Question 24.1: Measuring the Wavelength of a Light Source Goal Show how You...

(a) Determine the wavelength of the light.

Solve Equation 24.5 for the wavelength and substitute m=2, y_{2}=4.50 \times 10^{-2} \mathrm{~m}, L=1.20 \mathrm{~m}, and d=3.00 \times 10^{-5} \mathrm{~m}:

\begin{aligned} \lambda & =\frac{y_{2} d}{m L}=\frac{\left(4.50 \times 10^{-2} \mathrm{~m}\right)\left(3.00 \times 10^{-5} \mathrm{~m}\right)}{2(1.20 \mathrm{~m})} \\ & =5.63 \times 10^{-7} \mathrm{~m}=563 \mathrm{~nm} \end{aligned}

(b) Determine the distance between adjacent bright fringes.

Use Equation 24.5 to find the distance between any adjacent bright fringes (here, those characterized by m and m+1) :

\begin{aligned} \Delta y & =y_{m+1}-y_{m}=\frac{\lambda L}{d}(m+1)-\frac{\lambda L}{d} m=\frac{\lambda L}{d} \\ & =\frac{\left(5.63 \times 10^{-7} \mathrm{~m}\right)(1.20 \mathrm{~m})}{3.00 \times 10^{-5} \mathrm{~m}}=2.25 \mathrm{~cm} \end{aligned}

Remarks This calculation depends on the angle \theta being small, because the small-angle approximation was implicitly used. The measurement of the position of the bright fringes yields the wavelength of light, which in turn is a signature of atomic processes, as will be discussed in the chapters on modern physics. This kind of measurement, therefore, helped open the world of the atom.