Name: Thin Film Interference
Uploaded: 2019-07-19
Duration: 7 min 3 s
Description: Thin-film interference is a natural phenomenon in which light waves reflected by a thin film's upper and lower boundaries interfere with one another. Learn m...

Question

Describe Young's double-slit experiment and derive the expression for the fringe width.

Accepted Answer

Consider two pinholes S₁ and S₂ on an opaque screen. These are illuminated by another pinhole S which is lit by a bright source. Light waves spread out from S and fall on both S₁ and S₂ . These two holes behave like two coherent sources because light waves coming out from them are derived from the same single source. Any abrupt phase change in S will manifest in exactly similar phase changes in the light coming out from S₁and S₂. Thus, S₁and S₂are locked in phase. The spherical waves emanating from S₁and S₂will produce interference fringes on the screen GG^'.The arrangement for Young's double-slit experiment is shown below:

Let there be a bright fringe on a point P at a distance $x$ from 'O' (midpoint of GG^'). Hence, the condition for maxima must be satisfied there and we can write the path difference:

$S_{2} P - S_{1} P = n λ; n = 0, 1, 2 . . .$

Now:

${(S_{2} P)}^{2} - {(S_{1} P)}^{2} = [D^{2} + {(x + \frac{d}{2})}^{2}] - [D^{2} + {(x - \frac{d}{2})}^{2}] = 2 x d$

where $S_{1} S_{2} = d and OP = x$ . Thus:

$S_{2} P - S_{1} P = \frac{2 x d}{S_{2} P + S_{1} P}$

If $x, d < < D then (S_{2} P + S_{1} P)$ can be replaced by $2 D$ . Hence:

$S_{2} P - S_{1} P = \frac{x d}{D}$

So we will have constructive interference resulting in bright fringe when:

$\frac{x_{n} d}{D} = n λ$

$\Rightarrow x_{n} = \frac{n D λ}{d}$

Dark and bright fringes are equally spaced. Hence, the distance between two consecutive bright fringes or the distance between the two consecutive dark fringes:

$β = x_{n + 1} - x_{n} = \frac{D λ}{d}$

This is the required expression for the fringe width.

Describe Young's double-slit experiment and derive the expression for the fringe width.

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