The electric field associated with an EM wave in vacuum is given by `vec(E)=40 cos(k z-6 xx 10^(8) t) hat(i)`, respectively. The value of wave vector `k` (in `m^(-1)` ) is

Answer

To find the value of the wave vector \( k \) from the given electric field equation of the electromagnetic wave, we can follow these steps:

### Step 1: Identify the given electric field equation
The electric field is given by:
\[ \vec{E} = 40 \cos(kz - 6 \times 10^8 t) \hat{i} \]
From this equation, we can identify the angular frequency \( \omega \) and the wave vector \( k \).

### Step 2: Identify the angular frequency \( \omega \)
In the equation, the term \( 6 \times 10^8 t \) corresponds to the angular frequency \( \omega \). Thus, we have:
\[ \omega = 6 \times 10^8 \, \text{rad/s} \]

### Step 3: Use the relation between wave vector \( k \) and angular frequency \( \omega \)
The relationship between the wave vector \( k \), angular frequency \( \omega \), and the speed of light \( c \) in vacuum is given by:
\[ \omega = k \cdot c \]
where \( c \) (the speed of light in vacuum) is approximately \( 3 \times 10^8 \, \text{m/s} \).

### Step 4: Rearrange the equation to find \( k \)
We can rearrange the equation to solve for \( k \):
\[ k = \frac{\omega}{c} \]

### Step 5: Substitute the values of \( \omega \) and \( c \)
Now, substituting the values we have:
\[ k = \frac{6 \times 10^8 \, \text{rad/s}}{3 \times 10^8 \, \text{m/s}} \]

### Step 6: Calculate the value of \( k \)
Calculating the above expression:
\[ k = \frac{6}{3} = 2 \, \text{m}^{-1} \]

### Final Answer
Thus, the value of the wave vector \( k \) is:
\[ \boxed{2 \, \text{m}^{-1}} \]

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