The electric field in an elctromagnetic wave is given by $\stackrel{\to <!--\; \to \; -->}{\mathrm{E}}=\stackrel{^<!--\; ^\; -->}{\mathrm{i}}40\cos <!--\; \; -->\mathrm{\omega <!--\; \omega \; -->}(\mathrm{t}-<!--\; -\; -->\frac{\mathrm{z}}{\mathrm{c}}){\mathrm{N}\mathrm{C}}^{-<!--\; -\; -->1}$. The magnetic field induction of this wave is (in SI unit):

The correct answer is Option 4: $\overrightarrow{B}=\hat{j}\frac{40}{c}\cos \omega (t-\frac{z}{c})$

We are given the electric field of an electromagnetic wave:

$\overrightarrow{E}=\hat{i}40\cos \omega (t-\frac{z}{c}){\mathrm{NC}}^{-1}$

We need to find the corresponding magnetic field $\overrightarrow{B}$. Let us work through this step-by-step.

Step 1: Identify the direction of wave propagation.

The wave phase is $\omega (t-\frac{z}{c})$. The term $\frac{z}{c}$ tells us the wave is travelling in the +z direction, i.e., the propagation direction is $\hat{k}$.

Step 2: Identify the direction of the Electric Field.

From the given expression, the electric field oscillates along the +x direction (represented by $\hat{i}$).

Step 3: Determine the direction of the Magnetic Field using the right-hand rule.

In an electromagnetic wave, the electric field $\overrightarrow{E}$, the magnetic field $\overrightarrow{B}$, and the direction of propagation $\hat{k}$ are mutually perpendicular and obey the right-hand rule:

$\hat{k}=\frac{\overrightarrow{E}\times \overrightarrow{B}}{|\overrightarrow{E}\times \overrightarrow{B}|}$

So we need the direction such that $\hat{i}\times \hat{B}=\hat{k}$.

Using the standard cross product identity: $\hat{i}\times \hat{j}=\hat{k}$

Therefore, $\stackrel{��}{B}$ must point in the +y direction (represented by $\hat{j}$).

Step 4: Determine the magnitude of the Magnetic Field.

For an electromagnetic wave travelling in free space, the ratio of the magnitude of the electric field to the magnitude of the magnetic field equals the speed of light $c$:

$\frac{E_0}{B_0}=c$

Here, the amplitude of the electric field is $E_0=40$ N/C.

Therefore, the amplitude of the magnetic field is:

$B_0=\frac{E_0}{c}=\frac{40}{c}\text{T}$

Step 5: Write the complete expression for the Magnetic Field.

Since the magnetic field oscillates with the same phase as the electric field (they are in phase in an EM wave), and it points in the $\hat{j}$ direction with amplitude $\frac{40}{c}$:

$\overrightarrow{B}=\hat{j}\frac{40}{c}\cos \omega (t-\frac{z}{c})\text{T}$

This confirms the correct answer. The three key principles used were: (1) the wave propagates in the +z direction, (2) $\overrightarrow{E}$, $\overrightarrow{B}$, and $\hat{k}$ form a right-handed orthogonal triad, and (3) the ratio $\frac{E_0}{B_0}=c$.