Animated Solution for Physics - Electromagnetic Waves: The electric field of a plane electromagnetic wave propagating along the x-direction in vacuum is $\mathbf{E}=E_0\hat{\mathbf{j}}\cos (\omega t-kx)$. The magnetic field $\mathbf{B}$, at the moment $t=0$ is

The universe is bathed in light, a magnificent dance of electric and magnetic fields propagating through the void of space. James Clerk Maxwell, in one of the greatest triumphs of human intellect, showed us that these fields are inextricably linked. In this problem, we are given the electric field of a plane electromagnetic wave and tasked with uncovering its magnetic counterpart at a specific moment in time. Let's embark on this journey to decode the anatomy of an electromagnetic wave.

We are given the electric field vector of the wave:

This single equation holds a wealth of information. First, the unit vector $\hat{\mathbf{j}}$ tells us that the electric field is oscillating strictly along the y-axis. Second, the phase term $(\omega t-kx)$ is the signature of a traveling wave. Because the spatial variable $x$ is paired with a negative sign, we know unequivocally that the wave is propagating in the positive x-direction. Therefore, the direction of wave propagation, which we will denote as $\hat{\mathbf{c}}$, is simply $\hat{\mathbf{i}}$.

In a plane electromagnetic wave propagating through a vacuum (or any isotropic medium), the electric field $\mathbf{E}$, the magnetic field $\mathbf{B}$, and the direction of propagation $\mathbf{c}$ form a mutually orthogonal triad. This geometric relationship is elegantly captured by the cross product:

We already know that $\hat{\mathbf{E}}=\hat{\mathbf{j}}$ and $\hat{\mathbf{c}}=\hat{\mathbf{i}}$. Substituting these into our master equation gives:

Now, imagine your right hand. Point your fingers in the direction of the y-axis ($\hat{\mathbf{j}}$). You need to curl them in a direction such that your thumb points along the positive x-axis ($\hat{\mathbf{i}}$). The only way to achieve this is if you curl your fingers towards the positive z-axis. Mathematically, the standard cross product rules dictate that $\hat{\mathbf{j}}\times \hat{\mathbf{k}}=\hat{\mathbf{i}}$. Thus, the magnetic field must oscillate along the z-axis:

We have the direction, but what about the magnitude? Maxwell's equations reveal that the amplitudes of the electric and magnetic fields in a vacuum are not independent; they are locked together by the speed of light, $c$. The relationship is:

Furthermore, the speed of light itself is a fundamental property of the vacuum, determined by its electric permittivity (${ϵ}_0$) and magnetic permeability (${\mu }_0$):

By substituting this expression for $c$ into our amplitude ratio, we can solve for the magnetic field amplitude $B_0$:

We now possess all the necessary pieces: the amplitude, the direction, and the knowledge that in a vacuum, the electric and magnetic fields oscillate perfectly in phase. We can construct the general time-dependent equation for the magnetic field vector:

Substituting our expression for $B_0$, this becomes:

The problem specifically asks for the magnetic field at the exact moment $t=0$. Let's substitute $t=0$ into our phase term:

Here, we must recall a fundamental property of trigonometry. The cosine function is an even function, which means it is symmetric about the y-axis. Mathematically, $\cos (-\theta )=\cos (\theta )$. Applying this property, the negative sign inside the argument vanishes:

This perfectly matches option (d). By systematically breaking down the wave's geometry and applying the fundamental constants of nature, we have successfully reconstructed the magnetic field from the electric field. This interplay between electricity and magnetism is the very foundation of modern optics and wireless communication.