In Young's double-slit experiment, fringe separation \( \Delta y \) is inversely proportional to wavelength \( \lambda \). The formula for fringe separation is:
\[
\Delta y = \frac{\lambda D}{d}
\]
Where:
- \( \lambda \) represents the wavelength of light.
- \( D \) is the distance from the slits to the screen.
- \( d \) is the separation between the slits.
The number of fringes per centimeter is the inverse of the fringe separation:
\[
\text{Number of fringes per cm} = \frac{1}{\Delta y}
\]
For two distinct wavelengths, \( \lambda_1 \) and \( \lambda_2 \), the relationship is:
\[
\frac{\text{Number of fringes per cm with } \lambda_2}{\text{Number of fringes per cm with } \lambda_1} = \frac{\lambda_1}{\lambda_2}
\]
Given \( \lambda_1 = 5600 \, \text{\AA} \) and \( \lambda_2 = 7000 \, \text{\AA} \), the number of fringes with \( \lambda_2 \) is calculated as:
\[
\frac{15}{\frac{7000}{5600}} = 12
\]
Therefore, there are 12 fringes per cm when the wavelength is \( \lambda_2 = 7000 \, \text{\AA} \).