I'm learning with Kenneth Krane Modern Physics 3ed.
28-(b)
When white light travels through glass, the phase velocity of each wavelength depends on the wavelength. (This is
the origin of dispersion and the breaking up of white light
into its component colors—different wavelengths travel at
different speeds and have different indices of refraction.)
How does vphase depend on λ? Is dvphase/dλ positive or
negative? Therefore, is vgroup >vphase or < vphase?
solution
The index of refraction n for light in glass decreases as λ increases (shorter
wavelengths are refracted more than longer wavelengths); that is dn / dλ < 0 .
Because n = c / vphase , dn / dλ and dvphase / dλ have opposite signs and so dvphase / dλ > 0. Thus vgroup > vphase.
I don't understand the last formula [vgroup > vphase].
I think it should be [vgroup < vphase] because of the equation in problem 28-(a)[vgroup=vphase – λ(dvphase/dλ)]. Is the solution wrong?
Best Answer
The solution does indeed look wrong if everything you have supplied is correct. If the equation $$v_\text{group}=v_\text{phase} - \lambda\frac{dv_\text{phase}}{dλ}$$ holds true, and $$\frac{dv_\text{phase}}{d\lambda}\gt 0$$ then it is certainly true that $$\boxed{v_\text{phase}\gt v_\text{group}}$$
where of course $\lambda\gt 0$. It is true (in general) that for dispersive mediums like glass, the group velocity is usually smaller than the phase velocity.