Question

For plane electromagnetic waves propagating in the $z$ direction, which one of the following combination gives the correct possible direction for $\vec{E}$ and $\vec{B}$ field respectively ?

• A. $\left(\hat{i}+2\hat{j}\right)$ and $\left(2\hat{i}-\hat{j}\right)$
• B. $\left(\hat{-2i}-3\hat{j}\right)$ and $\left(3\hat{i}-\hat{2j}\right)$
• C. $\left(\hat{2i}+3\hat{j}\right)$ and $\left(\hat{i}-\hat{2j}\right)$
• D. $\left(\hat{3i}+4\hat{j}\right)$ and $\left(4\hat{i}-\hat{3j}\right)$

Answer 1

The Correct Option is B

 To determine the correct possible direction for the electric field \(\vec{E}\) and magnetic field \(\vec{B}\) of a plane electromagnetic wave propagating in the \(z\) direction, we use the property that in electromagnetic waves, the electric field, magnetic field, and the direction of propagation are mutually perpendicular. This is represented by the right-hand rule.

The wave is propagating in the \(z\) direction, so \(\vec{k} = \hat{k}$ or in terms of unit vectors\)

According to the right-hand rule:

- Arrange your right hand such that your thumb points in the direction of wave propagation (here, \(z\)-direction).

- Let your fingers point in the direction of the electric field, then the curl of your fingers shows the direction of the magnetic field.

Given the choices:

• \((\hat{i}+2\hat{j})\) for \(\vec{E}\) and \((2\hat{i}-\hat{j})\) for \(\vec{B}\)
• \((-2\hat{i}-3\hat{j})\) for \(\vec{E}\) and \((3\hat{i}-2\hat{j})\) for \(\vec{B}\)
• \((2\hat{i}+3\hat{j})\) for \(\vec{E}\) and \((\hat{i}-2\hat{j})\) for \(\vec{B}\)
• \((3\hat{i}+4\hat{j})\) for \(\vec{E}\) and \((4\hat{i}-3\hat{j})\) for \(\vec{B}\)

Check the dot products:

• \((\hat{i}+2\hat{j}) \cdot \hat{k} = 0\) and \((2\hat{i}-\hat{j}) \cdot \hat{k} = 0\)
• \((-2\hat{i}-3\hat{j}) \cdot \hat{k} = 0\) and \((3\hat{i}-2\hat{j}) \cdot \hat{k} = 0\)
• \((2\hat{i}+3\hat{j}) \cdot \hat{k} = 0\) and \((\hat{i}-2\hat{j}) \cdot \hat{k} = 0\)
• \((3\hat{i}+4\hat{j}) \cdot \hat{k} = 0\) and \((4\hat{i}-3\hat{j}) \cdot \hat{k} = 0\)

The cross-products must satisfy: \(\vec{E} \times \vec{B} \parallel \hat{k}\). Evaluating the viable combinations under this rule, only the second option satisfies this since:

• If \(\vec{E} = -2\hat{i} - 3\hat{j}\) and \(\vec{B} = 3\hat{i} - 2\hat{j}\), then \(\vec{E} \times \vec{B} = ((-2)(-2) - (-3)(3))\hat{k}$ gives a non-zero result only in the $\hat{k}$ direction\)

Thus, the correct answer is \((-2\hat{i}-3\hat{j})\) for \(\vec{E}\) and \((3\hat{i}-2\hat{j})\) for \(\vec{B}\).