91Ó°ÊÓ

The relationship between the wavelength and the index of redfraction for the constructive interference of reflected light is,

\(2nd = \dfrac{{3\lambda }}{2}\)

\(\begin{aligned}\lambda &= 380{\rm{ nm}}\\ &= \dfrac{{380}}{{{{10}^9}}}\,{\rm{m}}\\ &= 380 \times {10^{ - 9}}{\rm{ m}}\end{aligned}\).

a. To find required minimum thickness, use relation for constructive interference of reflected light –

\(2nd = \dfrac{{3\lambda }}{2}\)

Afterward simplifying above relation, the relation for finding minimum thickness is obtained –

\(\begin{aligned}2nd &= \dfrac{3}{2}\lambda \\4nd &= 3\lambda \\\dfrac{{4nd}}{4} &= \dfrac{{3\lambda }}{4}\\nd &= \dfrac{{3\lambda }}{4}\\\dfrac{{nd}}{n} &= \dfrac{{3\lambda }}{{4n}}\\d &= \dfrac{{3\lambda }}{{4n}}\,........................\,\left( 1 \right)\end{aligned}\)

Now, plug in values in equation\(\left( 1 \right)\)for minimum thickness and solve this equation –

\(\begin{aligned}d &= \dfrac{{3\lambda }}{{4n}} \cdot \dfrac{1}{4}\\d &= \dfrac{{3\lambda }}{{16n}}\\d &= \dfrac{{3 \cdot 380 \cdot {{10}^{ - 9}}{\rm{ m}}}}{{16 \cdot 1.33}}\\d &= \dfrac{{1140 \cdot {{10}^{ - 9}}{\rm{ m}}}}{{21.28}}\\d &= 53.5 \cdot {10^{ - 9}}{\rm{ m}}\end{aligned}\)

Therefore, the value for thickness is obtained as \(d = 53.5 \cdot {10^{ - 9}}{\rm{ m}}\).

b. As the thickness of the soap bubble is obtained as \(d = 53.5 \cdot {10^{ - 9}}{\rm{ m}}\), it can be seen the soap bubble has very small thickness i.e., less than zero. So, the soap bubble is very fragile.

Therefore, the soap bubble is extremely fragile.