The magnetic field of a plane electromagnetic wave is
`vecB = 3 xx 10^(-8) sin[200 pi (y + ct)] hati T`
Where `c = 3 xx 10^(8) ms^(-1)` is the speed of light. The corresponding electric field is:

Options

A. `vecE = -9 sin[200 pi(y - ct)] hatk V//m`
B. `vecE = 9 sin[200 pi(y - ct)] hatk V//m`
C. `vecE = -10^(-6) sin[200 pi(y - ct)] hatk V//m`
D. `vecE = 3 xx 10^(-8) sin[200 pi(y - ct)] hatk V//m`

Explanation

To find the corresponding electric field of the given magnetic field of a plane electromagnetic wave, we can follow these steps:

### Step 1: Identify the given magnetic field
The magnetic field is given as:
\[ \vec{B} = 3 \times 10^{-8} \sin[200 \pi (y + ct)] \hat{i} \, \text{T} \]
where \( c = 3 \times 10^8 \, \text{m/s} \).

### Step 2: Determine the magnitude of the electric field
The relationship between the electric field \( \vec{E} \) and the magnetic field \( \vec{B} \) in an electromagnetic wave is given by:
\[ E_0 = c B_0 \]
where \( E_0 \) is the amplitude of the electric field and \( B_0 \) is the amplitude of the magnetic field.

From the magnetic field, we can identify:
\[ B_0 = 3 \times 10^{-8} \, \text{T} \]

Now, substituting the values:
\[ E_0 = (3 \times 10^8 \, \text{m/s}) \times (3 \times 10^{-8} \, \text{T}) = 9 \, \text{V/m} \]

### Step 3: Determine the direction of the electric field
In an electromagnetic wave, the electric field \( \vec{E} \), magnetic field \( \vec{B} \), and the direction of wave propagation \( \vec{k} \) are mutually perpendicular, following the right-hand rule.

1. The wave is propagating in the direction of \( y + ct \), which implies the wave is moving in the negative \( y \)-direction. Therefore, the wave vector \( \vec{k} \) is in the direction of \( -\hat{j} \).
2. The magnetic field \( \vec{B} \) is in the \( \hat{i} \) direction (x-direction).

Using the right-hand rule:
- Point your fingers in the direction of \( \vec{B} \) (which is \( \hat{i} \)).
- Curl your fingers in the direction of wave propagation (which is \( -\hat{j} \)).
- Your thumb will point in the direction of \( \vec{E} \).

Thus, the direction of \( \vec{E} \) is in the \( -\hat{k} \) direction (z-direction).

### Step 4: Write the electric field equation
The electric field can be expressed as:
\[ \vec{E} = E_0 \sin[200 \pi (y + ct)] \hat{k} \]
Since the direction is \( -\hat{k} \), we can write:
\[ \vec{E} = 9 \sin[200 \pi (y + ct)] (-\hat{k}) \, \text{V/m} \]
or
\[ \vec{E} = -9 \sin[200 \pi (y + ct)] \hat{k} \, \text{V/m} \]

### Final Answer
The corresponding electric field is:
\[ \vec{E} = -9 \sin[200 \pi (y + ct)] \hat{k} \, \text{V/m} \]