Animated Solution for Physics - Electromagnetic Waves: For a plane electromagnetic wave, the magnetic field at a point $x$ and time $t$ is $\mathbf{B}(x,t)=[1.2\times 10^{-7}\sin (0.5\times 10^{3}x+1.5\times 10^{11}t)\hat{\mathbf{k}}]\text{ T}$. The instantaneous electric field $\mathbf{E}$ corresponding to $\mathbf{B}$ is (Given, speed of light, $c=3\times 10^8\text{ m/s}$)

Imagine you are standing in the vast emptiness of a vacuum, and suddenly, a wave of pure energy washes over you. This isn't a wave of water or sound; it is an electromagnetic wave—a perfectly synchronized dance of electric and magnetic fields propagating through space at the ultimate speed limit of the universe.

In this problem, we are given the mathematical blueprint of the magnetic field component of such a wave, and our mission is to reconstruct its electric field counterpart. Let's embark on this thrilling journey of decoding Maxwell's legacy!

We are presented with the magnetic field equation:

By comparing this to the standard wave equation $\mathbf{B}(x,t)=B_0\sin (kx+\omega t)\hat{\mathbf{n}}$, we can immediately extract two vital pieces of information. First, the maximum amplitude of the magnetic field is $B_0=1.2\times 10^{-7}\text{ T}$. Second, the phase of the wave—the heartbeat that dictates its oscillation—is $(0.5\times 10^{3}x+1.5\times 10^{11}t)$.

Because the electric and magnetic fields in a plane electromagnetic wave are perfectly in phase, the electric field will share this exact same sine term. No phase shifts, no delays. They rise and fall together.

Now, we need to find the amplitude of the electric field, $E_0$. According to Maxwell's equations, the ratio of the electric field amplitude to the magnetic field amplitude in a vacuum is exactly equal to the speed of light, $c$. This gives us the elegant relation:

Let's substitute the values we know. We plug in $3\times 10^8\text{ m/s}$ for the speed of light and $1.2\times 10^{-7}\text{ T}$ for $B_0$:

Multiplying the numbers, $3\times 1.2$ gives $3.6$. And $10^8$ multiplied by $10^{-7}$ leaves us with $10^1$. Therefore, the amplitude of the electric field is exactly $36\text{ V/m}$.

Next, we must determine the direction in which this wave is traveling. The secret lies within the phase term: $kx+\omega t$.

In wave mechanics, a positive sign between the spatial term ($kx$) and the temporal term ($\omega t$) indicates that the wave is propagating in the negative direction of the spatial axis. Since our spatial variable is $x$, the wave is traveling along the negative x-axis.

Thus, the unit vector for the velocity of propagation is:

A fundamental property of electromagnetic waves is that the electric field ($\mathbf{E}$), the magnetic field ($\mathbf{B}$), and the direction of propagation ($\mathbf{v}$) are all mutually perpendicular. Their relationship is strictly governed by the cross product:

From our given equation, we know the magnetic field oscillates along the positive z-axis, so $\hat{\mathbf{B}}=\hat{\mathbf{k}}$. We also just found that $\hat{\mathbf{v}}=-\hat{\mathbf{i}}$. Substituting these into our cross product relation yields:

Now, we rely on standard vector algebra. We know that $\hat{\mathbf{j}}\times \hat{\mathbf{k}}=\hat{\mathbf{i}}$. Therefore, to obtain a negative $\hat{\mathbf{i}}$, our electric field direction must be negative $\hat{\mathbf{j}}$.

This tells us the electric field is oscillating along the negative y-axis.

Finally, we bring all our hard-earned pieces together to construct the complete equation for the instantaneous electric field. We combine the amplitude ($36\text{ V/m}$), the identical phase, and the direction ($-\hat{\mathbf{j}}$):

And there we have it! By carefully dissecting the magnetic field and applying the fundamental laws of electromagnetism, we have successfully derived the electric field. This is the beauty of physics—every piece of the puzzle is connected by elegant, unbreakable rules.