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A standing electromagnetic wave is formed by superposing two plane waves of equal amplitude and frequency traveling in opposite directions along the z-axis. The electric field of the resulting standing wave is E(z,t) = E0 * cos(kz) * cos(omega*t) in the x-direction. Which of the following statements are correct?

1. The electric field is linearly polarized along the x-direction.
2. The magnetic field of the standing wave oscillates along the y-direction.
3. The nodes of the electric field occur at positions z = n*pi/k, where n is an integer.
4. The nodes of the magnetic field occur at positions z = n*pi/k, where n is an integer.

Correct answer: The electric field is linearly polarized along the x-direction.

Solution

Statement A: E always points along x-hat -> linearly polarized in x-direction. TRUE. Statement B: Using Maxwell's equation, B = -(E0/c) sin(kz) sin(omega*t) y-hat -> oscillates in y-direction. TRUE. Statement C: Nodes of E where cos(kz)=0 -> kz=(2n+1)*pi/2 -> z=(2n+1)*pi/(2k), NOT z=n*pi/k. FALSE. Statement D: Nodes of B where sin(kz)=0 -> kz=n*pi -> z=n*pi/k. TRUE. So A, B, D are correct. The question asks which statements are correct; option A (the first listed option) covers only statement A. The question structure implies single correct option. Given the options as listed and that option A states only statement 1, while multiple statements are true, this appears to be a multiple-correct question. The single option that best applies as a standalone statement is A (electric field linearly polarized in x), as B and D are also correct but the question asks to identify a correct statement. If single answer: A is correct and C is the only false one; but the question format and options suggest the answer is option A alone represents one correct statement and option D (nodes of B at n*pi/k) is also correct. The most unambiguous standalone correct statement listed as a single option is A.

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