Step 1: Understanding the Concept:
Electromagnetic waves consist of time-varying electric and magnetic fields that are coupled together and travel through space.
One of the most defining characteristics of these waves is their transverse nature.
This means that the oscillating electric field vector \(\vec{E}\) and the oscillating magnetic field vector \(\vec{B}\) are not only perpendicular to each other but also mutually perpendicular to the direction in which the wave is moving.
In physics, the directional flow of energy in an electromagnetic field is described by the Poynting vector, which determines the propagation direction.
Step 2: Key Formula or Approach:
The Poynting vector, denoted as \(\vec{S}\), is mathematically defined by the cross product of the electric and magnetic field vectors:
\[ \vec{S} = \frac{1}{\mu_0} (\vec{E} \times \vec{B}) \]
Since \(\mu_0\) (the permeability of free space) is a positive scalar constant, the direction of the Poynting vector \(\vec{S}\) is identical to the direction of the vector cross product \(\vec{E} \times \vec{B}\).
The wave propagates in the direction of \(\vec{S}\).
Step 3: Detailed Explanation:
Let's analyze the properties of the vector cross product to verify why option (B) is correct.
According to the problem, the wave is moving along the positive x-axis, represented by the unit vector \(\hat{i}\).
In a standard electromagnetic wave, if we assume the electric field \(\vec{E}\) oscillates along the y-axis (\(\hat{j}\)) and the magnetic field \(\vec{B}\) oscillates along the z-axis (\(\hat{k}\)), we apply the right-hand rule for cross products.
Using the cyclic property of unit vectors:
\[ \hat{j} \times \hat{k} = \hat{i} \]
This matches the propagation direction given in the problem.
Now, let's examine why the other options are incorrect:
- Option (A): \(\vec{B} \times \vec{E}\) would result in a vector pointing in the opposite direction (\(-\hat{i}\)) because the cross product is anti-commutative (\(\vec{A} \times \vec{B} = - \vec{B} \times \vec{A}\)).
- Option (C): \(\vec{E} \cdot \vec{B}\) is a scalar dot product. Since \(\vec{E}\) and \(\vec{B}\) are perpendicular, their dot product is zero and provides no directional information.
- Option (D): \(\vec{E} + \vec{B}\) is a vector sum that lies in the plane of the fields, which is perpendicular to the actual direction of propagation.
Step 4: Final Answer:
The direction of propagation of a plane electromagnetic wave is uniquely and correctly defined by the vector expression \(\vec{E} \times \vec{B}\).