Formula Sheet
Complete Reference
40+ Formulas
The Formula Bible for Wave Optics
Every equation you'll ever need for YDSE, diffraction, polarization, and interference - organized and explained.
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1. Interference Formulas
Path Difference and Phase Difference
$$\Delta x = \frac{\lambda}{2\pi}\Delta\phi$$
Where:
- $\Delta x$ = Path difference
- $\Delta\phi$ = Phase difference
- $\lambda$ = Wavelength of light
Condition for Constructive Interference
$$\Delta x = n\lambda \quad \text{or} \quad \Delta\phi = 2n\pi$$
Where:
- $n = 0, 1, 2, 3, \ldots$ (Order of maximum)
- Bright fringe condition
Condition for Destructive Interference
$$\Delta x = (2n+1)\frac{\lambda}{2} \quad \text{or} \quad \Delta\phi = (2n+1)\pi$$
Where:
- $n = 0, 1, 2, 3, \ldots$ (Order of minimum)
- Dark fringe condition
Resultant Intensity
$$I = I_1 + I_2 + 2\sqrt{I_1I_2}\cos\delta$$
Where:
- $I_1, I_2$ = Intensities of individual waves
- $\delta$ = Phase difference between waves
- For equal intensities: $I = 4I_0\cos^2\left(\frac{\delta}{2}\right)$
2. Young's Double Slit Experiment (YDSE)
Fringe Width
$$\beta = \frac{\lambda D}{d}$$
Where:
- $\beta$ = Fringe width (distance between consecutive bright/dark fringes)
- $\lambda$ = Wavelength of light
- $D$ = Distance between slits and screen
- $d$ = Distance between two slits
Position of nth Bright Fringe
$$y_n = \frac{n\lambda D}{d}$$
Where:
- $y_n$ = Distance of nth bright fringe from central maximum
- $n = 0, 1, 2, 3, \ldots$ (Order of fringe)
Position of nth Dark Fringe
$$y_n = \frac{(2n-1)\lambda D}{2d}$$
Where:
- $y_n$ = Distance of nth dark fringe from central maximum
- $n = 1, 2, 3, \ldots$ (Order of fringe)
Angular Position of Fringes
$$\theta_n \approx \frac{n\lambda}{d} \quad \text{(for small angles)}$$
Where:
- $\theta_n$ = Angular position of nth fringe
- Valid for $\theta < 10^\circ$
Path Difference at Point P on Screen
$$\Delta x = \frac{yd}{D}$$
Where:
- $y$ = Distance of point P from central maximum
- Valid for $y \ll D$
Effect of Placing Thin Film
$$\Delta x_{\text{film}} = (\mu - 1)t$$
Where:
- $\mu$ = Refractive index of film
- $t$ = Thickness of film
- Shift in fringes: $\Delta y = \frac{D}{d}(\mu - 1)t$
3. Diffraction Formulas
Single Slit Diffraction - Minima
$$a\sin\theta = n\lambda$$
Where:
- $a$ = Width of the slit
- $\theta$ = Angle of diffraction
- $n = \pm 1, \pm 2, \pm 3, \ldots$ (Order of minimum)
- Condition for destructive interference
Single Slit Diffraction - Intensity
$$I = I_0\left(\frac{\sin\beta}{\beta}\right)^2$$
Where:
- $I_0$ = Maximum intensity at central maximum
- $\beta = \frac{\pi a\sin\theta}{\lambda}$
- Central maximum at $\theta = 0$
Angular Width of Central Maximum
$$\theta = \frac{\lambda}{a}$$
Where:
- $\theta$ = Half angular width
- Full angular width = $2\theta = \frac{2\lambda}{a}$
Diffraction Grating - Principal Maxima
$$(a + b)\sin\theta = n\lambda$$
Where:
- $a$ = Width of slit
- $b$ = Width of opaque portion
- $a + b$ = Grating element
- $n = 0, 1, 2, 3, \ldots$ (Order of spectrum)
Resolving Power of Grating
$$R = \frac{\lambda}{\Delta\lambda} = nN$$
Where:
- $R$ = Resolving power
- $\Delta\lambda$ = Smallest wavelength difference that can be resolved
- $n$ = Order of spectrum
- $N$ = Total number of rulings on grating
Missing Orders in Double Slit
$$\frac{a + b}{a} = \frac{n}{m}$$
Where:
- $n$ = Order of interference maximum
- $m$ = Order of diffraction minimum
- When ratio is integer, that interference maximum is missing
4. Polarization Formulas
Brewster's Law
$$\mu = \tan i_p$$
Where:
- $\mu$ = Refractive index of medium
- $i_p$ = Polarizing angle (Brewster's angle)
- At this angle, reflected light is completely polarized
Malus' Law
$$I = I_0\cos^2\theta$$
Where:
- $I$ = Intensity of transmitted light
- $I_0$ = Intensity of incident polarized light
- $\theta$ = Angle between transmission axis and plane of polarization
Polarization by Reflection
$$i_p + r = 90^\circ$$
Where:
- $i_p$ = Angle of incidence (Brewster's angle)
- $r$ = Angle of refraction
- Reflected and refracted rays are perpendicular
Polarizing Angle and Refractive Index
$$\mu = \frac{\sin i_p}{\sin r} = \frac{\sin i_p}{\sin(90^\circ - i_p)} = \tan i_p$$
Where:
- Alternative derivation of Brewster's Law
- Using Snell's Law: $\mu = \frac{\sin i_p}{\sin r}$
📋 Quick Reference - Important Constants & Values
Common Wavelengths
- Red light: ~700 nm
- Green light: ~550 nm
- Blue light: ~450 nm
- Sodium D-line: 589.3 nm
Typical Values in Problems
- Slit separation (d): 0.1 - 1 mm
- Screen distance (D): 1 - 2 m
- Fringe width (β): 0.5 - 5 mm
- Wavelength (λ): 400 - 700 nm
🎯 Memory Tips & Tricks
⚡
Fringe Width Formula
Remember: $\beta = \frac{\lambda D}{d}$ - "Lambda D over d"
🔍
Bright vs Dark Fringes
Bright: $n\lambda$, Dark: $(2n-1)\frac{\lambda}{2}$ (for position from center)
📐
Brewster's Law
$\mu = \tan i_p$ - "The tangent of polarizing angle gives refractive index"
🎪
Malus' Law
$I = I_0\cos^2\theta$ - Intensity depends on cosine squared
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