Question

Repeat Exercise\({\rm{27}}{\rm{.71}}\), but take the light to be incident at a \({\rm{4}}{{\rm{5}}^ \circ }\)angle.

Short answer

The wavelength of reflected light is\({\rm{952\;nm}}\)and it will produce black color.

Step-by-step solution

Definition of Concept 

Wavelength : The distance between identical points (adjacent crests) in adjacent cycles of a waveform signal propagated in space or along a wire is defined as the wavelength. This length is typically specified in wireless systems in metres (m), centimetres (cm), or millimetres (mm) .

Find the wavelength of reflected light

Considering the given information,

Angle of incidence,\(\theta = {45^ \circ }\)

Thickness of the oil slick,\(t = 120\;nm\)

Redfractive index of water,\(n = 1.40\)

Apply the formula,

The formula is used to compute the thickness of the coating that causes constructing interference :

\(t = \dfrac{{\lambda \sin \theta }}{{4n}}\)

Here,

The wavelength of the reflected light is denoted by\(\lambda \).

The redfractive index is n.

The angel of incidence is\(\theta \).

The thickness of the layer causing interference is t.

The wavelength of reflected light is determined using the formula:

\(t = \dfrac{{\lambda \sin \theta }}{{4n}}\)

Putting the values,

\(\begin{aligned}120 \times {10^{ - 9}} &= \dfrac{{\lambda \sin {{45}^ \circ }}}{{4 \times 1.40}}\\\lambda &= 950\;nm\end{aligned}\)

The wavelength in the range\((700\;nm - 1\;mm)\)is in the infrared spectrum, which we cannot see with our naked eyes; thus, the light reflected from the oil slick is black.

Therefore, the requiredwavelength of reflected light is\(952\;\,nm\)and it will produce black color.