Properties of light and refractive index of materials Why does the refractive index of a material dependent on the wavelenght of light incident on it.
 A: The refractive index is defined by the ratio
$$n=\frac{v}{c},$$
between the speed of the light in the medium and the speed of the light in the vacuum. From the wave equation we read the velocity $v$ in terms of the permittivity and permeability of the medium,
$$v=\frac{1}{\sqrt{\epsilon\mu}},$$
and, in particular, for the vacuum we have
$$c=\frac{1}{\sqrt{\epsilon_0\mu_0}}.$$
From the three equations above and the definition of relative permittivity and relative permeability we can write the refraction index as
$$n=\sqrt{\epsilon_r \mu_r}.$$
For simplicity, let us consider non magnetic materials (non metals would be enough), so that
$$n=\sqrt{\epsilon_r}.$$
This relative permittivity depends on the frequency of the incident wave (or rather its wavelength), so $n=n(\omega)$ (or $n=n(\lambda)$). 
To get an intuition on this frequency dependency we have to recall that from a microscopic point of view, the electronic permittivity actually measures how well an electronic cloud can be polarized by an electromagnetic field. The greater the distortion, the greater the dipole moment, the atomic polarizability, and the permittivity. The botton line is that we can understand how the refraction index depend on frequency if we understand how the electronic polarization depends on frequency. 
A simple model to explain that consists on considering the centre of mass of the electronic cloud a driven oscillator. The equation of motion is
$$\frac{d^2x}{dt^2}+\gamma\frac{dx}{dt}+\omega_0^2x=qE,$$
where $qE$, the electric force, is the driven term. The constant $\omega_0$ is the cloud's natural frequency of oscillation and gamma is same damping. The amplitude of oscillation (for the steady solution) is frequency dependent, it follows curves like those in the figure

where the different curves are associated to different damping. As we can see, light with different wavelengths (different frequencies) put the electronic cloud to oscillate with different amplitudes leading to different refractive index. Of course this is a crude model, but it can predict refraction index that agrees with the experimental result up to the fourth decimal place.
