Animated Solution for Physics - Electromagnetic Waves: A plane electromagnetic wave of frequency $100\text{ MHz}$ is travelling in vacuum along the x-direction. At a particular point in space and time, $\mathbf{B}=2.0\times 10^{-8}\hat{\mathbf{k}}\text{ T}$ (where, $\hat{\mathbf{k}}$ is unit vector along z-direction). What is $\mathbf{E}$ at this point?

Imagine standing on the shore of a cosmic ocean. The waves that wash over you aren't made of water, but of pure, oscillating energy. These are electromagnetic waves—the very fabric of light, radio signals, and X-rays. In the late 19th century, James Clerk Maxwell discovered something profound: a changing electric field creates a magnetic field, and a changing magnetic field creates an electric field. They dance together in a perfect, self-sustaining embrace, hurtling through the vacuum of space at the ultimate speed limit of the universe: the speed of light, $c$.

In this problem, we are given a snapshot of this cosmic dance. We are told that a plane electromagnetic wave is traveling through a vacuum along the positive x-direction. At a specific, frozen moment in time and space, we measure the magnetic field vector to be $\mathbf{B}=2.0\times 10^{-8}\hat{\mathbf{k}}\text{ T}$. Our mission is to deduce the exact state of the electric field vector, $\mathbf{E}$, at that very same moment.

You might notice that the problem provides the frequency of the wave: $100\text{ MHz}$. This is a classic distractor! While the frequency tells us what kind of wave this is (in this case, it falls in the FM radio/VHF television band), it has absolutely no bearing on the instantaneous relationship between the amplitudes of the electric and magnetic fields. The relationship between $\mathbf{E}$ and $\mathbf{B}$ is a fundamental geometric and physical constraint that holds true regardless of how fast the wave is oscillating.

To find the electric field, we must first determine its magnitude. In a vacuum, the magnitudes of the electric field ($E$) and the magnetic field ($B$) are inextricably linked by the speed of light ($c$). This relationship is elegantly simple:

Why is this true? It stems from the fact that the energy density of the electric field must perfectly balance the energy density of the magnetic field in a propagating electromagnetic wave. The speed of light acts as the conversion factor between these two domains.

Let's execute the calculation. We know the speed of light in a vacuum is approximately $3\times 10^8\text{ m/s}$. Substituting our known values:

Notice how beautifully the powers of ten cancel out. The $10^8$ and $10^{-8}$ annihilate each other, leaving us with a straightforward multiplication:

So, the magnitude of the electric field is exactly $6.0\text{ V/m}$. But a vector is not complete without its direction.

This is where the geometry of electromagnetic waves comes into play. An electromagnetic wave is a transverse wave. This means that both the electric field and the magnetic field oscillate perpendicularly to the direction the wave is traveling. Furthermore, they are perpendicular to each other.

Mathematically, the direction of propagation of the wave (let's call its unit vector $\hat{\mathbf{c}}$) is always given by the cross product of the direction of the electric field ($\hat{\mathbf{E}}$) and the direction of the magnetic field ($\hat{\mathbf{B}}$):

This is a direct consequence of the Poynting vector, which describes the directional energy flux of the electromagnetic field.

Let's plug in what we know about the directions. The wave is traveling along the x-direction, so $\hat{\mathbf{c}}=\hat{\mathbf{i}}$. The magnetic field is pointing along the z-direction, so $\hat{\mathbf{B}}=\hat{\mathbf{k}}$. Our equation becomes:

Now, we must ask ourselves: what unit vector, when crossed with $\hat{\mathbf{k}}$, yields $\hat{\mathbf{i}}$?

Recall the cyclic properties of the standard Cartesian unit vectors: - $\hat{\mathbf{i}}\times \hat{\mathbf{j}}=\hat{\mathbf{k}}$ - $\hat{\mathbf{j}}\times \hat{\mathbf{k}}=\hat{\mathbf{i}}$ - $\hat{\mathbf{k}}\times \hat{\mathbf{i}}=\hat{\mathbf{j}}$

Looking at our cyclic rules, it is immediately clear that $\hat{\mathbf{E}}$ must be $\hat{\mathbf{j}}$. If you prefer a more tactile approach, you can use the right-hand rule: point your thumb in the direction of propagation ($\hat{\mathbf{i}}$), and your middle finger in the direction of the magnetic field ($\hat{\mathbf{k}}$). Your index finger will naturally point along the y-axis ($\hat{\mathbf{j}}$), which is the direction of the electric field.

We have successfully decoded both the magnitude and the direction of the electric field. By combining them, we construct the complete vector:

This problem is a beautiful demonstration of how physical laws and spatial geometry intertwine. By understanding the fundamental constraints of electromagnetic waves, we can deduce the entire state of the system from just a few pieces of information.