Complete Question
A thin film of cooking oil (n=1.43) is spread on a puddle of water (n=1.34). What is the minimum thickness Dmin of the oil that will strongly reflect blue light having a wavelength in air of 451 nm, at normal incidence? Dmin= nm What are the next three thicknesses that will also strongly reflect blue light of the same wavelength, at normal incidence?
a) 158 nm, 237 nm, and 315 nm
b) 237 nm, 394 nm, and 552 nm
c) 473 nm, 788 nm, and 1100 nm
d) 361 nm, 602 nm, and 842 nm
e) 315 nm, 473 nm, and 631 nm
Answer:
The minimum thickness is [tex]t= 78.8nm[/tex]
The correct option is B
Explanation:
From the question we are told that
The refractive index of cooking oil is [tex]n _c = 1.43[/tex]
The refractive index of water is [tex]n_w = 1.34[/tex]
The wavelength of reflection is is [tex]\lambda _ r = 451nm[/tex]
The formula for the thickness of the oil film is mathematically represented as
[tex]2 n t = (m + \frac{1}{2} ) \lambda[/tex]
Where n is the refractive index of oil
m is the integer number of fringe
t is the thickness
for a minimum reflection m= 0
Now making t the subject of the formula
[tex]t = \frac{(m + \frac{1}{2} \lambda ) }{2 n}[/tex]
Substituting value
[tex]t = \frac{(0 + 0.5) * 451 *10^{-9}}{2 * 1.43}[/tex]
[tex]t= 78.8nm[/tex]
For the next thickness m = 1
so we have
[tex]t_1 = \frac{(1 + 0.5 ) * 481}{2 * 1.43}[/tex]
[tex]= 237nm[/tex]
For the next thickness m = 2
so we have that
[tex]t_2 = \frac{(2 +0.5) *451 *10^{-9}}{2 * 1.43}[/tex]
[tex]= 394nm[/tex]
For the next thickness m = 3
so we have that
[tex]t_2 = \frac{(3 +0.5) *451 *10^{-9}}{2 * 1.43}[/tex]
[tex]= 552nm[/tex]
According to the question,
(a)
→ [tex]2nt = (m+\frac{1}{2} ) \lambda[/tex]
By substituting the values, we get
→ [tex]2\times 1.43\times t = (0+\frac{1}{2} )481[/tex]
→ [tex]t = 84.1 \ nm[/tex]
(b)
The next 3 thicknesses are:
→ [tex]2\times 1.43\times t = (1+\frac{1}{2} ) 481[/tex]
[tex]t = 252 \ nm[/tex]
→ [tex]2\times 1.43\times t = (2+\frac{1}{2} )481[/tex]
[tex]t = 420 \ nm[/tex]
→ [tex]2\times 43\times t = (3+\frac{1}{2} ) 481[/tex]
[tex]t = 588 \ nm[/tex]
Thus the above approach is correct.
Learn more:
https://brainly.com/question/13051021
