In a double-slit experiment, two coherent sources have slightly different intensities I and (I + δI), such that δ << I, show that resultant intensity at maxima is near 4I, while that at minima is nearly (δl)²/4l.

Solution

Given, Young's double slit experiment with two coherent sources.
Intensity of first source, I₁ = I
Intensity of second source, I₂ = I + δI

Now, using the formula for resultant intensity,

For maxima,

$I = I_{1} + I_{2} + 2 \sqrt{I_{1} I_{2}} \cos ϕ$

$I_{\max} = I_{1} + I_{2} + 2 \sqrt{I_{1} I_{2}} \cos 0 ° = I + (I + δI) + 2 (\sqrt{I (I + δI)}$

As $δI < < I,$ therefore,

   $I_{\max} = I + I + 2 I = 4 I$

For minima,

$I = I_{1} + I_{2} + 2 \sqrt{I_{1} I_{2}} \cos ϕ$

$I_{\min} = I + (I + δI) + 2 \sqrt{I (I + δI)} \cos 180 °$

$I_{\min} = 2 I + δI - 2 I {(1 + \frac{δI}{I})}^{1 / 2}$
   $= 2 I + δI - 2 I [1 + \frac{1}{2} \frac{δI}{I} + \frac{\frac{1}{2} (\frac{1}{2} - 1)}{2} {(\frac{δI}{I})}^{2}]$

$= 2 I + δI - 2 I - δI + \frac{1}{4} I {(\frac{δI}{I})}^{2}$
$∴$ $I_{\min} = \frac{{(δI)}^{2}}{4 I} .$

Hence proved.

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Wave Optics

In a double-slit experiment, two coherent sources have slightly different intensities I and (I + δI), such that δ << I, show that resultant intensity at maxima is near 4I, while that at minima is nearly (δl)²/4l.

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Is the speed of light in glass independent of the colour of light? If not, which of the two colours red and violet travels slower in a glass prism?

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For Daily Free Study Material Join wiredfaculty

Wave Optics

In a double-slit experiment, two coherent sources have slightly different intensities I and (I + δI), such that δ << I, show that resultant intensity at maxima is near 4I, while that at minima is nearly (δl)2/4l.

Some More Questions From Wave Optics Chapter

Is the speed of light in glass independent of the colour of light? If not, which of the two colours red and violet travels slower in a glass prism?

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

In a double-slit experiment, two coherent sources have slightly different intensities I and (I + δI), such that δ << I, show that resultant intensity at maxima is near 4I, while that at minima is nearly (δl)²/4l.