Magnetic field in a plane electromagnetic wave is given by
`bar(B) = B_(0)"sin"(kx + omegat)hat(j)T`
Expression for corresponding electric field will be

Options

A. `vec(E) oo -B_(0)c"sin" (kx +omegat)hat(k)V//m`
B. `vec(E) = B_(0)c"sin"(kx - omegat)hat(k)V//m`
C. `vec(E) = B_(0)c"sin"(kx + omegat)hat(k) V//m`
D. `vec(E) = (B_(0))/(c)"sin" (kx + omegat) hat(k)V//m`

Answer

To find the expression for the corresponding electric field \( \mathbf{E} \) given the magnetic field \( \mathbf{B} \) in a plane electromagnetic wave, we can follow these steps:

### Step 1: Write down the given magnetic field
The magnetic field is given as:
\[ \mathbf{B} = B_0 \sin(kx + \omega t) \hat{j} \]

### Step 2: Recall the relationship between electric field and magnetic field in electromagnetic waves
In electromagnetic waves, the electric field \( \mathbf{E} \) and magnetic field \( \mathbf{B} \) are related by the equation:
\[ \mathbf{E} = c \mathbf{B} \]
where \( c \) is the speed of light in vacuum.

### Step 3: Determine the direction of the electric field
The electric field \( \mathbf{E} \) is always perpendicular to both the magnetic field \( \mathbf{B} \) and the direction of wave propagation. Given that \( \mathbf{B} \) is in the \( \hat{j} \) direction (y-direction), and the wave is propagating in the x-direction (as indicated by the \( kx \) term), the electric field must be in the z-direction (i.e., \( \hat{k} \)).

### Step 4: Calculate the magnitude of the electric field
The magnitude of the electric field can be calculated using the relationship:
\[ E = cB \]
Substituting \( B = B_0 \sin(kx + \omega t) \):
\[ E = c \cdot B_0 \sin(kx + \omega t) \]

### Step 5: Write the expression for the electric field vector
Since the electric field is in the \( \hat{k} \) direction, we can write:
\[ \mathbf{E} = B_0 c \sin(kx + \omega t) \hat{k} \]

### Final Expression
Thus, the expression for the corresponding electric field is:
\[ \mathbf{E} = B_0 c \sin(kx + \omega t) \hat{k} \quad \text{(in volts per meter)} \]