Given magnetic field equation is`B= 3xx10^(-8) sin (omega t +kx +phi)hatj`then appropriate equation for electric field (E) will be :

Options

A. `20xx10^(-9) sin (omega t +kx +phi ) hatk`
B. `9 sin ( omega t +kx + phi ) hatk`
C. ` 16xx10^(-9) sin (omega t+ kx +phi)hatk`
D. `3xx10^(-9)siin (omega t +kx +phi)hatk`

Answer

To find the appropriate equation for the electric field (E) given the magnetic field equation \( B = 3 \times 10^{-8} \sin(\omega t + kx + \phi) \hat{j} \), we can follow these steps:

### Step 1: Identify the given magnetic field equation
The magnetic field is given as:
\[ B = 3 \times 10^{-8} \sin(\omega t + kx + \phi) \hat{j} \]
Here, the amplitude of the magnetic field \( B_0 = 3 \times 10^{-8} \) T, and it oscillates in the \( \hat{j} \) direction.

### Step 2: Use the relationship between electric field and magnetic field in electromagnetic waves
In electromagnetic waves, the relationship between the electric field \( E \) and the magnetic field \( B \) is given by:
\[ E_0 = c B_0 \]
where \( c \) is the speed of light in vacuum, approximately \( 3 \times 10^8 \) m/s.

### Step 3: Calculate the amplitude of the electric field
Substituting the values we have:
\[ E_0 = (3 \times 10^8 \, \text{m/s}) \times (3 \times 10^{-8} \, \text{T}) = 9 \, \text{V/m} \]

### Step 4: Determine the direction of the electric field
Since the magnetic field is in the \( \hat{j} \) direction, the electric field will be perpendicular to both the magnetic field and the direction of wave propagation. For a wave propagating in the \( \hat{i} \) direction, the electric field will be in the \( \hat{k} \) direction (or \( \hat{j} \) direction if the wave is propagating in the \( \hat{i} \) direction).

Thus, the electric field can be expressed as:
\[ E = 9 \sin(\omega t + kx + \phi) \hat{i} \]

### Step 5: Write the final equation for the electric field
The appropriate equation for the electric field is:
\[ E = 9 \sin(\omega t + kx + \phi) \hat{i} \]