A plane electromagnetic wave of frequency 50 MHz travels in free space along the positive $$x$$-direction. At a particular point in space and time, $$\vec{E} = 6.3 \hat{j}$$ V/m. The corresponding magnetic field $$\vec{B}$$, at that point will be:

The electromagnetic wave is stated to be travelling in free space along the positive $$x$$-axis. For every plane wave in vacuum we always have three mutually perpendicular vectors:

$$\vec E \perp \vec B \perp \vec k,$$

where $$\vec k$$ (or $$\vec v$$) denotes the direction of propagation. The right-hand rule applies: if the fingers of the right hand go from $$\vec E$$ to $$\vec B$$, the thumb points along the direction of propagation. Symbolically we write

$$\vec E \times \vec B \; \propto \; \vec k.$$

In the present problem the wave moves along $$+x$$, i.e. $$\vec k = \hat i.$$ At a particular point we are told

$$\vec E = 6.3 \,\hat j \text{ V/m}.$$

Because $$\vec E$$ is along $$+\hat j$$ and the wave goes along $$+\hat i$$, the magnetic field must point along $$+\hat k$$ so that $$\hat j \times \hat k = \hat i.$$ Hence the unit-vector direction of $$\vec B$$ is $$\hat k.$$

Next we evaluate the magnitude of $$\vec B$$. In free space the magnitudes of the electric and magnetic fields of a plane wave are related by the universal relation

$$E = c\,B,$$

where $$c = 3.0 \times 10^{8}\ \text{m/s}$$ is the speed of light. Stating the same formula in the form we need:

$$B = \dfrac{E}{c}.$$

Now we substitute the given numerical value $$E = 6.3\ \text{V/m}:$$

$$B = \dfrac{6.3\ \text{V/m}}{3.0 \times 10^{8}\ \text{m/s}}.$$

Simplifying the fraction step by step, first write the denominator with only one significant figure for clarity, then divide:

$$B = \dfrac{6.3}{3.0} \times 10^{-8}\ \text{T}.$$

The quotient $$\dfrac{6.3}{3.0}$$ is $$2.1$$, so we have

$$B = 2.1 \times 10^{-8}\ \text{T}.$$

We already established that the vector points along $$+\hat k$$, hence

$$\vec B = 2.1 \times 10^{-8}\ \hat k\ \text{T}.$$

Among the given options, this corresponds exactly to Option A.

Hence, the correct answer is Option A.