The magnetic field in a plane electromagnetic wave is $$B_y = (3.5 \times 10^{-7}) \sin(1.5 \times 10^3 x + 0.5 \times 10^{11} t) \text{ T}$$. The corresponding electric field will be :

Problem Analysis

Given magnetic field equation:

$$B_y = (3.5 \times 10^{-7}) \sin(1.5 \times 10^3 x + 0.5 \times 10^{11} t) \, \text{T}$$

From the equation, the amplitude of the magnetic field is:

$$B_0 = 3.5 \times 10^{-7} \, \text{T}$$

Step-by-Step Solution

Step 1: Find the Amplitude of the Electric Field ($$E_0$$)

Using the relation between the amplitudes of electric and magnetic fields ($E_0 = B_0 \cdot c$):

$$E_0 = (3.5 \times 10^{-7}) \times (3 \times 10^8)$$$$E_0 = 10.5 \times 10^1 = 105 \, \text{V/m}$$

Step 2: Determine the Direction of the Electric Field

1. The wave is propagating along the negative x-axis (since both x and t coefficients have the same sign in $$\sin(kx + \omega t)$$). Therefore, the direction of propagation vector $$\hat{k} = -\hat{i}$$.
2. The magnetic field is along the y-axis ($$\hat{B} = \hat{j}$$).
3. In an electromagnetic wave, the direction of propagation is given by the Poynting vector cross product: $$\hat{E} \times \hat{B} \parallel \text{Direction of propagation}$$

$$\hat{E} \times \hat{j} = -\hat{i}$$

Using the unit vector cross-multiplication rule ($$\hat{k} \times \hat{j} = -\hat{i}$$), the electric field must be along the z-axis ($$\hat{E} = \hat{k}$$). Thus, the field is $$E_z$$.

Final Equation

Matching the phase component directly from the given $$B_y$$:

$$E_z = 105 \sin(1.5 \times 10^3 x + 0.5 \times 10^{11} t) \, \text{V/m}$$

Correct Option: (4)