An EM wave propagating in x-direction has a wavelength of 8mm. The electric field vibrating y-direction has maximum magnitude of `60Vm^(-1)`. Choose the correct equations for electric and magnetic fields if the EM wave is propagating in vacuum :

Options

A. `E_(y)=60sin[(pi)/(4)xx10^(3)(x-3xx10^(8)t)]hatjVm^(-1)`
`B_(z)=2sin[(pi)/(4)xx10^(3)(x-3xx10^(8)t)]hatkT`
B. `E_(y)=60sin[(pi)/(4)xx10^(3)(x-3xx10^(8)t)]hatjVm^(-1)`
`B_(z)=2xx10^(-7)sin[(pi)/(4)xx10^(3)(x-3xx10^(8)t)]hatkT`
C. `E_(y)=2xx10^(-7)sin[(pi)/(4)xx10^(3)(x-3xx10^(8)t)hatjVm^(-1)`
`B_(z)=60sin[(pi)/(4)xx10^(3)(x-3xx10^(8)t)]hatkT`
D. `E_(y)=2xx10^(-7)sin[(pi)/(4)xx10^(4)(x-4xx10^(8)t)]hatjVm^(-1)`
`B_(z)=60sin[(pi)/(4)xx10^(4)(x-4xx10^(8)t)]hatkT`

Answer

To solve the problem of finding the correct equations for the electric and magnetic fields of an electromagnetic (EM) wave propagating in the x-direction, we can follow these steps:

### Step 1: Identify the Given Information
- Wavelength (\(\lambda\)) = 8 mm = \(8 \times 10^{-3}\) m
- Maximum magnitude of the electric field (\(E_0\)) = 60 V/m
- The wave propagates in the x-direction, and the electric field oscillates in the y-direction.

### Step 2: Determine the Wave Speed
In vacuum, the speed of an electromagnetic wave is equal to the speed of light (\(c\)):
\[ c = 3 \times 10^8 \text{ m/s} \]

### Step 3: Calculate the Wave Number (\(k\))
The wave number \(k\) is given by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of \(\lambda\):
\[ k = \frac{2\pi}{8 \times 10^{-3}} = \frac{\pi}{4 \times 10^{-3}} \text{ m}^{-1} \]

### Step 4: Calculate the Angular Frequency (\(\omega\))
The relationship between the speed of the wave, wave number, and angular frequency is given by:
\[ c = \frac{\omega}{k} \implies \omega = c \cdot k \]
Substituting the values of \(c\) and \(k\):
\[ \omega = (3 \times 10^8) \cdot \left(\frac{\pi}{4 \times 10^{-3}}\right) = \frac{3\pi \times 10^{11}}{4} \text{ rad/s} \]

### Step 5: Write the Electric Field Equation
The electric field (\(E\)) of an EM wave can be expressed as:
\[ E = E_0 \sin(kx - \omega t) \]
Since the electric field oscillates in the y-direction, we can write:
\[ E = 60 \sin\left(\frac{\pi}{4 \times 10^{-3}} x - \frac{3\pi \times 10^{11}}{4} t\right) \hat{j} \text{ V/m} \]

### Step 6: Calculate the Maximum Magnetic Field (\(B_0\))
The maximum magnetic field \(B_0\) is related to the electric field by:
\[ B_0 = \frac{E_0}{c} \]
Substituting the values:
\[ B_0 = \frac{60}{3 \times 10^8} = 2 \times 10^{-7} \text{ T} \]

### Step 7: Write the Magnetic Field Equation
The magnetic field (\(B\)) can be expressed as:
\[ B = B_0 \sin(kx - \omega t) \]
Since the magnetic field oscillates in the z-direction, we can write:
\[ B = 2 \times 10^{-7} \sin\left(\frac{\pi}{4 \times 10^{-3}} x - \frac{3\pi \times 10^{11}}{4} t\right) \hat{k} \text{ T} \]

### Final Equations
Thus, the equations for the electric and magnetic fields are:
1. Electric Field:
\[ E = 60 \sin\left(\frac{\pi}{4 \times 10^{-3}} x - \frac{3\pi \times 10^{11}}{4} t\right) \hat{j} \text{ V/m} \]
2. Magnetic Field:
\[ B = 2 \times 10^{-7} \sin\left(\frac{\pi}{4 \times 10^{-3}} x - \frac{3\pi \times 10^{11}}{4} t\right) \hat{k} \text{ T} \]