Formula Bank & Dimensional Analysis

Every formula with conditions, dimensions, and when to apply

1. Basic Wave Properties

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c = νλ

Where:

c = speed of light in vacuum = 3 × 10⁸ m/s
ν = frequency (Hz)
λ = wavelength (m)

Dimensional Formula: [LT⁻¹] = [T⁻¹] × [L]

v = c/n = νλₘ

Where:

v = speed in medium
n = refractive index (dimensionless)
λₘ = wavelength in medium = λ/n

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Critical Point

Frequency never changes when light enters different medium. Only speed and wavelength change.

2. Interference - Young's Double Slit Experiment

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Path Difference & Phase Difference

Δx = S₂P - S₁P = (d sin θ) ≈ dy/D

Where:

Δx = path difference
d = distance between slits
D = distance from slits to screen
y = distance of point P from central maximum

Phase difference: δ = (2π/λ) × Δx

Dimensional Formula: δ is dimensionless (radians)

Conditions for Maxima and Minima

Bright Fringes (Maxima)

Δx = nλ
δ = 2nπ
y = nλD/d

Where n = 0, ±1, ±2, ±3, ...

Dark Fringes (Minima)

Δx = (2n + 1)λ/2
δ = (2n + 1)π
y = (2n + 1)λD/2d

Where n = 0, ±1, ±2, ±3, ...

Fringe Width

β = λD/d

Where: β = fringe width (distance between two consecutive bright or dark fringes)

Dimensional Formula: [L]

Units: Usually expressed in mm or cm

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Exam Insight

Fringe width β is independent of order n. All fringes (bright and dark) are equally spaced in YDSE. This is the most tested property.

Angular Fringe Width

θ = λ/d

Dimensional Formula: Dimensionless (radians)

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Strategy Tip

If problem asks for "angular position" or mentions "small angle approximation," use angular formulas. If it mentions screen distance D explicitly, use linear position formulas.

3. Intensity Distribution

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Resultant Intensity

I = I₁ + I₂ + 2√(I₁I₂) cos δ

Where:

I = resultant intensity at point P
I₁, I₂ = intensities from individual sources
δ = phase difference = (2π/λ)Δx

Special Cases

If I₁ = I₂ = I₀:
I = 4I₀ cos²(δ/2)

At Maxima

δ = 2nπ

I_max = 4I₀

At Minima

δ = (2n + 1)π

I_min = 0

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Conceptual Depth

I_max = 4I₀, not 2I₀. This factor of 4 comes from constructive interference: amplitudes add (A + A = 2A), and intensity ∝ amplitude² → (2A)² = 4A² = 4I₀. This confuses many students.

4. Single Slit Diffraction

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Condition for Minima

a sin θ = nλ

Where:

a = slit width
θ = angular position of nth minimum
n = 1, 2, 3, ... (n ≠ 0, as θ = 0 gives central maximum)

Width of Central Maximum

Angular width: 2θ = 2λ/a
Linear width: 2y = 2λD/a

Where: D = distance from slit to screen

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Common Mistake Alert

Central maximum width = 2λD/a (distance between first minima on both sides). Students often write λD/a which is only half-width. Factor of 2 is crucial.

Width of Secondary Maxima

Width of each secondary maximum = λD/a

Note: Secondary maxima are half the width of central maximum

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Exam Insight

Questions often ask: "How does central maximum width change if slit width is doubled?" Answer: Width ∝ 1/a, so doubling 'a' halves the width. Inversely proportional relationship.

5. Polarization

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Malus's Law

I = I₀ cos² θ

Where:

I = transmitted intensity
I₀ = incident polarized light intensity
θ = angle between transmission axes of polarizer and analyzer

θ = 0°

I = I₀

Maximum transmission

θ = 45°

I = I₀/2

Half intensity

θ = 90°

I = 0

Complete blocking

Brewster's Law

tan θ_B = n₂/n₁ = μ

Where:

θ_B = Brewster's angle (polarizing angle)
μ = refractive index of medium

At Brewster's angle: θ_B + θ_r = 90°

Where θ_r is angle of refraction

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Strategy Tip

For glass (n ≈ 1.5), Brewster's angle ≈ 56.3°. Memorize this value—it appears frequently in numerical problems as a checksum for your calculation.

🧮 Quick Calculator

Formula: Width = 2λD/a

Use fringe width calculator with 'a' instead of 'd' and multiply result by 2.

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