There exist media in which the product of phase velocity and group velocity is a constant. If such a medium possesses normal dispersion, how does the phase velocity depend on wavelength? (Electromagnetic wave propagation in ionized gases and in hollow-pipe waveguides is an example of this situation.)

Short Answer

Expert verified
Answer: For a medium with normal dispersion and a constant product of phase and group velocities, the phase velocity is inversely proportional to the square root of the cube of the wavelength, i.e., v_p ∝ 1/(λ^(3/2)).

Step by step solution

01

Calculate the Group Velocity

To find the group velocity (v_g), we need to differentiate the refractive index (n) with respect to the angular frequency (ω) and multiply it by -1. From the equation for phase velocity, v_p: v_p = c / n and ω = ck/n. We can solve for the refractive index n as: n = ck/ω. Now, we differentiate n with respect to ω: dn/dω = - (ck)/(ω^2). Remember that v_g = - (dω/dk), so we need to take the inverse of the derivative and multiply by -1 to obtain v_g: v_g = -1/(dn/dω) = ω^2/ck.
02

Apply the Condition of Constant Product of Phase Velocity and Group Velocity

The condition given in the exercise states that the product of the phase and group velocities is constant: v_p * v_g = constant. Substituting the expressions of v_p and v_g, we get: (c / n) * (ω^2 / (ck)) = constant. The speed of light (c) is also constant. We know that the angular frequency ω and the wavenumber k are related by the speed of light as ω = ck/n. Now, we can rewrite the previous equation in terms of wavelength λ: (c / (c/λ)) * ((c/λ)^2 / (c*((c/λ)/ω))) = constant. This simplifies to: λ * ω^2 = constant.
03

Find the Dependence of Phase Velocity on Wavelength

We know that the phase velocity v_p is related to the angular frequency ω and the wavenumber k by v_p = ω/k. We substitute this into the relation we found in Step 2: λ * (v_p * k)^2 = constant. This relation indicates that: λ^3 * v_p^2 = constant. Therefore, the phase velocity depends on wavelength as: v_p ∝ 1/(λ^(3/2)). In conclusion, for media where the product of the phase and group velocities is constant and it possesses normal dispersion, the phase velocity is inversely proportional to the square root of the cube of the wavelength.

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