An electromagnetic wave travel along `z`-axis. Which of the following pair of space and time varying fields would generate such a wave?

Options

A. `E_(x^(1))B_(y)`
B. `E_(y^(1))B_(x)`
C. `E_(x^(1))B_(z)`
D. `E_(x^(1))B_(x)`

Explanation

To determine which pair of space and time varying fields would generate an electromagnetic wave traveling along the z-axis, we can follow these steps:

### Step 1: Understand the Direction of Propagation
Electromagnetic waves consist of electric (E) and magnetic (B) fields that are mutually perpendicular to each other and to the direction of wave propagation.

### Step 2: Identify the Propagation Direction
Since the wave is traveling along the z-axis, we can denote the direction of propagation as the unit vector \( \hat{k} \) in the z-direction.

### Step 3: Determine the Orientation of the Electric and Magnetic Fields
For an electromagnetic wave traveling in the z-direction:
- The electric field (E) can be oriented along the x-axis.
- The magnetic field (B) can be oriented along the y-axis.

### Step 4: Use the Right-Hand Rule
According to the right-hand rule, if you point your thumb in the direction of the wave propagation (z-axis), your index finger in the direction of the electric field (x-axis), your middle finger will point in the direction of the magnetic field (y-axis). This confirms that:
- \( \vec{E} \) is along the x-axis.
- \( \vec{B} \) is along the y-axis.

### Step 5: Conclusion
Thus, the pair of space and time varying fields that would generate an electromagnetic wave traveling along the z-axis are:
- Electric field \( \vec{E} \) in the x-direction.
- Magnetic field \( \vec{B} \) in the y-direction.

### Final Answer
The correct answer is: Electric field \( \vec{E} \) along the x-axis and magnetic field \( \vec{B} \) along the y-axis.

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