Answer:
For constructive interference
The first three thicknesses are
[tex]t_1 =85nm[/tex]
[tex]t_2 =254nm[/tex]
[tex]t_3 =423nm[/tex]
For destructive interference
The first three thicknesses are
[tex]t_D__{1}} = 169nm[/tex]
[tex]t_D__{2}} = 338nm[/tex]
[tex]t_D__{3}} = 507nm[/tex]
Explanation:
The diagram for this question is shown on the first uploaded image
From the question we are told that
The refractive index of oil is [tex]n_o = 1.55[/tex]
The refractive index of water is [tex]n_w = 1.33[/tex]
The average wavelength is [tex]\lambda_a = 524 nm = 524*10^{-9}m[/tex]
For constructive interference the thickness is mathematically represented as
[tex]t = (m + \frac{1}{2} ) \frac{\lambda }{2 n_w}[/tex]
Where m is the order of the interference with a value from [tex]m = 0,1 , -1 , 2 , -2 , ...[/tex]
For m = 0
The thickness of the oil slick would be
[tex]t_1 = 0 + \frac{1}{2} * \frac{\lambda }{2 * n_w}[/tex]
Substituting value
[tex]t_1 = 0 + \frac{1}{2} * \frac{524 *10^{-9}}{2 * 1.55}[/tex]
[tex]t_1 =85nm[/tex]
For m = 1
The thickness of the oil slick would be
[tex]t_2 = 1 + \frac{1}{2} * \frac{\lambda }{2 * n_w}[/tex]
Substituting value
[tex]t_2 = 1 + \frac{1}{2} * \frac{524 *10^{-9}}{2 * 1.55}[/tex]
[tex]t_2 =254nm[/tex]
For m = 2
The thickness of the oil slick would be
[tex]t_3 = 2 + \frac{1}{2} * \frac{\lambda }{2 * n_w}[/tex]
Substituting value
[tex]t_3= 2 + \frac{1}{2} * \frac{524 *10^{-9}}{2 * 1.55}[/tex]
[tex]t_3 =423nm[/tex]
For destructive interference the thickness is mathematically represented as
[tex]t = \frac{m \lambda }{2 n_w}[/tex]
For m = 1
[tex]t_D__{1}} = \frac{\lambda }{2 * n_w}[/tex]
Substituting value
[tex]t_D__{1}} = \frac{524 *10^{-9} }{2 * 1.55}[/tex]
[tex]t_D__{1}} = 169nm[/tex]
For m = 2
[tex]t_D__{2}} = \frac{\lambda }{2 * n_w}[/tex]
Substituting value
[tex]t_D__{2}} = \frac{ 2 * 524 *10^{-9} }{2 * 1.55}[/tex]
[tex]t_D__{2}} = 338nm[/tex]
For m = 3
[tex]t_D__{3}} = \frac{\lambda }{2 * n_w}[/tex]
Substituting value
[tex]t_D__{3}} = \frac{ 3 * 524 *10^{-9} }{2 * 1.55}[/tex]
[tex]t_D__{3}} = 507nm[/tex]
