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**WAVE MOTION**

**Interference of Light**

conditions |

__Conditions for interference__

1. The waves from light sources must be **coherent** with each other. This means that they must be of the same frequency, with a constant phase difference between them.

2. The amplitude (maximum displacement) of interfering waves must have the same magnitude. Slight variations produce lack of contrast in the interference pattern.

__ Young's Double Slit Experiment - Apparatus __

It is important to realise that the diagram is **not** to scale.

Typically the distance (**D**) between the double slits and the screen is ~ 0.2 m (20 cm).

The distance (**a**) between the double slits is ~ 10^{-3}m (1mm).

The preferred monochromatic light source is a sodium lamp.

__Young's Double Slit Experiment - Display__

The image above is taken from the **central maximum** area of a display.

You will notice some dimming in the image from the centre travelling outwards. This is because the regular light-dark bands are superimposed on the light pattern from the single slit.

The intensity pattern is in effect a combination of both the single-slit diffraction pattern and the double slit interference pattern.

In other words, t he amplitude of the diffraction pattern **modulates** the interference pattern.

The diffraction pattern acts like an *'envelope'* containing the interference pattern.

__Young's Double Slit Experiment - theory__

The separation (**y**) of bright/dark fringes can be calculated using simple trigonometry and algebra.

Consider two bright fringes at **C** and **D**.

For the fringe at **C**, the method is to find the **path difference** between the two rays **S _{1}C** and

**S**. This is then equated to an exact number of wavelengths

_{2}C**n**.

A similar expression is found for the fringe at **D**, but for the number of wavelengths **n+1** .

The two expressions are then combined to exclude **n** .

With reference to triangle **CAS _{2}** , using Pythagoras' Theorem:

substituting for **AC** and **S _{2}A** in terms of x

_{C},

**a**and

**D**,

(i

also, with reference to triangle **CBS _{1}**

(ii

Subtracting equation (ii from equation (i ,

Using 'the difference of two squares' to expand the LHS,

hence,

The path difference **S _{2}C** -

**S**is therefore given by:

_{1}C

In reality, **a** ~ 10^{-3}m and **D** ~ 0.2 m .

So the length **a** is much smaller than **D** (approx 1/200 th **D**).

The two rays **S _{2}C** and

**S**are roughly horizontal and approximate to

_{1}C**D**.

So,

cancelling the 2's,

For a bright fringe at point **C** the path difference **S _{2}C** -

**S**must be a whole number (

_{1}C**n**) of wavelengths (

**λ**).

Hence,

Rearranging to make **x**** _{C}** the subject,

Similarly, for the next bright fringe at** D**, when the path difference is one wavelength longer, that is equal to **(n+1) **wavelengths ,

hence the fringe separation ** x _{D} - x_{C}** is given by,

assigning the fringe separation the letter * y* ,

or with wavelength * λ* the subject,

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