A plane electromagnetic wave propagating in the x-direction has wavelength of `6.0 mm`. The electric field is in the y-direction and its maximum magnitude of `33 Vm^(-1)`. The equation for the electric field as function of x and l is

Options

A. `11 sin pi(t-x//c)`
B. `33 sin pi xx 10^(11)(t-x//c)`
C. `33 sin pi(t-x//c)`
D. `11 sin pi xx 10^(11) (t-x//c)`

Explanation

To find the equation for the electric field of a plane electromagnetic wave propagating in the x-direction, we can follow these steps:

### Step 1: Identify the General Form of the Electric Field Equation
The general equation for the electric field \( E \) of a plane electromagnetic wave is given by:
\[ E(x, t) = E_0 \sin(\omega t - kx) \]
where:
- \( E_0 \) is the maximum electric field (amplitude),
- \( \omega \) is the angular frequency,
- \( k \) is the wave number.

### Step 2: Determine the Wave Number \( k \)
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Given that the wavelength \( \lambda = 6.0 \, \text{mm} = 6.0 \times 10^{-3} \, \text{m} \), we can calculate \( k \):
\[ k = \frac{2\pi}{6.0 \times 10^{-3}} \approx \frac{2\pi}{0.006} \approx 1047.2 \, \text{m}^{-1} \]

### Step 3: Determine the Angular Frequency \( \omega \)
The angular frequency \( \omega \) is related to the speed of light \( c \) and the wavelength \( \lambda \) by the formula:
\[ \omega = \frac{2\pi c}{\lambda} \]
Using \( c = 3 \times 10^8 \, \text{m/s} \):
\[ \omega = \frac{2\pi (3 \times 10^8)}{6.0 \times 10^{-3}} = \frac{6\pi \times 10^8}{6.0 \times 10^{-3}} = \pi \times 10^{11} \, \text{s}^{-1} \]

### Step 4: Substitute Values into the Electric Field Equation
Now we can substitute \( E_0 = 33 \, \text{V/m} \), \( \omega = \pi \times 10^{11} \, \text{s}^{-1} \), and \( k \approx 1047.2 \, \text{m}^{-1} \) into the general equation:
\[ E(x, t) = 33 \sin\left(\pi \times 10^{11} t - 1047.2 x\right) \]

### Final Equation
Thus, the equation for the electric field as a function of \( x \) and \( t \) is:
\[ E(x, t) = 33 \sin\left(\pi \times 10^{11} t - 1047.2 x\right) \]