A narrow monochromatic beam of light of intensity I is incident on a glass plate A as shown in (Fig. a). Another identical glass plate B is kept close to A and parallel to it. Each glass plate reflects 25% of the light intensity incident on it and transmits the remaining. Find the ratio of the minimum and maximum intensities in the interference pattern formed by the two beams obtained after one reflection at each plate.

Solution

A beam of light of intensity I is incident on plate A. Since the plate reflects 25% of I, the intensity of the reflected beam I (see Fig. b) is

I_{1} = I \times \frac{25}{100} = \frac{I}{4}

The remaining intensity 3I/4 falls on plate B which reflects 25% of the intensity incident on it. Hence, intensity of beam reflected from B is

\frac{3 I}{4} \times \frac{25}{100} = \frac{3 I}{16}

A beam of intensity 3 I/16 falls on plate A which transmits 75% of this intensity. Hence the intensity of beam 2 is

I_{2} = \frac{3 I}{16} \times \frac{75}{100} = \frac{9 I}{64}

Now, intensity ∝(amplitude)².

If a₁ and a₂ are the amplitudes of beam 1 and 2 respectively, then

I_{1} = k a_{1}^{2} and I_{2} = {ka}_{2}^{2}

\Rightarrow

\frac{{a_{1}}^{2}}{{a_{2}}^{2}} = \frac{I_{1}}{I_{2}} = \frac{I / 4}{9 I / 64} = \frac{16}{9}

\Rightarrow

\frac{a_{1}}{a_{2}} = \frac{4}{3}

i.e.,

a_{1} = 4 units and a_{2} = 3 units .

The resultant amplitude at maxima = a₁ + a₂ and at minima = a₁- a₂.

Therefore, the ratio of the intensities at maxima and minima of the interference pattern is

\frac{I_{\max}}{I_{\min}} = {(\frac{a_{1} + a_{2}}{a_{1} - a_{2}})}^{2} = {(\frac{4 + 3}{4 - 3})}^{2} = 49 : 1

Some More Questions From Wave Optics Chapter

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Wave Optics

Some More Questions From Wave Optics Chapter

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Wave Optics

Some More Questions From Wave Optics Chapter

What is the shape of the wavefront in each of the following cases: Light diverging from a point source.

What is the shape of the wavefront in each of the following cases: Light emerging out of a convex lens when a point source is placed at its focus.

What is the shape of the wavefront in each of the following cases:The portion of the wavefront of light from a distant star intercepted by the Earth.

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

What is the shape of the wavefront in each of the following cases:
Light diverging from a point source.

What is the shape of the wavefront in each of the following cases:
Light emerging out of a convex lens when a point source is placed at its focus.

What is the shape of the wavefront in each of the following cases:
The portion of the wavefront of light from a distant star intercepted by the Earth.