An electromagnetic wave is propagating in vacuum along x-axis, which is produced by oscillating charge of frequency `3 xx 10^(10)` Hz. The amplitude of magnetic field, `1 xx 10^(-7)`T along z-axis find equation for oscillating electric field

Options

A. `vec(E ) =30 sin (6pi xx 10^(10) t - 628 X) hat(j) (N)/(C )`
B. 2) `vec(E ) = sin (6 pi xx 10^(10) t - 628 X ) hat(J) (N)/(C ) `
C. 3) `vec(E ) = 30 sin ( 6 pi xx 10^(10) t - X)hat(j) (N)/(C ) `
D. 4) `vec(E ) = 30 tan (6 pi xx 10^(10) t - X ) hat(j) (N)/(C ) `

Answer

To find the equation for the oscillating electric field of an electromagnetic wave propagating in a vacuum along the x-axis, we will follow these steps:

### Step 1: Identify Given Values
- Frequency (f) = \(3 \times 10^{10} \, \text{Hz}\)
- Amplitude of magnetic field (B₀) = \(1 \times 10^{-7} \, \text{T}\)

### Step 2: Calculate Wavelength (λ)
The wavelength (λ) can be calculated using the formula:
\[ \lambda = \frac{c}{f} \]
where \(c\) is the speed of light in vacuum, approximately \(3 \times 10^8 \, \text{m/s}\).

Substituting the values:
\[ \lambda = \frac{3 \times 10^8}{3 \times 10^{10}} = 10^{-2} \, \text{m} \]

### Step 3: Calculate the Amplitude of Electric Field (E₀)
The relationship between the electric field amplitude (E₀) and the magnetic field amplitude (B₀) in an electromagnetic wave is given by:
\[ E₀ = B₀ \cdot c \]

Substituting the values:
\[ E₀ = (1 \times 10^{-7}) \cdot (3 \times 10^8) = 3 \times 10^1 = 30 \, \text{V/m} \]

### Step 4: Calculate Angular Frequency (ω)
The angular frequency (ω) is given by:
\[ \omega = 2\pi f \]
Substituting the frequency:
\[ \omega = 2\pi \cdot (3 \times 10^{10}) = 6\pi \times 10^{10} \, \text{rad/s} \]

### Step 5: Calculate Wave Number (k)
The wave number (k) is given by:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the wavelength:
\[ k = \frac{2\pi}{10^{-2}} = 200\pi \, \text{rad/m} \]

### Step 6: Write the Equation for the Electric Field
The electric field (E) of an electromagnetic wave propagating in the x-direction can be expressed as:
\[ E(x,t) = E₀ \sin(\omega t - kx) \]

Substituting the values of \(E₀\), \(\omega\), and \(k\):
\[ E(x,t) = 30 \sin(6\pi \times 10^{10} t - 200\pi x) \]

### Final Equation
Thus, the equation for the oscillating electric field is:
\[ E(x,t) = 30 \sin(6\pi \times 10^{10} t - 200\pi x) \, \text{V/m} \]

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