A plane electromagnetic wave of frequency $20 ~\mathrm{MHz}$ propagates in free space along $\mathrm{x}$-direction. At a particular space and time, $\overrightarrow{\mathrm{E}}=6.6 \hat{j} \mathrm{~V} / \mathrm{m}$. What is $\overrightarrow{\mathrm{B}}$ at this point?

In free space, the relationship between the electric field (E) and the magnetic field (B) in an electromagnetic wave is given by: <br/><br/> $B = \frac{E}{c}$ <br/><br/> where c is the speed of light in a vacuum, approximately equal to $3 \times 10^8 \mathrm{~m} / \mathrm{s}$. We are given that the electric field is $\overrightarrow{\mathrm{E}}=6.6 \hat{j} \mathrm{~V} / \mathrm{m}$. To find the magnetic field, we can first calculate the magnitude of B: <br/><br/> $$ B = \frac{6.6 \mathrm{~V} / \mathrm{m}}{3 \times 10^8 \mathrm{~m} / \mathrm{s}} = 2.2 \times 10^{-8} \mathrm{~T} $$ <br/><br/> Now, we need to find the direction of the magnetic field. Since the electromagnetic wave propagates in the x-direction, and the electric field is in the y-direction (j), the magnetic field should be in the z-direction (k) to satisfy the right-hand rule for electromagnetic waves. The direction of the magnetic field will be positive k (counter-clockwise rotation from x to y). <br/><br/> So, $\overrightarrow{\mathrm{B}} = 2.2 \times 10^{-8} \hat{k} \mathrm{T}$.