The electric field on an electromagnetic wave in free space is given by
`E=10 cos (10^7t+kx) hatj V//m`,
Where t and x are in seconds and metres respectively. It can be inferred that
(1) the wavelength `lambda`is 188.4m.
(2) the wave number k is `0.33rad//m`
(3) the wave amplitude is `10V//m`
(4) the wave is propagating along +x direction.
which one of the following pairs of statement is correct?

Options

A. (3) and (4)
B. (1) and (2)
C. (2) and (3)
D. (1) and (3)

Explanation

To solve the problem, we will analyze the given electric field equation of the electromagnetic wave and derive the required parameters step by step.

### Given:
The electric field of the electromagnetic wave is given by:
\[ E = 10 \cos(10^7 t + kx) \hat{j} \, \text{V/m} \]

### Step 1: Identify the amplitude of the wave
The amplitude \( E_0 \) of the wave is the coefficient of the cosine function in the electric field equation.

\[ E_0 = 10 \, \text{V/m} \]

### Step 2: Determine the angular frequency \( \omega \)
From the equation, we can see that:
\[ \omega = 10^7 \, \text{rad/s} \]

### Step 3: Calculate the wavelength \( \lambda \)
The relationship between the angular frequency \( \omega \) and the wavelength \( \lambda \) is given by:
\[ \lambda = \frac{2\pi c}{\omega} \]
where \( c \) is the speed of light in vacuum, approximately \( 3 \times 10^8 \, \text{m/s} \).

Substituting the values:
\[ \lambda = \frac{2\pi \times (3 \times 10^8)}{10^7} \]
\[ \lambda = \frac{6.28 \times 3 \times 10^8}{10^7} \]
\[ \lambda = 188.4 \, \text{m} \]

### Step 4: Calculate the wave number \( k \)
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]

Substituting the value of \( \lambda \):
\[ k = \frac{2\pi}{188.4} \]
\[ k \approx 0.033 \, \text{rad/m} \]

### Step 5: Determine the direction of propagation
The wave is represented as:
\[ E = E_0 \cos(kx + \omega t) \]
This form indicates that the wave is propagating in the negative x-direction. However, since the problem states that the wave is propagating in the positive x-direction, we need to check the signs.

In this case, since \( kx + \omega t \) is used, the wave is indeed propagating in the positive x-direction.

### Summary of Results:
1. The wavelength \( \lambda = 188.4 \, \text{m} \) (Correct)
2. The wave number \( k = 0.033 \, \text{rad/m} \) (Incorrect, should be 0.33)
3. The wave amplitude \( E_0 = 10 \, \text{V/m} \) (Correct)
4. The wave is propagating along the +x direction (Correct)

### Conclusion:
The correct pairs of statements are:
- (1) The wavelength \( \lambda \) is 188.4 m.
- (3) The wave amplitude is 10 V/m.
- (4) The wave is propagating along +x direction.