Question

If the electric field of an electromagnetic wave is given by,
𝐸⃗ = (4𝑥̂ + 3𝑦̂)𝑒 ^{𝑖(𝜔𝑡+𝑎𝑥−600𝑦)} 
then the value of a is: (all values are in the SI units)

• A. 450
• B. -450
• C. 800
• D. −800

Answer 1

The Correct Option is A

From the exponential term \( e^{i(\omega t + ax - 600y)} \), the wave vector \( \mathbf{k} \) is: \[ \mathbf{k} = a\hat{x} - 600\hat{y}. \] The direction of propagation of the electromagnetic wave is given by the wave vector \( \mathbf{k} \), and it must be perpendicular to the direction of the electric field \( \mathbf{E} \) because \( \mathbf{E} \cdot \mathbf{k} = 0 \) (transverse wave condition).

The electric field is: \[ \mathbf{E} = 4\hat{x} + 3\hat{y}. \] The dot product \( \mathbf{E} \cdot \mathbf{k} = 0 \) gives: \[ (4)(a) + (3)(-600) = 0. \] Simplify: \[ 4a - 1800 = 0. \] Solve for \( a \): \[ a = \frac{1800}{4} = 450. \]

Final Answer:

The value of \( a \) is: 450.