Kinetics And Dynamics

Step 1: Understanding the Concept
The relationship between the electric field ( ) and magnetic field ( ) of a plane electromagnetic wave is governed by Maxwell's equations. Specifically, Faraday's law of induction, , connects the spatial variation of to the temporal variation of .
Step 2: Key Formula or Approach
We will use Faraday's Law in differential form: From this, we can find by integrating with respect to time.
For a plane wave, this relationship can be written as , where is the wave vector.
Step 3: Detailed Explanation
Given the electric field: . This can be written as , where .
First, let's compute the curl of : Now, calculate the partial derivatives: Substitute these back into the curl expression: Now, use Faraday's Law: To find , we integrate with respect to time : Finally, we need to find the relationship between and . The phase of the wave is , which is .
So, the wave vector is .
The magnitude of the wave vector is .
For an electromagnetic wave in vacuum, the dispersion relation is .
Step 4: Final Answer
Substitute the value of back into the expression for : This matches option (A).