Question

a thin film of titanium dioxide (n = 2.61) is placed on glass. for what thickness will blue light (wavelength = 470 nm) interfere constructively when m = 2? ? nm

Explanation:

Step1: Recall constructive interference formula

For a thin film where the refractive index of the film ($n_{film}=2.61$) is greater than the substrate (glass, $n_{glass}\approx1.5$) and air ($n_{air}=1$), there is a 180-degree phase shift at both surfaces. The condition for constructive interference is:
$$2nt = m\lambda$$
where $n$ = refractive index of film, $t$ = thickness, $m$ = order, $\lambda$ = wavelength in vacuum.

Step2: Rearrange for thickness

Isolate $t$ from the formula:
$$t = \frac{m\lambda}{2n}$$

Step3: Substitute given values

Plug in $m=2$, $\lambda=470\ \text{nm}$, $n=2.61$:
$$t = \frac{2\times470}{2\times2.61} = \frac{470}{2.61} \approx 179.31\ \text{nm}$$

Answer:

179.31 nm