Animated Solution for Physics - Electromagnetic Waves: A plane electromagnetic wave is propagating along the direction $\frac{\hat{i}+\hat{j}}{\sqrt{2}}$ with its polarisation along the direction $\hat{k}$. The correct form of the magnetic field of the wave would be (here $B_0$ is an appropriate constant)

Imagine you are standing in a 3D coordinate system, visualizing an invisible wave of energy passing through space. We are given a plane electromagnetic wave, and our mission is to deduce the exact mathematical expression for its magnetic field.

Let's break down the clues provided in the problem. First, we are told the direction of propagation of the wave. It travels along the vector:

This tells us the wave is moving diagonally across the $xy$-plane. Next, we are given its polarization. In physics, the polarization of an electromagnetic wave is conventionally defined by the direction of its electric field vector. So, the electric field points straight up along the $z$-axis:

E, B, and c

Now, how do we find the magnetic field? This is where the beautiful symmetry of electromagnetism comes into play. The electric field ($\mathbf{E}$), the magnetic field ($\mathbf{B}$), and the direction of wave propagation ($\mathbf{c}$) are always mutually perpendicular in a vacuum or isotropic medium.

They form a right-handed orthogonal system. Mathematically, this relationship is expressed by the cross product:

Because $\hat{E}$, $\hat{B}$, and $\hat{c}$ form an orthogonal right-handed triad, we can cyclically permute them to isolate the magnetic field direction. This means the direction of the magnetic field, $\hat{B}$, is simply the cross product of the propagation direction and the electric field direction:

Let's substitute the vectors we know into this relation. We plug in our $\hat{c}$ and $\hat{E}$ vectors. Don't rush through this cross product; let's expand it carefully to avoid any silly mistakes.

Distributing the cross product, we get:

Recall your standard unit vector cross products. The cross product $\hat{i}\times \hat{k}$ gives us $-\hat{j}$, and $\hat{j}\times \hat{k}$ gives us positive $\hat{i}$. Watch out for the minus sign here! Substituting these back, we find:

This perfectly aligns with our visual representation. The magnetic field oscillates in the $xy$-plane, perpendicular to the direction of propagation.

We have the direction, but what about the wave's phase? The phase of a traveling wave dictates how it oscillates in space and time. For a wave traveling in the positive direction of a unit vector $\hat{c}$, the spatial part of the phase is given by $-\mathbf{k}\cdot \mathbf{r}$, where $\mathbf{k}$ is the wave vector.

The wave vector $\mathbf{k}$ is simply the wave number $k$ multiplied by the direction of propagation $\hat{c}$:

So, the complete phase term becomes $\omega t-k\left(\frac{\hat{i}+\hat{j}}{\sqrt{2}}\right)$.

Combining the amplitude $B_0$, the direction we just found, and the correct phase, we get our final expression for the magnetic field:

This matches option (c) perfectly. By trusting the fundamental geometry of electromagnetic waves, we've elegantly arrived at the solution!