Young's Interference Experiment — Question 4

Question 4

A plate over the upper slit moves the central fringe toward that slit

The upper and lower slits are at $y=+d/2$ and $-d/2$. A thin plate over the upper slit adds optical path $D=(n-1)t$. At screen height y, the outgoing upper-minus-lower path difference is $-dy/L$ in the paraxial limit. Let $d=0.50\,\mathrm{mm}$, $L=2.0\,\mathrm m$, $n=1.50$, $t=3.0\,\mu\mathrm m$, and $\lambda=600\,\mathrm{nm}$.

1. Find the new location of the zero-total-path-difference fringe.

2. Express the displacement in fringe spacings and determine whether the original screen center is bright or dark.

Question 4 - Solution

A plate over the upper slit moves the central fringe toward that slit

(a) Method and calculation

$$\Delta\mathcal L=D-dy/L,\quad D=0.50(3.0)=1.50\,\mu\mathrm m.$$$$\Delta\mathcal L=0\Longrightarrow\boxed{y_0=LD/d=+6.00\,\mathrm{mm}}.$$

(b) Interpretation and further calculation

$$\Delta y=\lambda L/d=2.40\,\mathrm{mm},\quad\boxed{y_0/\Delta y=D/\lambda=2.50}.$$$$y=0:\ \delta=2\pi D/\lambda=5\pi\Longrightarrow\boxed{\text{dark at the original center}}.$$

Physical check

The shift is upward: shortening the outgoing path from the upper slit compensates for the plate delay.