To find the wavelength of the refracted light in a medium with a refractive index of 2, we can use the relationship between frequency, wavelength, and the speed of light. The frequency ($f$) of the light remains the same when it enters the medium.

The formula relating frequency, wavelength, and velocity is:

$ v = f \lambda $

Where:

$ v $ is the speed of light in the medium,

$ f $ is the frequency of the light,

$ \lambda $ is the wavelength of the light.

The wavelength of light in the medium ($\lambda_{\text{medium}}$) can be found using the refractive index ($\mu$) and the wavelength in a vacuum ($\lambda_{\text{vacuum}}$) as follows:

$ \lambda_{\text{medium}} = \frac{\lambda_{\text{vacuum}}}{\mu} $

Given:

Frequency of the light, $ f = 5 \times 10^{14} \, \text{Hz} $

Speed of light in a vacuum, $ v_{\text{vacuum}} = 3 \times 10^{8} \, \text{m/s} $

Refractive index of the medium, $ \mu = 2 $

First, calculate $\lambda_{\text{vacuum}}$:

$ \lambda_{\text{vacuum}} = \frac{v_{\text{vacuum}}}{f} = \frac{3 \times 10^{8}}{5 \times 10^{14}} $

Now calculate $\lambda_{\text{medium}}$:

$ \lambda_{\text{medium}} = \frac{\lambda_{\text{vacuum}}}{2} = \frac{3 \times 10^{8}}{2 \times 5 \times 10^{14}} $

$ \lambda_{\text{medium}} = 0.3 \times 10^{-6} \, \text{m} $

Converting to nanometers:

$ 0.3 \times 10^{-6} \, \text{m} = 300 \, \text{nm} $

Therefore, the wavelength of the refracted light in the medium is 300 nm.