Young's Interference Experiment — Question 8

Question 8

A focal-plane fringe pattern is independent of the later screen-distance formula

Two narrow slits are followed immediately by a thin lens with focal length $f=0.30\,\mathrm m$. A detector lies in its rear focal plane. For small angles $y=f\theta$. The slit spacing is $d=0.15\,\mathrm{mm}$ and wavelength $\lambda=600\,\mathrm{nm}$.

1. Derive and evaluate the fringe spacing in the focal plane.

2. Find the slit spacing required to fit exactly eight bright-fringe intervals across a $6.0\,\mathrm{mm}$ detector width with the same lens and wavelength.

Question 8 - Solution

A focal-plane fringe pattern is independent of the later screen-distance formula

(a) Method and calculation

$$d\theta_m=m\lambda,\quad y_m=f\theta_m\Longrightarrow\boxed{\Delta y=f\lambda/d}.$$$$\boxed{\Delta y=1.20\,\mathrm{mm}}.$$

(b) Interpretation and further calculation

$$8\Delta y=6.0\,\mathrm{mm}\Longrightarrow\Delta y=0.750\,\mathrm{mm}.$$$$\boxed{d=\frac{f\lambda}{\Delta y}=0.240\,\mathrm{mm}}.$$

Physical check

The relevant scale is the lens focal length because the detector is measuring outgoing angle, not free-space propagation from the slit plane.