JEE Advanced Physics: Modern Physics / Wave Mechanics questions with solutions

Q1. In a string-theory model, the extra hidden dimension y is compactified on the surface of a thin cylinder of radius b. An electromagnetic wave on the surface takes the form E(x, y) = A*sin(kₓ * x)*cos(k_y * y), where y is the coordinate around the cylinder. Find the allowed values of k_y (m = 1, 2, 3,...).

1. k_y = m/b
2. k_y = m/(2b)
3. k_y = m/(pi*b)
4. k_y = 2m/(pi*b)

Answer: k_y = m/b

The coordinate y is periodic with period 2*pi*b (circumference of the cylinder). For the wave to be single-valued, k_y*(2*pi*b) must be a multiple of 2*pi, giving k_y = m/b.