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 parameters
The given magnetic field \( B \) is expressed as:
\[ B = B_0 \sin(\omega t + kx + \phi) \hat{j} \]
where \( B_0 = 3 \times 10^{-8} \) T (Tesla).

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

### Step 3: Calculate the magnitude of the electric field
Rearranging the equation gives us:
\[ E_0 = B_0 \cdot c \]
Substituting the values we have:
\[ E_0 = (3 \times 10^{-8}) \cdot (3 \times 10^8) \]
Calculating this gives:
\[ E_0 = 9 \, \text{N/C} \]

### Step 4: Determine the direction of the electric field
The magnetic field is in the \( \hat{j} \) direction (y-direction). In electromagnetic waves, the electric field \( E \), magnetic field \( B \), and the direction of propagation are mutually perpendicular. If the magnetic field is in the \( \hat{j} \) direction, the electric field will be in the \( \hat{k} \) direction (z-direction).

### Step 5: Write the electric field equation
Thus, the electric field can be expressed as:
\[ E = E_0 \sin(\omega t + kx + \phi) \hat{k} \]
Substituting \( E_0 = 9 \) N/C, we get:
\[ E = 9 \sin(\omega t + kx + \phi) \hat{k} \]

### Final Answer
The appropriate equation for the electric field \( E \) is:
\[ E = 9 \sin(\omega t + kx + \phi) \hat{k} \]

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