Question

Consider an electromagnetic plane wave $\vec{E}=E_0(\hat{t}+b\hat{y})\cos\left[\frac{2\pi}{\lambda}(ct-(x-\sqrt{3}y))\right]$, where $\lambda$ is the wavelength and $c$ is the speed of light. The value of $b$ is ............. (Specify answer up to two digits after the decimal point.)
 

Hint:

For EM waves, $\vec{E}$ is always perpendicular to the propagation direction $\vec{k}$.

Answer 1

Correct Answer: 0.55

Step 1: Understand E-field polarization.
The electric field direction is given by the vector $(1, b)$. The wave propagates in direction $(1, -\sqrt{3})$.

Step 2: Use orthogonality of $\vec{E$ and $\vec{k}$.}
$\vec{E}\cdot \vec{k} = 1\cdot 1 + b(-\sqrt{3}) = 0$.

Step 3: Solve for $b$.
$1 - b\sqrt{3} = 0 $\Rightarrow$ b = \frac{1}{\sqrt{3}} \approx 0.577$.

Step 4: Convert to magnitude of full polarization vector.
Total magnitude = $\sqrt{1^2 + b^{-2}} = \sqrt{3} = 1.73$.