JEE Mains Physics · Formula sheet

Atoms formulas

8 formulas, 1 reference table and 29 common traps for JEE Mains Physics Atoms, grouped by subtopic.

Rutherford Scattering and Bohr's Postulates

Rutherford scattering and the distance of closest approach

Closest approach and impact parameter

r_0 = \frac{1}{4\pi\epsilon_0}\,\frac{2Ze^2}{K}, \qquad b = \frac{r_0}{2}\cot\frac{\theta}{2}

Bohr's postulates and quantised angular momentum

Bohr's quantisation and frequency condition

L = mvr = \frac{nh}{2\pi}, \qquad h\nu = E_{\text{upper}} - E_{\text{lower}}

Common traps

The alpha carries charge 2e

The potential energy at closest approach is k(2e)(Ze)/r₀. Writing ke²Z/r₀ drops the factor 2 and halves the distance, and that halved value is usually an option.

Convert MeV to joules before using k = 9 × 10⁹

1 MeV = 1.6 × 10⁻¹³ J. Mixing MeV with SI constants gives an answer off by a power of ten. Working in MeV fm with e²/4πε₀ = 1.44 MeV fm avoids the conversion.

Closest approach is an upper limit on the radius

The alpha stops before touching the nucleus, so the nuclear radius is at most r₀. Read whether the question wants a radius or a diameter: the diameter is twice the radius.

Rare rebounds are not caused by faster alphas

All alphas in the beam have the same energy. A few bounce back because the nucleus is tiny, so only a few come in with an impact parameter near zero.

A multiple of h/2π, not of h

Bohr's rule is L = nh/2π. A statement that says angular momentum is an integral multiple of h is false, even though h and L have the same dimensions.

Read n off L before anything else

If L = 5h/π, rewrite it as 10h/2π: the orbit is n = 10. Then use the energy or radius formula. Plugging L straight into an energy formula has no meaning.

Higher orbit minus lower orbit

Bohr's frequency condition is hν = E_upper − E_lower for both emission and absorption. Writing E_lower − E_upper gives a negative frequency.

Bohr's model is for one electron only

It works for H, He⁺, Li²⁺ and other hydrogen-like ions. It leaves out the repulsion between electrons, so it fails for neutral helium and heavier atoms.

Bohr Orbits: Radius, Speed and Energy

Learn this subtopic in the notes

Bohr radius and orbital speed in hydrogen-like ions

Bohr radius and speed

r_n = 0.529\,\frac{n^2}{Z}\ \text{Å}, \qquad v_n = 2.19 \times 10^{6}\,\frac{Z}{n}\ \text{m/s}

Energy levels and the kinetic–potential split

Energy of level n

E_n = -13.6\,\frac{Z^2}{n^2}\ \text{eV}, \qquad K = -E, \quad U = 2E

Common traps

Radius goes as n², speed as 1/n

Swapping the powers is the commonest slip. The radius of the third orbit is 9 times the first; the speed is one third of it.

A larger Z makes the orbit smaller

r = a₀n²/Z, so Li²⁺ orbits are a third the size of hydrogen's for the same n. The speed, on the other hand, is three times as large.

Build a derived quantity from r and v

Frequency is v/2πr, current is ef, magnetic moment is evr/2, field at the centre is μ₀I/2r. Write each in terms of n and Z step by step; guessing the power is where marks go.

Z is squared in the energy

E = −13.6 Z²/n² eV. Using Z instead of Z² gives He⁺ half its true ground-state energy. The radius has Z to the first power; the energy has Z².

Excited-state numbers are one behind n

The first excited state is n = 2 and the second excited state is n = 3. Reading second excited as n = 2 is wrong in every such question.

Kinetic energy falls as the electron moves out

Total and potential energy rise towards zero in a higher orbit, but the kinetic energy falls, because K = −E. Saying all three increase is a standard wrong option.

Spectral Series and Wavelength Ratios

Learn this subtopic in the notes

Ratios of two spectral lines

Rydberg formula and combining lines

\frac{1}{\lambda} = RZ^2\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right), \qquad \frac{1}{\lambda_{AC}} = \frac{1}{\lambda_{AB}} + \frac{1}{\lambda_{BC}}

Hydrogen spectral series, first lines and series limits

Series	Lower level	Region	Longest line (first member)	Shortest line (series limit)
Lyman	1	Ultraviolet	2 → 1: $4/3R$ , about 122 nm	∞ → 1: $1/R$ , about 91 nm
Balmer	2	Visible	3 → 2: $36/5R$ , about 656 nm	∞ → 2: $4/R$ , about 365 nm Balmer's first lines are visible, but its limit, 365 nm, is just into the ultraviolet.
Paschen	3	Infrared	4 → 3: $144/7R$ , about 1875 nm	∞ → 3: $9/R$ , about 820 nm
Brackett	4	Infrared	5 → 4: $400/9R$ , about 4050 nm	∞ → 4: $16/R$ , about 1458 nm
Pfund	5	Far infrared	6 → 5: $900/11R$ , about 7460 nm	∞ → 5: $25/R$ , about 2280 nm

For a hydrogen-like ion divide every wavelength by Z².

Common traps

The series limit is the shortest wavelength

The limit comes from n = ∞, the biggest jump into that level, so it carries the most energy and the shortest wavelength. The first member is the longest.

Each series has its own lower level

Only Lyman ends on n = 1. The Balmer limit is 4/R, not 1/R, and the Paschen limit is 9/R. Fix n_f from the series name before substituting.

Count members up from the longest line

The first Balmer member is 3 → 2, the second 4 → 2, the third 5 → 2. Counting down from the series limit gives the wrong jump.

Turn the bracket ratio upside down for wavelength

The bracket is 1/λ. A larger bracket means a shorter wavelength, so the wavelength ratio is the inverse of the bracket ratio. Energy, frequency and momentum ratios are not inverted.

Energies add; wavelengths do not

For levels A > B > C, E_AC = E_AB + E_BC. Adding the wavelengths instead gives a longer wavelength for the bigger jump, which is impossible.

Z cancels only within one ion

Two lines of the same ion share Z², so it cancels. The same jump in two different ions scales as Z², and forgetting it is off by a factor of 4 or 9.

Transition Energies, Excitation and X-rays

Learn this subtopic in the notes

Photon energy and wavelength of a transition

Energy and wavelength of a transition

\Delta E = 13.6\,Z^2\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)\ \text{eV}, \qquad \lambda\,(\text{nm}) = \frac{1240}{\Delta E\,(\text{eV})}

Excitation, absorption and the number of spectral lines

Lines from level n and the gap from the ground state

N = \frac{n(n-1)}{2}, \qquad E_n - E_1 = 13.6\,Z^2\left(1 - \frac{1}{n^2}\right)\ \text{eV}

Recoil of the emitting atom and X-ray photons

Photon momentum, recoil and the X-ray cut-off

p = \frac{E}{c} = \frac{h}{\lambda}, \qquad v_{\text{recoil}} = \frac{E}{Mc}, \qquad \lambda_{\min} = \frac{hc}{eV}

Common traps

Use the constants the question gives

hc = 1240 eV nm is the default, but a question that states h = 4 × 10⁻¹⁵ eV s means hc = 1200 eV nm, and its answer is built on that. Options are often 2–3% apart.

The gap scales as Z²

The same jump in He⁺ releases 4 times the energy it does in hydrogen, and in Li²⁺ 9 times. Leaving Z out treats every ion as hydrogen.

Big frequency means a jump near the ground state

The 2 → 1 gap is larger than any jump between higher levels, such as 5 → 4. The higher the levels, the closer they sit, the smaller the photon.

A photon must match; an electron need not

A 12.5 eV photon passes through ground-state hydrogen untouched, because no gap equals 12.5 eV. A 12.5 eV electron excites it to n = 3 and keeps the rest of its energy.

A sample and a single atom count differently

A sample excited to level n shows n(n − 1)/2 lines, because different atoms fall by different routes. One atom falling step by step gives at most n − 1 photons.

Balmer lines need n = 3 first

Lifting the atom to n = 2 is not enough: from there it can only fall to n = 1, a Lyman line. The least energy for any Balmer line is the gap to n = 3.

Recoil uses the atom's mass

The whole atom recoils, so divide the photon's momentum by the atom's mass, about 1.67 × 10⁻²⁷ kg for hydrogen. Dividing by the electron's mass gives a speed nearly 2000 times too large.

The cut-off wavelength ignores the target

λ_min = hc/eV depends only on the tube voltage. The target metal sets the characteristic lines such as Kα, not the cut-off.

Kα energy is a difference of two hole energies

The photon carries the energy of the K-hole state minus the energy of the L-hole state. Taking the K-hole energy alone as the photon energy is the usual slip.

Atoms formulas

Rutherford Scattering and Bohr's Postulates

Rutherford scattering and the distance of closest approach

Bohr's postulates and quantised angular momentum

Common traps

Bohr Orbits: Radius, Speed and Energy

Bohr radius and orbital speed in hydrogen-like ions

Energy levels and the kinetic–potential split

Common traps

Spectral Series and Wavelength Ratios

Ratios of two spectral lines

Hydrogen spectral series, first lines and series limits

Common traps

Transition Energies, Excitation and X-rays

Photon energy and wavelength of a transition

Excitation, absorption and the number of spectral lines

Recoil of the emitting atom and X-ray photons

Common traps

More JEE Mains Physics formula sheets