### Related Formula For a first order dark fringe (minima) to form, the net optical path difference between the interfering paths must be an odd multiple of half-wavelength: Delta x = fraclambda2$\Delta x = \frac{\lambda}{2}$ ### Core Logic From the symmetric setup diagram geometry: * Total path length 1 = textSource rightarrow S_1 rightarrow P = sqrtD^2 + d^2 + sqrtD^2 + d^2 = 2sqrtD^2 + d^2$\text{Source} \rightarrow S_1 \rightarrow P = \sqrt{D^2 + d^2} + \sqrt{D^2 + d^2} = 2\sqrt{D^2 + d^2}$ * Total path length 2 = textSource rightarrow S_2 rightarrow P = D + D = 2D$\text{Source} \rightarrow S_2 \rightarrow P = D + D = 2D$ Hence, the exact geometric path difference is: Delta x = 2sqrtD^2 + d^2 - 2D$\Delta x = 2\sqrt{D^2 + d^2} - 2D$ ### Step 1: Set up Path Difference Balance Equating path difference to the minima criterion: 2sqrtD^2 + d^2 - 2D = fraclambda2$2\sqrt{D^2 + d^2} - 2D = \frac{\lambda}{2}$ sqrtD^2 + d^2 - D = fraclambda4 implies sqrtD^2 + d^2 = D + fraclambda4$\sqrt{D^2 + d^2} - D = \frac{\lambda}{4} \implies \sqrt{D^2 + d^2} = D + \frac{\lambda}{4}$ ### Step 2: Expand and Isolate Slit Space d Squaring both sides: D^2 + d^2 = D^2 + fraclambda^216 + frac2 D lambda4$D^2 + d^2 = D^2 + \frac{\lambda^2}{16} + \frac{2 D \lambda}{4}$ d^2 = fracD lambda2 + fraclambda^216$d^2 = \frac{D \lambda}{2} + \frac{\lambda^2}{16}$ ### Step 3: Approximate and Substitute Values Since lambda$\lambda$ (400 mathrm~nm$400 \mathrm{~nm}$) is tiny compared to D$D$ (0.2 mathrm~m$0.2 \mathrm{~m}$), the term fraclambda^216$\frac{\lambda^2}{16}$ is completely negligible: d^2 approx fracD lambda2$d^2 \approx \frac{D \lambda}{2}$ Substituting values: d^2 = frac0.2 times 400 times 10^-92 = 400 times 10^-10 mathrm~m^2$d^2 = \frac{0.2 \times 400 \times 10^{-9}}{2} = 400 \times 10^{-10} \mathrm{~m^2}$ d = sqrt400 times 10^-10 = 20 times 10^-5 mathrm~m = 0.20 times 10^-3 mathrm~m = 0.20 mathrm~mm$d = \sqrt{400 \times 10^{-10}} = 20 \times 10^{-5} \mathrm{~m} = 0.20 \times 10^{-3} \mathrm{~m} = 0.20 \mathrm{~mm}$ Therefore, the minimum distance between the slits is 0.20 mathrm~mm$0.20 \mathrm{~mm}$. ### Pattern Recognition This non-standard double slit setup calculates the primary total path directly from geometric lines rather than relying on standard angular small angle assumptions (y d / D$y d / D$). Recognizing when secondary higher-order differentials (lambda^2$\lambda^2$) can be safely dropped keeps calculations clean. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Wave Optics