# Young Double-Slit Experiment ## Requirements for Observable Interference - For clear interference fringes to appear, light sources must satisfy two critical conditions: ### Coherent Sources - Coherent sources maintain a **constant phase difference** between them - Without coherence, the interference pattern constantly shifts and becomes impossible to observe - Think of two dancers maintaining perfect synchronisation—they must keep the same rhythm throughout ### Monochromatic Sources - Monochromatic means **single wavelength** light - White light contains many wavelengths which would create overlapping patterns - Using one colour (wavelength) ensures the pattern is sharp and measurable

Coherent sources have the same frequency and a constant phase difference

## Path Difference - Path difference describes the extra distance one wave travels compared to another - When waves from two slits meet at a point on a screen, they have travelled different distances - This difference in journey length determines whether interference is constructive or destructive ### Path Difference and Interference - Path difference is measured in **multiples of wavelength** ($\lambda$) **Constructive Interference:** - Occurs when path difference = $n\lambda$ (where $n = 0, 1, 2, 3...$) - Example: path difference of $2\lambda$ or $5\lambda$ - Waves arrive in phase and reinforce each other **Destructive Interference:** - Occurs when path difference = $(n + \frac{1}{2})\lambda$ - Example: path difference of $\frac{3\lambda}{2}$ or $\frac{7\lambda}{2}$ - Waves arrive in anti-phase and cancel each other out

Constructive interference creates bright fringes; destructive interference creates dark fringes

## Phase Difference - Two waves with different path differences also have different **phase** - Phase difference is expressed as an angle in radians or degrees - Related to path difference through wavelength ### Phase Difference and Interference **Constructive Interference:** - Phase difference = even multiple of $\pi$ (e.g. $2\pi$, $4\pi$, $6\pi$) - Waves are **in phase** **Destructive Interference:** - Phase difference = odd multiple of $\pi$ (e.g. $\pi$, $3\pi$, $5\pi$) - Waves are **in anti-phase** ## Experimental Setup - Young's experiment uses a coherent light source to create an interference pattern on a screen - A monochromatic source (typically a laser) illuminates a single slit first - Light diffracts through the single slit, creating a coherent wavefront - This wavefront then passes through two narrow slits close together - Each double slit acts as a separate coherent source - The diffracted waves from both slits overlap and interfere - An interference pattern of alternating bright and dark fringes appears on the screen ### Why Use a Single Slit First? - The single slit ensures both double slits receive light from the **same original wavefront** - This guarantees the two sources are coherent - Without the single slit, the sources would not maintain constant phase difference ## The Interference Pattern - The pattern consists of **bright fringes** and **dark fringes** - Bright fringes occur where constructive interference happens - Dark fringes occur where destructive interference happens - The central bright fringe (zeroth order, $n = 0$) is the brightest - Brightness decreases as you move away from the centre - Monochromatic light produces sharp, clear fringes - Using visible light from a laser is common due to its coherence

White light produces a coloured pattern because different wavelengths interfere at different positions

### Intensity Distribution The graph below shows how light intensity varies across the screen, illustrating the characteristic pattern of bright and dark fringes:

- Notice the central maximum has the highest intensity - Subsidiary maxima decrease in intensity as distance from centre increases - Dark fringes have zero intensity where destructive interference occurs ## Historical Significance: Wave Nature of Light - Before Young's experiment, scientists debated whether light was a wave or a stream of particles - Newton proposed light consisted of tiny particles called **corpuscles** - However, particles cannot explain interference patterns - Young's experiment in 1801 demonstrated that light produces interference—a distinctly wave-like behaviour - This provided strong evidence for the **wave theory of light** - Later, Huygens' wave theory explained diffraction and refraction using wavelets - Each point on a wavefront acts as a source of secondary wavelets - These wavelets spread out at the same speed as the original wave ## Double-Slit Equation - The fringe spacing, wavelength, slit separation, and screen distance are related by: $$\lambda = \frac{ax}{D}$$ Where: | Symbol | Meaning | Unit | |--------|---------|------| | $\lambda$ | Wavelength of light | m | | $a$ | Distance between the centres of the double slits | m | | $x$ | Fringe width (separation between adjacent bright or dark fringes) | m | | $D$ | Distance from double slit to screen | m | ### Important Condition - This equation only applies when $a \ll D$ - In other words, the slit separation must be **much smaller** than the screen distance - Typical values: $a \approx 0.5$ mm, $D \approx 2$ m ### Interpreting the Equation - **Larger wavelength** → wider fringe spacing - **Smaller slit separation** → wider fringe spacing - **Greater screen distance** → wider fringe spacing - These relationships help determine unknown quantities experimentally ### Relationship: Fringe Width and Wavelength The graph below illustrates how fringe width increases linearly with wavelength (for constant $a = 0.3$ mm and $D = 2.0$ m):

- This linear relationship is why red light (longer wavelength) produces wider fringes than blue light - The gradient equals $\frac{D}{a}$ - Different colours produce different fringe spacings, which is why white light creates coloured patterns ## Worked Example: Calculating Slit Separation A green laser with frequency $5.5 \times 10^{14}$ Hz is directed at a double slit. The screen is positioned $3.2$ m from the slits. The distance between the second bright fringe on either side of the central maximum is measured as $28$ mm. Calculate the separation between the two slits. **Answer:** **Step 1: Calculate the wavelength using the wave equation** $$c = f\lambda$$ $$\lambda = \frac{c}{f} = \frac{3.0 \times 10^8}{5.5 \times 10^{14}} = 5.45 \times 10^{-7} \text{ m}$$ **Step 2: Determine the fringe width** - The measurement of $28$ mm is across **4 fringe widths** (from $n = -2$ to $n = +2$) - Therefore, one fringe width: $x = \frac{28}{4} = 7.0$ mm $= 7.0 \times 10^{-3}$ m **Step 3: Rearrange the double-slit equation for $a$** $$a = \frac{\lambda D}{x}$$ **Step 4: Substitute values** $$a = \frac{5.45 \times 10^{-7} \times 3.2}{7.0 \times 10^{-3}}$$ $$a = 2.49 \times 10^{-4} \text{ m} = 0.25 \text{ mm}$$

Always measure across multiple fringes and divide to reduce measurement uncertainty

## Key Differences from Progressive Waves - The double-slit experiment creates a **stationary interference pattern** - Unlike progressive waves that transfer energy, the pattern stores energy at fixed locations - Bright fringes represent regions of maximum intensity (constructive interference) - Dark fringes represent regions of zero intensity (destructive interference) - The pattern doesn't move—only the amplitude varies with position ## Practical Considerations ### Maximising Fringe Visibility - Use a monochromatic laser for the clearest pattern - Narrow slits increase diffraction and fringe visibility - Small slit separation $a$ increases fringe spacing - Large screen distance $D$ also increases fringe spacing - Darken the room to see faint fringes clearly ### Common Issues - If fringes are too close together, increase $D$ or decrease $a$ - If fringes are too dim, increase light intensity or use narrower slits - Ensure slits are parallel and evenly illuminated - Vibrations can blur the pattern—use a stable setup