Question:
An electromagnetic wave has electric and magnetic fields given by \(\vec{E}(t) = \vec{E}_m \sin(kx - \omega t)\), \(\vec{B}(t) = \vec{B}_m \sin(kx - \omega t)\). If the direction of \(\vec{E}_m\) and \(\vec{B}_m\) are \((i + 2j)\) and \((-i + \frac{1}{2} j)\) respectively, then the direction of propagation of the wave is:
An electromagnetic wave has electric and magnetic fields given by \(\vec{E}(t) = \vec{E}_m \sin(kx - \omega t)\), \(\vec{B}(t) = \vec{B}_m \sin(kx - \omega t)\). If the direction of \(\vec{E}_m\) and \(\vec{B}_m\) are \((i + 2j)\) and \((-i + \frac{1}{2} j)\) respectively, then the direction of propagation of the wave is:
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The propagation direction of an EM wave is always along \(\vec{E} \times \vec{B}\).
Updated On: Jul 18, 2026
- \(\mathbf{k}\)
- \(-\mathbf{k}\)
- \(i + j\)
- \(i - j\)
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The Correct Option is A
Solution and Explanation
Step 1: Recall propagation direction formula.
For an electromagnetic wave, the direction of propagation is given by: \[ \vec{k} \propto \vec{E} \times \vec{B} \] where \(\vec{k}\) is the wave vector, \(\vec{E}\) and \(\vec{B}\) are the electric and magnetic field vectors.
Step 2: Write given vectors.
\[ \vec{E}_m = i + 2j, \quad \vec{B}_m = -i + \frac{1}{2} j \]
Step 3: Compute cross product.
\[ \vec{E} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 0 \\ -1 & 1/2 & 0 \end{vmatrix} \]
Step 4: Evaluate determinant.
\(\hat{i}\) component: \[ (2 \cdot 0 - 0 \cdot 1/2) = 0 \] \(\hat{j}\) component: \[ -(1 \cdot 0 - 0 \cdot (-1)) = 0 \] \(\hat{k}\) component: \[ 1 \cdot 1/2 - 2 \cdot (-1) = 0.5 + 2 = 2.5 \]
Step 5: Determine direction.
\[ \vec{k} \propto \vec{E} \times \vec{B} = 2.5 \hat{k} \] Hence, wave propagates along positive \(\hat{k}\) direction.
Step 6: Final conclusion.
\[ \boxed{\mathbf{k}} \]
For an electromagnetic wave, the direction of propagation is given by: \[ \vec{k} \propto \vec{E} \times \vec{B} \] where \(\vec{k}\) is the wave vector, \(\vec{E}\) and \(\vec{B}\) are the electric and magnetic field vectors.
Step 2: Write given vectors.
\[ \vec{E}_m = i + 2j, \quad \vec{B}_m = -i + \frac{1}{2} j \]
Step 3: Compute cross product.
\[ \vec{E} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 0 \\ -1 & 1/2 & 0 \end{vmatrix} \]
Step 4: Evaluate determinant.
\(\hat{i}\) component: \[ (2 \cdot 0 - 0 \cdot 1/2) = 0 \] \(\hat{j}\) component: \[ -(1 \cdot 0 - 0 \cdot (-1)) = 0 \] \(\hat{k}\) component: \[ 1 \cdot 1/2 - 2 \cdot (-1) = 0.5 + 2 = 2.5 \]
Step 5: Determine direction.
\[ \vec{k} \propto \vec{E} \times \vec{B} = 2.5 \hat{k} \] Hence, wave propagates along positive \(\hat{k}\) direction.
Step 6: Final conclusion.
\[ \boxed{\mathbf{k}} \]
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