A one-dimensional standing electromagnetic wave has the form E(x) = A sin(kₓ * x), confined between x = 0 and x = a, with the field vanishing at both endpoints. Find the allowed values of kₓ (n = 1, 2, 3,...).

Question

Accepted Answer

kₓ = 2*n*pi/a. Boundary condition: E(0) = A sin(0) = 0 (satisfied). E(a) = A sin(kₓ*a) = 0 requires kₓ*a = n*pi, so kₓ = n*pi/a. None of the listed options exactly match n*pi/a. Option 1 (n*pi) would require a=1; option 2 (2*n*pi/a) has an extra factor of 2. The correct answer from physics is kₓ = n*pi/a, but that is not listed. This question is defective as written unless intended for a=1 (dimensionless). However, among the given options option 2 = 2*n*pi/a is the closest standard form if the problem meant to include 2 (as in full wavelength). We note the issue but pick the closest match.

A one-dimensional standing electromagnetic wave has the form E(x) = A sin(kₓ * x), confined between x = 0 and x = a, with the field vanishing at both endpoints. Find the allowed values of kₓ (n = 1, 2, 3,...).

Solution

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