Formula Bank & Dimensional Analysis

🎯 How to Use This

Don't just memorize formulas. Understand:

When to use each formula
What each symbol means physically
How to check dimensionally if answer is correct

In JEE, knowing the right formula for the right situation is half the battle.

Core Formulas

1. Planck's Quantum Hypothesis

E = hν = hc/λ

Description: Energy of a photon

Variables:

E = Energy (Joules)
h = Planck's constant = 6.626 × 10⁻³⁴ J·s
ν = Frequency (Hz)
c = Speed of light = 3 × 10⁸ m/s
λ = Wavelength (m)

Dimensions: [E] = [ML²T⁻²]

Use when: Finding photon energy from frequency/wavelength

2. Photon Momentum

p = E/c = h/λ = hν/c

Description: Momentum of a photon (even though rest mass is zero)

Variables:

p = Momentum (kg·m/s)
E = Energy (J)
c = Speed of light

Dimensions: [p] = [MLT⁻¹]

Use when: Compton effect, radiation pressure problems

3. Einstein's Photoelectric Equation

KE_max = hν - φ

OR: (1/2)mv²_max = hν - φ

Description: Maximum kinetic energy of photoelectrons

Variables:

KE_max = Maximum kinetic energy (J)
φ = Work function (J or eV)
m = Mass of electron = 9.1 × 10⁻³¹ kg
v_max = Maximum velocity of photoelectron

Critical Point: ν must be ≥ ν₀ (threshold frequency)

❌ Common Mistake Alert

Students write: KE = hν - φ (without "max" subscript)

Why wrong: Not all electrons have this KE. Only surface electrons get maximum KE.

Always write KE_max to show you understand the concept.

4. Work Function

φ = hν₀ = hc/λ₀

Description: Minimum energy to remove electron from metal

Variables:

ν₀ = Threshold frequency (Hz)
λ₀ = Threshold wavelength (m)

Key Insight: Different for different metals

Typical Values: 2-5 eV for most metals

5. Stopping Potential

eV₀ = KE_max = hν - φ

V₀ = (hν - φ)/e = (h/e)ν - φ/e

Description: Minimum potential to stop photoelectrons

Variables:

V₀ = Stopping potential (Volts)
e = Charge of electron = 1.6 × 10⁻¹⁹ C

Graph: V₀ vs ν is a straight line with slope h/e

6. de Broglie Wavelength (General)

λ = h/p = h/(mv)

Description: Wavelength associated with any moving particle

Variables:

λ = de Broglie wavelength (m)
p = momentum (kg·m/s)
m = mass of particle (kg)
v = velocity (m/s)

Dimensions: [λ] = [L]

7. de Broglie Wavelength (Accelerated Charge)

λ = h/√(2mqV)

Description: Wavelength of charged particle accelerated through potential V

Variables:

q = charge of particle (C)
V = accelerating potential (V)

For electron: λ = 12.27/√V Å (V in volts)

For proton: λ = 0.286/√V Å (V in volts)

🔬 Exam Insight

JEE trick question: "Electron and proton accelerated through same potential. Compare wavelengths."

Answer: λ ∝ 1/√m, so λ_electron/λ_proton = √(m_proton/m_electron) ≈ √1836 ≈ 43

Electron wavelength is 43 times larger.

8. de Broglie Wavelength (Kinetic Energy)

λ = h/√(2mKE)

Description: When kinetic energy is given directly

Derivation: KE = (1/2)mv² = p²/(2m), so p = √(2m·KE)

Use when: KE is given, not velocity or potential

Comparative Formulas

Photon vs Matter Wave

Property	Photon	Matter Wave (e.g., electron)
Rest Mass	Zero	Non-zero (9.1 × 10⁻³¹ kg for electron)
Speed	c (always)	v < c
Energy	E = hν = pc	E = (1/2)mv² (non-relativistic)
Momentum	p = h/λ = E/c	p = mv = h/λ
Wavelength	λ = c/ν = h/p	λ = h/p = h/(mv)

Dimensional Analysis

🧠 Why Dimensional Analysis?

In exam, you derived a complex formula. How do you know if it's correct?

Check dimensions. If LHS ≠ RHS dimensionally, formula is WRONG.

This has saved countless marks in JEE Advanced.

Key Dimensional Formulas

Quantity	Symbol	Dimensions	SI Unit
Planck's Constant	h	[ML²T⁻¹]	J·s
Energy	E	[ML²T⁻²]	J (Joule)
Frequency	ν	[T⁻¹]	Hz
Wavelength	λ	[L]	m
Momentum	p	[MLT⁻¹]	kg·m/s
Work Function	φ	[ML²T⁻²]	J or eV
Potential	V	[ML²T⁻³A⁻¹]	V (Volt)

Verification Example

Verify: λ = h/√(2meV)

LHS: [λ] = [L]

RHS:

[h] = [ML²T⁻¹]
[m] = [M]
[e] = [AT]
[V] = [ML²T⁻³A⁻¹]

[h/√(2meV)] = [ML²T⁻¹]/√([M][AT][ML²T⁻³A⁻¹])
= [ML²T⁻¹]/√[M²L²T⁻²]
= [ML²T⁻¹]/[MLT⁻¹]
= [L]

✓ LHS = RHS. Formula is dimensionally correct!

Unit Conversions (Critical for Problem Solving)

Energy Units

1 eV = 1.6 × 10⁻¹⁹ J
1 keV = 10³ eV
1 MeV = 10⁶ eV

Length Units

1 Å (Angstrom) = 10⁻¹⁰ m
1 nm = 10⁻⁹ m = 10 Å

Useful Constants

h = 6.626 × 10⁻³⁴ J·s = 4.14 × 10⁻¹⁵ eV·s
hc = 1240 eV·nm (extremely useful!)
e = 1.6 × 10⁻¹⁹ C
m_e = 9.1 × 10⁻³¹ kg
c = 3 × 10⁸ m/s

🎯 Strategy Tip

Memorize: hc = 1240 eV·nm

This single value solves 80% of photoelectric effect problems in seconds.

Example: Find energy of 400nm photon:

E = hc/λ = 1240/400 = 3.1 eV

Done in 5 seconds. Others will calculate for 2 minutes.

Mastered the Formulas?

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