot x - 2 \times 10^{14} t \right) $$

Alternatively, using $$\lambda = 1.5 \times 10^{-6}$$ m and $$f = 2 \times 10^{14}$$ Hz, we can write the argument as $$2\pi \left( \frac{x}{\lambda} - f t \right)$$:

$$ kx - \omega t = 2\pi \left( \frac{x}{1.5 \times 10^{-6}} - (2 \times 10^{14}) t \right) $$

Therefore, the magnetic field expression becomes:

$$ \vec{B}(x, t) = (9 \times 10^{-8}) \hat{k} \sin \left[ 2\pi \left( \frac{x}{1.5 \times 10^{-6}} - 2 \times 10^{14} t \right) \right] $$

Now, comparing with the options:

Option A: $$\vec{B}(x, t) = (9 \times 10^{-8} \text{ T}) \hat{j} \sin[1.5 \times 10^{-6} x - 2 \times 10^{14} t]$$ has the wrong direction ($$\hat{j}$$ instead of $$\hat{k}$$) and incorrect argument (no $$2\pi$$ factor and wrong coefficient for $$x$$).

Option B: $$\vec{B}(x, t) = (9 \times 10^{-8} \text{ T}) \hat{i} \sin[2\pi(1.5 \times 10^{-8} x - 2 \times 10^{14} t)]$$ has the wrong direction ($$\hat{i}$$) and incorrect wave number (coefficient $$1.5 \times 10^{-8}$$ instead of $$\frac{1}{1.5 \times 10^{-6}}$$).

Option C: $$\vec{B}(x, t) = (3 \times 10^{-8} \text{ T}) \hat{j} \sin 2\pi\left[\frac{x}{1.5 \times 10^{-6}} - 2 \times 10^{14} t\right]$$ has the wrong amplitude ($$3 \times 10^{-8}$$ T instead of $$9 \times 10^{-8}$$ T) and wrong direction ($$\hat{j}$$).

Option D: $$\vec{B}(x, t) = (9 \times 10^{-8} \text{ T}) \hat{k} \sin 2\pi\left[\frac{x}{1.5 \times 10^{-6}} - 2 \times 10^{14} t\right]$$ matches exactly: correct amplitude $$9 \times 10^{-8}$$ T, correct direction $$\hat{k}$$, and correct argument $$2\pi \left( \frac{x}{\lambda} - f t \right)$$ with $$\lambda = 1.5 \times 10^{-6}$$ m and $$f = 2 \times 10^{14}$$ Hz.

Hence, the correct answer is Option D.