Given below are two statements : Statement I : Phase velocity of an electromagnetic wave in a bounded medium can be greater than the speed of light in an unbounded medium. Statement II : Frequency of electromagnetic wave is affected by the permittivity of the medium. In the light of the above statements, choose the correct answer from the options given below :
Statement I is true but Statement II is false
Statement I — "Phase velocity in a bounded medium can exceed the speed of light in an unbounded medium." TRUE. In a bounded medium such as a waveguide the wave travels as a mode, zig-zagging between the walls, and its phase fronts advance along the axis faster than the wave itself moves:
\(v_p=\dfrac{c}{\sqrt{1-\left(\dfrac{f_c}{f}\right)^{2}}} \gt c\)
No physical law is broken, because the phase velocity is only the speed at which a point of constant phase appears to move; it carries no energy and no information. Those travel at the group velocity
\(v_g=c\sqrt{1-\left(\dfrac{f_c}{f}\right)^{2}} \lt c, \qquad v_p v_g=c^{2}\)
so the product is fixed and the group velocity always stays below c, exactly as relativity requires.
Statement II — "Frequency of an electromagnetic wave is affected by the permittivity of the medium." FALSE. Frequency is set by the source and is invariant when a wave crosses into another medium — the fields on the two sides of a boundary must oscillate in step, otherwise the boundary conditions could not be satisfied at every instant. What the permittivity changes is the velocity, and hence the wavelength:
\(v=\dfrac{c}{\sqrt{\varepsilon_r}}, \qquad \lambda=\dfrac{v}{f}=\dfrac{\lambda_0}{\sqrt{\varepsilon_r}}\)
So light entering glass slows down and its wavelength shortens, while its colour (frequency) is unchanged — the familiar fact behind refraction.
Combining: Statement I true, Statement II false.
Hence, Statement I is true but Statement II is false.
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