The wave function (in `S.I unit`) for an electromagnetic wave is given as- `psi(x,t)=10^(3)sinpi(3xx10^(6)x-9xx10^(14)t)`
The speed of the wave is

Options

A. `3 xx 10^(8) m//s`
B. `2 xx 10^(8) m//s`
C. `10^(8)m`
D. `6 xx 10^(8) m//s`

Explanation

To find the speed of the electromagnetic wave given the wave function \( \psi(x,t) = 10^3 \sin\left(\pi (3 \times 10^6 x - 9 \times 10^{14} t)\right) \), we will follow these steps:

### Step 1: Identify the wave function form
The wave function is given in the form:
\[ \psi(x,t) = A \sin(kx - \omega t) \]
where:
- \( A \) is the amplitude,
- \( k \) is the wave number,
- \( \omega \) is the angular frequency.

### Step 2: Extract \( k \) and \( \omega \)
From the given wave function:
\[ k = \pi \times 3 \times 10^6 \quad \text{and} \quad \omega = \pi \times 9 \times 10^{14} \]

### Step 3: Calculate the wavelength \( \lambda \)
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Rearranging gives:
\[ \lambda = \frac{2\pi}{k} \]
Substituting the value of \( k \):
\[ \lambda = \frac{2\pi}{\pi \times 3 \times 10^6} = \frac{2}{3 \times 10^6} \text{ meters} \]

### Step 4: Calculate the frequency \( \nu \)
The angular frequency \( \omega \) is related to the frequency \( \nu \) by the formula:
\[ \omega = 2\pi\nu \]
Rearranging gives:
\[ \nu = \frac{\omega}{2\pi} \]
Substituting the value of \( \omega \):
\[ \nu = \frac{\pi \times 9 \times 10^{14}}{2\pi} = \frac{9 \times 10^{14}}{2} \text{ Hertz} \]

### Step 5: Calculate the speed \( v \)
The speed \( v \) of the wave is given by the formula:
\[ v = \nu \lambda \]
Substituting the values of \( \nu \) and \( \lambda \):
\[ v = \left(\frac{9 \times 10^{14}}{2}\right) \left(\frac{2}{3 \times 10^6}\right) \]
Simplifying this:
\[ v = \frac{9 \times 10^{14} \times 2}{2 \times 3 \times 10^6} = \frac{9 \times 10^{14}}{3 \times 10^6} = 3 \times 10^8 \text{ meters per second} \]

### Final Answer
The speed of the wave is:
\[ \boxed{3 \times 10^8 \text{ m/s}} \]