A plane e.m. wave propagating in the x-direction has a wavelength 6.0 mm. The electric field is in the y-direction and its maximum magitude is `33Vm^_1`. Write suitable equation for the electric and magnetic fields as a function of x and t.

Explanation

To solve the problem of finding the equations for the electric and magnetic fields of a plane electromagnetic wave propagating in the x-direction, we will follow these steps:

### Step 1: Identify Given Information
- Wavelength (λ) = 6.0 mm = 6.0 × 10^-3 m
- Maximum electric field (E₀) = 33 V/m
- The wave propagates in the x-direction.
- The electric field is in the y-direction.

### Step 2: Calculate the Angular Frequency (ω) and Wave Number (k)
1. **Calculate the speed of light (c)**:
\[ c = 3 \times 10^8 \text{ m/s} \]

2. **Calculate the wave number (k)**:
\[ k = \frac{2\pi}{\lambda} = \frac{2\pi}{6.0 \times 10^{-3}} \approx 1047.2 \text{ rad/m} \]

3. **Calculate the angular frequency (ω)** using the relationship \( c = \lambda f \) (where f is the frequency):
\[ f = \frac{c}{\lambda} = \frac{3 \times 10^8}{6.0 \times 10^{-3}} \approx 5 \times 10^{13} \text{ Hz} \]
\[ \omega = 2\pi f \approx 2\pi \times 5 \times 10^{13} \approx 3.14 \times 10^{14} \text{ rad/s} \]

### Step 3: Write the Equation for the Electric Field (E)
The electric field (E) of a plane wave propagating in the x-direction can be expressed as:
\[ E_y = E_0 \sin(\omega t - kx) \]
Substituting the values we have:
\[ E_y = 33 \sin(3.14 \times 10^{14} t - 1047.2 x) \]

### Step 4: Calculate the Maximum Magnetic Field (B₀)
Using the relationship between the electric field and the magnetic field in an electromagnetic wave:
\[ B_0 = \frac{E_0}{c} = \frac{33}{3 \times 10^8} \approx 1.1 \times 10^{-7} \text{ T} \]

### Step 5: Write the Equation for the Magnetic Field (B)
The magnetic field (B) is perpendicular to both the electric field and the direction of wave propagation. Therefore, it will be in the z-direction:
\[ B_z = B_0 \sin(\omega t - kx) \]
Substituting the value of B₀:
\[ B_z = (1.1 \times 10^{-7}) \sin(3.14 \times 10^{14} t - 1047.2 x) \]

### Final Equations
1. **Electric Field**:
\[ E_y = 33 \sin(3.14 \times 10^{14} t - 1047.2 x) \text{ V/m} \]

2. **Magnetic Field**:
\[ B_z = (1.1 \times 10^{-7}) \sin(3.14 \times 10^{14} t - 1047.2 x) \text{ T} \]