A monochromatic light wave with wavelength λ1 and frequency ν1 in air enters another medium. If the angle of incidence and angle of refraction at the interface are 45° and 30° respectively, then the wavelength λ2 and frequency ν2 of the refracted wave are:

To solve this problem, we need to use two key principles: Snell's Law and the relationship between speed, frequency, and wavelength of a wave.

1. Snell's Law: This law states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the speed of light in the original medium to the speed of light in the refracting medium. Mathematically, it is expressed as:

   sin(i)/sin(r) = v1/v2

   where i is the angle of incidence, r is the angle of refraction, v1 is the speed of light in the original medium (air), and v2 is the speed of light in the refracting medium.

   Given that i = 45° and r = 30°, we can substitute these values into the equation:

   sin(45°)/sin(30°) = v1/v2

   Solving this equation will give us the ratio of the speeds of light in the two mediums.

2. Relationship between speed, frequency, and wavelength: The speed of a wave is equal to the product of its frequency and wavelength. This can be expressed as:

   v = λν

   where v is the speed of the wave, λ is the wavelength, and ν is the frequency.

   Since the frequency of a wave does not change when it enters a different medium, we can say that ν1 = ν2.

   Therefore, the wavelength of the wave in the refracting medium (λ2) can be found by rearranging the equation to:

   λ2 = v2/ν2

   Substituting the value of v2 from the equation obtained using Snell's Law and the value of ν2 = ν1, we can find the value of λ2.

In conclusion, the frequency of the refracted wave (ν2) will be the same as the frequency of the incident wave (ν1), and the wavelength of the refracted wave (λ2) can be found using the equations above.