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Two plane electromagnetic waves with equal amplitude E0 and angular frequency omega travel in opposite directions along the z-axis. Their electric field vectors are: E1(z,t) = E0 * cos(kz - omega*t) in the x-direction, E2(z,t) = E0 * cos(kz + omega*t + phi) in the x-direction. Their superposition forms a standing wave. Which of the following statements are correct? (A) When phi = pi/2, the resulting wave is elliptically polarized and the polarization changes with position along the z-axis. (B) When phi = pi, the resulting wave is linearly polarized along a fixed direction in the xy-plane. (C) The magnetic field vector of the resulting wave oscillates perpendicular to the plane of polarization at each point. (D) When phi = pi, nodes of the electric field occur at positions z = n*pi/k, where n is an integer.

1. When phi = pi/2, the resulting wave is elliptically polarized and polarization changes with z-position.
2. When phi = pi, the resulting wave is linearly polarized along a fixed direction in the xy-plane.
3. The magnetic field vector oscillates perpendicular to the plane of polarization at each point.
4. When phi = pi, electric field nodes occur at z = n*pi/k, where n is an integer.

Correct answer: When phi = pi, electric field nodes occur at z = n*pi/k, where n is an integer.

Solution

Both waves are polarized along the x-direction (same polarization direction), so the superposition is always linearly polarized along x at every point; it is NOT elliptically polarized for any phi. Statement A is incorrect (the combination of two co-polarized waves cannot produce elliptical polarization). Statement B: For phi = pi, E_total = E0[cos(kz-omega*t) - cos(kz+omega*t)] = 2*E0*sin(kz)*sin(omega*t). This is a standing wave linearly polarized along x (not a 'fixed direction in xy-plane' in a special sense, but it is linear). Statement B could be considered correct if linearly polarized along x is meant. Statement C: In a standing EM wave, E and B are not simple plane waves; the standard relation about B perpendicular to E in a single plane does not apply in the usual sense. Statement D: For phi = pi, E_total = 2*E0*sin(kz)*sin(omega*t). Nodes (amplitude = 0) occur when sin(kz) = 0, i.e., kz = n*pi, i.e., z = n*pi/k. Statement D is correct.

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