In Young's double-slit experiment, a student observes 8 fringes in a certain segment of the screen when a mono

Question

In Young's double-slit experiment, a student observes 8 fringes in a certain segment of the screen when a monochromatic light of 600 nm wavelength is used. If the wavelength of light is changed to 400 nm, then the number of fringes he would observe in the same region of the screen is:

Accepted Answer

1. **Identify Given Values:**
   - Initial number of fringes ($n_1$) = 8
   - Initial wavelength ($\lambda_1$) = $600 	ext{ nm}$
   - Final wavelength ($\lambda_2$) = $400 	ext{ nm}$
   - The segment of the screen (width $L$) remains constant.
2. **Formula:** The width of the region occupied by $n$ fringes is given by $L = n \beta$, where $\beta = \frac{\lambda D}{d}$ is the fringe width. Thus, $L = \frac{n \lambda D}{d}$.
3. **Relation:** Since $L, D,$ and $d$ are constant, the product of the number of fringes and the wavelength is constant:
   $$ n_1 \lambda_1 = n_2 \lambda_2 $$
4. **Calculation:**
   $$ 8 	imes 600 	ext{ nm} = n_2 	imes 400 	ext{ nm} $$
   $$ 4800 = n_2 	imes 400 $$
   $$ n_2 = \frac{4800}{400} = 12 $$
5. **Conclusion:** The number of fringes observed with the new wavelength is 12.

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