Atoms & Nuclei Formula Sheet — JEE Main Physics

Q: What are the most important Atoms & Nuclei formulas for JEE Main?

This Atoms & Nuclei formula sheet covers all the high-yield Physics formulas, definitions and theorems you need for JEE Main, across Alpha-particle scattering experiment, Rutherford's model of atom, Bohr model, Energy levels, Hydrogen spectrum — each shown with the key result and, where useful, a worked example.

Q: Is this Atoms & Nuclei formula sheet free?

Yes — the full chapter formula sheet is free to read online, no login or payment required.

Q: How should I revise Atoms & Nuclei formulas?

Blurt the Atoms & Nuclei formulas from memory, then check against this sheet to find your gaps — and practise a few previous-year questions on the chapter to make sure you can apply them under time pressure.

Every key Atoms & Nuclei formula, definition and theorem for JEE Main Physics in one place — with common examiner traps and worked examples. Free to read; blurt from memory, then check your gaps.

Syllabus — topics coveredNTA · 16 sub-topics

Alpha-particle scattering experiment
Rutherford's model of atom
Bohr model
Energy levels
Hydrogen spectrum
Composition and size of nucleus
Atomic masses
Isotopes, isobars, isotones
Radioactivity
Alpha, beta and gamma particles/rays and their properties
Radioactive decay law
Mass-energy relation
Mass defect
Binding energy per nucleon and its variation with mass number
Nuclear fission
Nuclear fusion

Atomic Models & Rutherford Scattering

Thomson to Rutherford

▸ $Thomson$ (plum-pudding): $+$ charge spread out, electrons embedded — fails large-angle scatter.
▸ $Rutherford$ (nuclear): all $+$ charge & most mass in a tiny central nucleus; electrons orbit.

Distance of closest approach

d = \frac{1}{4 π ε _{0}} \frac{2 Z e ^{2}}{K}

head-on: all KE

K \to

electric PE;

α =_{2}^{4}

He (

+ 2 e

); sets an upper limit on nuclear size. Impact parameter for scattering angle

θ

b = \frac{1}{4 π ε _{0}} \frac{Z e ^{2} cot ( θ /2 )}{K}

(smaller

b \Rightarrow

larger

θ

;

b = 0

is the head-on rebound)

Geiger-Marsden experiment: most alpha particles pass undeviated, a few deflect through large angles, about 1 in 8000 rebounds — pointing to a small dense positive nucleus — Impact parameter b fixes the scattering angle $θ$ .

Observations

▸Most $α$ pass nearly undeviated $\Rightarrow$ atom is mostly empty.
▸ $\sim 1$ in $8000$ scatter by $> 9 0^{\circ}$ $\Rightarrow$ small, dense, $+$ nucleus.
▸Nucleus $\sim 1 0^{- 15}$ m vs atom $\sim 1 0^{- 10}$ m.

🚫 Examiner Trap · Rutherford model

(1) Large-angle scatter (not the average) proves the nucleus is tiny & dense — most

α

feel almost nothing. (2)

d = 2 Z e^{2} /4 π ε_{0} K

is an

upper

limit on size, found by

K =

PE at closest approach. (3) Classical orbits FAIL: an accelerating electron must radiate, spiral in, and give a continuous spectrum — atoms don't.

Bohr Model of the Hydrogen Atom

Bohr's three postulates

▸Electrons revolve in $stationary orbits$ without radiating.
▸Angular momentum quantised: $L = mvr = \frac{nh}{2 π}$ .
▸Photon emitted/absorbed on a jump: $h ν = E_{i} - E_{f}$ .

Comparative: how quantities scale (

n

Z

)

Quantity	Formula	Scaling
Radius $r_{n}$	$\frac{n ^{2}}{Z} a_{0}$ ( $a_{0} = 0.529$ Å)	$\propto n^{2} / Z$
Speed $v_{n}$	$\frac{c}{137} \frac{Z}{n}$	$\propto Z / n$
Energy $E_{n}$	$- \frac{13.6 Z ^{2}}{n ^{2}}$ eV	$\propto Z^{2} / n^{2}$
Period $T$	—	$\propto n^{3} / Z^{2}$

Hydrogen energy-level diagram drawn to scale: levels at -13.6, -3.40, -1.51 eV converging to 0 eV at n=infinity; ionisation needs 13.6 eV from the ground state — Levels crowd toward $0$ eV as $n \to \infty$ .

Energy bookkeeping

K = - E_{n} = \frac{13.6 Z ^{2}}{n ^{2}} eV, U = 2 E_{n} = - 2 K

total

E = - K

; potential energy is twice the total

More Bohr results

▸Orbital frequency $f = \frac{v}{2 π r} \propto \frac{Z ^{2}}{n ^{3}}$ ; equivalent current $I = e f$ ; magnetic moment $μ = \frac{n e h}{4 πm} = n μ_{B}$ ( $μ_{B} = 9.27 \times 1 0^{- 24}$ A $m^{2}$ ).
▸Atom recoils when it emits: photon momentum $h ν / c = M v$ , so $v_{reco i l} = \frac{h ν}{M c}$ and recoil KE $= \frac{( h ν ) ^{2}}{2 M c ^{2}}$ (tiny).
▸Hydrogen-like systems with a heavy 'electron' (muonic atom, positronium): replace m by the reduced mass $μ = \frac{m M}{m + M}$ ; $E_{n} \propto μ$ , $r_{n} \propto \frac{1}{μ}$ (positronium: $E_{1} = - 6.8$ eV).

🚫 Examiner Trap · Bohr model

(1) Bohr works ONLY for

one-electron

(H-like) species — use

Z^{2}

for He

^{+}

, L

i^{2 +}

. (2)

E_{n} = - 13.6 Z^{2} / n^{2}

negative

(bound); ionisation energy

= + 13.6 Z^{2}

from

n = 1

. (3)

K = - E

U = 2 E

— the signs trip people up. (4) de Broglie standing wave

2 π r_{n} = nλ

gives the quantisation.

Hydrogen Spectrum & Spectral Series

Rydberg formula

\frac{1}{λ} = R Z^{2} (\frac{1}{n _{f}^{2}} - \frac{1}{n _{i}^{2}}), R = 1.097 \times 1 0^{7} m^{- 1}

n_{i} > n_{f}

(emission);

h c / λ = E_{i} - E_{f}

Series	$n_{f}$	$n_{i}$	Region
Lyman	1	2,3,…	UV
Balmer	2	3,4,…	Visible
Paschen	3	4,5,…	IR
Brackett	4	5,6,…	IR
Pfund	5	6,7,…	IR

Series limit & line count

\frac{1}{λ _{min}} = \frac{R Z ^{2}}{n _{f}^{2}} (n_{i} \to \infty); # lines = \frac{n ( n - 1 )}{2}

max lines when electrons fall from level

n

Schematic hydrogen energy levels with downward transitions: to n=1 give the Lyman (UV) series, to n=2 the Balmer (visible) series, to n=3 the Paschen (IR) series — Each landing level $n_{f}$ defines one series.

⚡ Shortcut · Franck–Hertz & X-rays

H: ionisation

= 13.6

eV, first excitation

E_{2} - E_{1} = 10.2

eV. X-ray cutoff

λ_{min} = \frac{12400}{V ( volt )}

Å; Moseley

ν = a (Z - b)

🚫 Examiner Trap · Hydrogen spectrum

(1)

Balmer

is the only series in the

visible

; Lyman is UV (to

n = 1

), Paschen+ are IR. (2) Number of spectral lines from level n is

n (n - 1) /2

, NOT n. (3) Largest

λ

in a series

=

smallest energy gap (

n_{f} + 1 \to n_{f}

); series limit is the shortest

λ

. (4) Use

n_{i} > n_{f}

1/ λ > 0

The Nucleus — Composition & Size

Nuclide

_{Z}^{A} X

Z =

protons (atomic number),

N =

neutrons,

A = Z + N =

mass number (total

nucleons

Comparative: nuclear families — not in JEE Main 2026 syllabus

Family	Same	Example
Isotopes	$Z$	$_{1}^{1}$ H, $_{1}^{2}$ H, $_{1}^{3}$ H
Isobars	$A$	$_{1}^{3}$ H, $_{2}^{3}$ He
Isotones	$N$	$_{79}^{197}$ Au, $_{80}^{198}$ Hg

Particle masses

m_{p} = 1.00728 u, m_{n} = 1.00867 u, m_{e} = 0.00055 u

free neutron unstable (

τ \approx 1000

s):

n \to p + e^{-} + \overset{ν}{ˉ}

Nuclear radius & density

R = R_{0} A^{1/3} (R_{0} = 1.2 fm); ρ \approx 2.3 \times 1 0^{17} kg/m^{3}

V \propto R^{3} \propto A \Rightarrow

density is the

same for all nuclei

Plot of nuclear radius R versus A to the one-third power: a straight line through the origin with slope R0=1.2 fm, illustrating constant nuclear density — $R \propto A^{1/3} \Rightarrow$ constant nuclear density.

Mass–energy unit

1 u = \frac{1}{12} m (_{6}^{12} C) = 1.66 \times 1 0^{- 27} kg = 931.5 MeV / c^{2}

Einstein

E = m c^{2}

🚫 Examiner Trap · Nucleus composition & size

(1) Nuclear density is

constant

(

\sim 2.3 \times 1 0^{17}

kg/m³) — independent of A (since

R \propto A^{1/3}

). (2) Isotopes(same Z)/isobars(same A)/isotones(same N) — don't confuse. (3) Convert mass defect via

1 u = 931.5

MeV. (4) The nucleus holds

> 99.9%

of the atom's mass in

\sim 1 0^{- 15}

of its volume.

Mass Defect, Binding Energy & Nuclear Force

Mass defect & binding energy

Δ M = [Z m_{p} + (A - Z) m_{n}] - M_{n u c}; E_{b} = Δ M c^{2} = Δ M (u) \times 931.5 MeV

nucleus is always lighter than its free nucleons;

E_{b} =

energy to break it apart

Binding energy per nucleon

E_{bn} = \frac{E _{b}}{A}

higher

E_{bn} \Rightarrow

more stable

Binding energy per nucleon versus mass number: rises steeply, peaks about 8.75 MeV near iron-56, then slowly falls; heavy nuclei release energy by fission, light nuclei by fusion — Peak $\approx 8.75$ MeV at $^{56}$ Fe (most stable).

Potential energy of two nucleons versus separation: strongly repulsive below r0 about 0.8 fm, a minimum at r0, attractive beyond, falling to zero within a few femtometres — Attractive for $r > r_{0}$ , repulsive for $r < r_{0} \approx 0.8$ fm.

Curve & force

▸ $E_{bn} \approx 8$ MeV, nearly flat for $30 < A < 170$ ; low for very light & very heavy nuclei.
▸Nuclear force: strongest known, $short-range$ (few fm), $saturates$ , $charge-independent$ .

🚫 Examiner Trap · Mass defect & BE

(1) The nucleus is

lighter

than its parts — the 'missing' mass is the binding energy (

Δ M c^{2}

). (2) Stability is set by

E_{b} / A

, NOT total

E_{b}

(a big unstable nucleus can have large

E_{b}

). (3) Both fission (heavy) and fusion (light) move toward the

^{56}

Fe peak and

release

energy. (4) Nuclear force saturates

\Rightarrow

E_{bn}

roughly constant.

Radioactivity — Decay Law & Half-life

★ Remember · Radioactivity — not in JEE Main 2026 syllabus

The 2026 Unit 18 syllabus stops at binding energy, fission and fusion; radioactivity, α/β/γ rays and the decay law were dropped. This page is kept because older papers and practice questions still use it.

Radioactivity: Spontaneous decay of an unstable nucleus (Becquerel, 1896) emitting

α

β

γ

. Independent of temperature, pressure or chemical state.

Decay law & activity

N = N_{0} e^{- λ t}; R = λ N = R_{0} e^{- λ t}

λ =

decay constant;

1 Bq = 1

decay/s,

1 C i = 3.7 \times 1 0^{10}

Half-life & mean life

T_{1/2} = \frac{0.693}{λ}, τ = \frac{1}{λ} = \frac{T _{1/2}}{0.693}

T_{1/2} = τ ln 2

;

τ > T_{1/2}

After

n

half-lives

N = N_{0} (\frac{1}{2})^{t / T_{1/2}}

fraction left

= (1/2)^{n}

n = t / T_{1/2}

Exponential decay curve: the number of undecayed nuclei halves every half-life, dropping to N0/2, N0/4, N0/8, N0/16 at one, two, three and four half-lives — Population halves each $T_{1/2}$ .

✎ Example · Fraction remaining

Tritium

T_{1/2} = 12.5

y. Fraction left after

25

$n = 25/12.5 = 2$ half-lives
fraction $= (1/2)^{2}$

∴

1/4

remains

🚫 Examiner Trap · Radioactive decay

(1) Decay is

exponential

, never linear — use

(1/2)^{n}

only when t is a whole number of half-lives, else

N_{0} e^{- λ t}

. (2) Activity

R = λ N

also decays with the SAME

T_{1/2}

. (3) Mean life

τ = 1/ λ = 1.44 T_{1/2}

(

> T_{1/2}

). (4)

λ

is independent of temperature/pressure/chemistry.

Decay Modes, Fission & Fusion

Comparative: decay modes — not in JEE Main 2026 syllabus

Decay	Emitted	$Z$	$A$
Alpha $α$	$_{2}^{4}$ He	$- 2$	$- 4$
Beta $^{-}$	$e^{-} + \overset{ν}{ˉ}$	$+ 1$	$0$
Beta $^{+}$	$e^{+} + ν$	$- 1$	$0$
Gamma $γ$	photon	$0$	$0$

Q-value

Q = (m_{reactants} - m_{products}) c^{2}

Q > 0

releases energy (fission, fusion, decay); shared as KE of the products. With binding energies:

Q = \sum E_{b} (products) - \sum E_{b} (reactants)

Soddy–Fajans (decay, not in 2026 syllabus)

▸ $α \to$ shift 2 left, mass $- 4$ ; $β^{-} \to$ 1 right; $β^{+} \to$ 1 left; $γ \to$ no shift.

Left: a neutron splits U-235 into two fragments plus three neutrons, releasing about 200 MeV. Right: deuterium and tritium fuse into helium-4 plus a neutron, requiring high temperature — Both fission and fusion move toward the BE-curve peak.

Fission

_{0}^{1} n +_{92}^{235} U \to_{56}^{144} Ba +_{36}^{89} Kr + 3_{0}^{1} n (\sim 200 MeV)

\sim 2.5

neutrons/fission

\to

chain reaction (reactor controlled, bomb not)

🎯 Exam · Reactor / fuel arithmetic

Fissions per second

= \frac{P}{E _{p er f i ss i o n}}

(convert MeV

\to

1 M e V = 1.6 \times 1 0^{- 13}

J). Mass of U-235 used per second

= \frac{fissions/s}{N _{A}} \times 0.235

kg.

1

kg of U-235 fully fissioned gives

\approx 8 \times 1 0^{13}

🚫 Examiner Trap · Decay, fission & fusion

(1) [decay, not in 2026 syllabus]

β^{-}

raises Z by 1 (

n \to p

);

β^{+}

lowers it — opposite directions. (2) [decay]

γ

changes neither Z nor A (just de-excitation). (3) Fusion needs

very high T

to beat Coulomb repulsion (Sun:

4_{1}^{1} H \to_{2}^{4} He + 26.7

MeV). (4) Per nucleon, fusion releases MORE energy than fission, though fission releases more per event.

More JEE Main Physics formula sheets

Frequently Asked Questions

What are the most important Atoms & Nuclei formulas for JEE Main?

This Atoms & Nuclei formula sheet covers all the high-yield Physics formulas, definitions and theorems you need for JEE Main, across Alpha-particle scattering experiment, Rutherford's model of atom, Bohr model, Energy levels, Hydrogen spectrum — each shown with the key result and, where useful, a worked example.

Is this Atoms & Nuclei formula sheet free?

Yes — the full chapter formula sheet is free to read online, no login or payment required.

How should I revise Atoms & Nuclei formulas?

Blurt the Atoms & Nuclei formulas from memory, then check against this sheet to find your gaps — and practise a few previous-year questions on the chapter to make sure you can apply them under time pressure.

Also useful: all formula sheetsJEE Main previous-year papersmost important chapters.