PhysicsISC Class 12 · Atoms

Bohr's Model of the Hydrogen Atom: Postulates, Orbit Radius and Electron Speed

Updated 15 October 2026 · Fact-checked

Bohr assumed the electron moves in circular orbits where its angular momentum is $mvr = \frac{nh}{2\pi}$, with the Coulomb force providing the centripetal force. Solving gives $r_n = \frac{\varepsilon_0 n^2h^2}{\pi me^2}$ and $v_n = \frac{e^2}{2\varepsilon_0 nh}$ for hydrogen, so $r_n \propto n^2$ and $v_n \propto \frac{1}{n}$, with $r_1 \approx 0.053$ nm.

Understand Bohr's Model: Radius and Speed of Orbits

Why a new model was needed. In Rutherford's model an electron circling the nucleus is accelerating, so classical theory says it should radiate continuously, spiral inwards and collapse, giving a continuous spectrum. Atoms are stable and hydrogen gives a line spectrum, so Bohr (1913) added quantum ideas.

Bohr's postulates (for hydrogen, Z = 1).
1. The electron revolves in certain circular orbits, called stationary orbits, without radiating energy.
2. In these orbits its angular momentum is an integral multiple of $\frac{h}{2 π}$ : $m v r = \frac{nh}{2 π}$ , n = 1, 2, 3, ... (n is the principal quantum number).
3. Radiation is emitted or absorbed only when the electron jumps between orbits: $h ν = E_{i} - E_{f}$ .

Derivation of the radius (asked in ISC 2024 Q19(i)(a) and 2025 Q19(i)(a)). The electrostatic attraction provides the centripetal force:
$\frac{m v ^{2}}{r} = \frac{1}{4 π ε _{0}} \frac{e ^{2}}{r ^{2}} \Rightarrow m v^{2} r = \frac{e ^{2}}{4 π ε _{0}} . (1)$
Quantisation: $v = \frac{nh}{2 π m r}$ . (2)
Substituting (2) in (1):
$m (\frac{nh}{2 π m r})^{2} r = \frac{e ^{2}}{4 π ε _{0}} \Rightarrow \frac{n ^{2} h ^{2}}{4 π ^{2} m r} = \frac{e ^{2}}{4 π ε _{0}},$
so
$r_{n} = \frac{ε _{0} n ^{2} h ^{2}}{π m e ^{2}} .$
Everything except n is constant, so $r_{n} \propto n^{2}$ . For n = 1 this gives the Bohr radius $r_{1} \approx 5.3 \times 1 0^{- 11}$ m = 0.053 nm, and $r_{n} = 0.053 n^{2}$ nm. (For a nucleus of charge Ze, $e^{2}$ becomes $Z e^{2}$ ; the syllabus uses Z = 1.)

Speed in the nth orbit. From (2), $v_{n} = \frac{nh}{2 π m r _{n}}$ ; substituting $r_{n}$ :
$v_{n} = \frac{e ^{2}}{2 ε _{0} nh}, v_{n} \propto \frac{1}{n},$
with $v_{1} \approx 2.19 \times 1 0^{6}$ m/s, about $\frac{c}{137}$ (slow enough to ignore relativity).

Other results.

Angular momentum $L_{n} = \frac{nh}{2 π} \propto n$ .
Time period $T_{n} = \frac{2 π r _{n}}{v _{n}} \propto n^{3}$ ; frequency of revolution $\propto \frac{1}{n ^{3}}$ .
Kinetic energy $K = \frac{1}{2} m v^{2} = \frac{e ^{2}}{8 π ε _{0} r}$ ; potential energy $U = - \frac{e ^{2}}{4 π ε _{0} r} = - 2 K$ ; total energy $E = - K = - \frac{e ^{2}}{8 π ε _{0} r}$ , which gives $E_{n} = - \frac{13.6}{n ^{2}}$ eV (developed in the energy-levels topic).

De Broglie's picture. Quantisation follows if the electron's matter wave fits the orbit as a standing wave: $2 π r_{n} = nλ$ with $λ = \frac{h}{m v}$ , which rearranges to $m v r = \frac{nh}{2 π}$ .

Bohr's Model: Radius and Speed of Orbits: formulas and key facts

Quantisation of angular momentum: $m v r = \frac{nh}{2 π}$; n = 1, 2, 3, ...
Force balance: $\frac{m v ^{2}}{r} = \frac{1}{4 π ε _{0}} \frac{e ^{2}}{r ^{2}}$; Coulomb force is centripetal.
Radius of nth orbit: $r_{n} = \frac{ε _{0} n ^{2} h ^{2}}{π m e ^{2}} = 0.053 n^{2} nm$; r ∝ n².
Speed in nth orbit: $v_{n} = \frac{e ^{2}}{2 ε _{0} nh} = \frac{2.19 \times 1 0 ^{6}}{n} m/s$; v ∝ 1/n.
Time period: $T_{n} = \frac{2 π r _{n}}{v _{n}} \propto n^{3}$; T₁ ≈ 1.52 × 10⁻¹⁶ s.
Energies: $K = - E, U = 2 E$; E = −13.6/n² eV.

How to solve Bohr's Model: Radius and Speed of Orbits questions

Most Bohr-orbit numericals are scaling problems from the ground state.

1Note the scaling: r ∝ n², v ∝ 1/n, L ∝ n, T ∝ n³, E ∝ 1/n².
2Use $r_{1} = 0.053$ nm and $v_{1} = 2.19 \times 1 0^{6}$ m/s unless the paper asks you to compute from constants.
3For derivations: write force balance, write quantisation, eliminate v, solve for r; then put Z = 1.
4For current due to the orbiting electron: $I = \frac{e}{T} = \frac{e v}{2 π r}$ .
5Give the final expression and the proportionality the question asks for (e.g. "hence $r_{n} \propto n^{2}$ ").

Bohr's Model: Radius and Speed of Orbits in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Giant atoms in space

Suppose radio astronomers detect hydrogen atoms in interstellar space whose electrons sit in the n = 100 orbit (so-called Rydberg atoms).

Estimate the orbit radius.

Try it first, then show the solutionSolution

$r_{100} = 0.053 \times 10 0^{2}$ nm $= 530$ nm.
That is ten thousand times the ground-state radius and comparable to the wavelength of visible light; such atoms survive only in the near-empty space between stars.

Answer: About 530 nm.

Memory hook: Radius grows as n squared: n = 100 means 10 000 times bigger.

Is the electron relativistic?

Imagine checking whether Bohr was right to use Newtonian mechanics for the electron in hydrogen's ground state.

Compare $v_{1}$ with the speed of light.

Try it first, then show the solutionSolution

$\frac{v _{1}}{c} = \frac{2.19 \times 1 0 ^{6}}{3 \times 1 0 ^{8}} \approx 7.3 \times 1 0^{- 3} \approx \frac{1}{137}$ .
Less than 1% of c, so relativistic effects are small; higher orbits are slower still.

Answer: About c/137: Newtonian mechanics is a good approximation.

Memory hook: One over one-three-seven.

Tricks to remember Bohr's Model: Radius and Speed of Orbits

Short memory aids. Say each one aloud once, then use it on a question the same day.

n-squared radius, n-inverse speed
r ∝ n², v ∝ 1/n, so L = mvr ∝ n.
Use it when: Scaling questions.
53 picometres
Bohr radius 0.053 nm = 53 pm.
Use it when: Numerical starting point.
Force, quantise, eliminate
Derivation in three moves: Coulomb = centripetal; mvr = nh/2π; remove v.
Use it when: Structuring the 3-mark derivation.
Kinetic is minus total
K = −E, U = 2E.
Use it when: Energy relations.

Bohr's Model: Radius and Speed of Orbits: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Speed in the second orbit · 1 mark · MCQBoard level

The speed of the electron in the second Bohr orbit of hydrogen, compared with the first, is:
(a) twice
(b) half
(c) four times
(d) one quarter

Try it first, then show the solutionSolution

$v_{n} \propto \frac{1}{n}$ : $\frac{v _{2}}{v _{1}} = \frac{1}{2}$ .

Answer: (b).

Example 2: The fourth orbit · 2 marks · Short answerBoard level

Find the radius of the fourth Bohr orbit of hydrogen and the electron's speed in it. ( $r_{1}$ = 0.053 nm, $v_{1} = 2.19 \times 1 0^{6}$ m/s)

Try it first, then show the solutionSolution

$r_{4} = 4^{2} \times 0.053 = 0.848$ nm. (1)
$v_{4} = \frac{2.19 \times 1 0 ^{6}}{4} \approx 5.48 \times 1 0^{5}$ m/s. (1)

Answer: 0.848 nm; ≈ $5.48 \times 1 0^{5}$ m/s.

Example 3: Derivation of the radius and speed · 4 marks · Long answerHarder board level

On the basis of Bohr's theory, derive expressions for the radius of the nth orbit and the speed of the electron in it, for a hydrogen atom. Show that $r_{n} \propto n^{2}$ .

Try it first, then show the solutionSolution

Centripetal force from Coulomb attraction: $\frac{m v ^{2}}{r} = \frac{e ^{2}}{4 π ε _{0} r ^{2}}$ , so $m v^{2} r = \frac{e ^{2}}{4 π ε _{0}}$ . (1)
Quantisation: $m v r = \frac{nh}{2 π}$ , so $v = \frac{nh}{2 π m r}$ . (1)
Substituting: $\frac{n ^{2} h ^{2}}{4 π ^{2} m r} = \frac{e ^{2}}{4 π ε _{0}}$ , giving $r_{n} = \frac{ε _{0} n ^{2} h ^{2}}{π m e ^{2}}$ ; all else constant, so $r_{n} \propto n^{2}$ . (1)
Then $v_{n} = \frac{nh}{2 π m r _{n}} = \frac{e ^{2}}{2 ε _{0} nh}$ , so $v_{n} \propto \frac{1}{n}$ . (1)

Answer: $r_{n} = \frac{ε _{0} n ^{2} h ^{2}}{π m e ^{2}}$ ; $v_{n} = \frac{e ^{2}}{2 ε _{0} nh}$ .

Example 4: Angular momentum in n = 2 · 2 marks · Short answerHarder board level

Find the angular momentum of the electron in the second orbit of hydrogen, and show that the product of $m$ , $v_{2}$ and $r_{2}$ gives the same value. ( $h = 6.63 \times 1 0^{- 34}$ J s, $m = 9.1 \times 1 0^{- 31}$ kg, $r_{2}$ = 0.212 nm, $v_{2} = 1.095 \times 1 0^{6}$ m/s)

Try it first, then show the solutionSolution

$L = \frac{2 h}{2 π} = \frac{h}{π} \approx 2.11 \times 1 0^{- 34}$ J s. (1)
$m v r = 9.1 \times 1 0^{- 31} \times 1.095 \times 1 0^{6} \times 0.212 \times 1 0^{- 9} \approx 2.11 \times 1 0^{- 34}$ J s. (1)

Answer: ≈ $2.11 \times 1 0^{- 34}$ J s by both routes.

Example 5: The orbiting electron as a current loop · 3 marks · Short answerHOTS / competency

For the ground state of hydrogen ( $r_{1}$ = 0.053 nm, $v_{1} = 2.19 \times 1 0^{6}$ m/s), find the frequency of revolution, the equivalent current, and the magnetic field this current produces at the nucleus.

Try it first, then show the solutionSolution

$f = \frac{v}{2 π r} = \frac{2.19 \times 1 0 ^{6}}{2 π \times 0.053 \times 1 0 ^{- 9}} \approx 6.58 \times 1 0^{15}$ Hz. (1)
$I = e f \approx 1.6 \times 1 0^{- 19} \times 6.58 \times 1 0^{15} \approx 1.05 \times 1 0^{- 3}$ A. (1)
$B = \frac{μ _{0} I}{2 r} = \frac{4 π \times 1 0 ^{- 7} \times 1.05 \times 1 0 ^{- 3}}{2 \times 0.053 \times 1 0 ^{- 9}} \approx 12.5$ T. (1)

Answer: ≈ $6.58 \times 1 0^{15}$ Hz; ≈ 1.05 mA; ≈ 12.5 T.

Example 6: Bohr radius from constants · 4 marks · Case-basedHOTS / competency

Suppose you compute Bohr's results from first principles using $h = 6.63 \times 1 0^{- 34}$ J s, $ε_{0} = 8.85 \times 1 0^{- 12}$ C² N⁻¹ m⁻², $m = 9.1 \times 1 0^{- 31}$ kg, $e = 1.6 \times 1 0^{- 19}$ C.
(i) Find $r_{1}$ . (1 mark)
(ii) Find $v_{1}$ . (1 mark)
(iii) Find $\frac{c}{v _{1}}$ . (1 mark)
(iv) For which n would the orbit radius reach about 1 µm? (1 mark)

Try it first, then show the solutionSolution

(i) $r_{1} = \frac{8.85 \times 1 0 ^{- 12} \times ( 6.63 \times 1 0 ^{- 34} ) ^{2}}{π \times 9.1 \times 1 0 ^{- 31} \times ( 1.6 \times 1 0 ^{- 19} ) ^{2}} \approx 5.32 \times 1 0^{- 11}$ m.
(ii) $v_{1} = \frac{( 1.6 \times 1 0 ^{- 19} ) ^{2}}{2 \times 8.85 \times 1 0 ^{- 12} \times 6.63 \times 1 0 ^{- 34}} \approx 2.18 \times 1 0^{6}$ m/s.
(iii) $\frac{3 \times 1 0 ^{8}}{2.18 \times 1 0 ^{6}} \approx 137$ .
(iv) $n^{2} = \frac{1 0 ^{- 6}}{5.3 \times 1 0 ^{- 11}} \approx 1.9 \times 1 0^{4}$ , so n ≈ 137.

Answer: (i) ≈ $5.32 \times 1 0^{- 11}$ m; (ii) ≈ $2.18 \times 1 0^{6}$ m/s; (iii) ≈ 137; (iv) n ≈ 137.

Where students lose marks in Bohr's Model: Radius and Speed of Orbits

The slips examiners see most often, and what to write instead.

Mistake: Not putting Z = 1
The ISC 2024 and 2025 examiners noted answers left with Z or K (the Coulomb constant) and not simplified for hydrogen.
Do this: Write the final $r_{n} = \frac{ε _{0} n ^{2} h ^{2}}{π m e ^{2}}$ with Z = 1.
Usually costs: 1 mark.
Mistake: Missing steps in the derivation
2024 Q19(i)(a): steps not mentioned or not simplified.
Do this: Show force balance, quantisation, substitution and the final expression.
Usually costs: 1 to 2 marks.
Mistake: Not linking to n²
2025 Q19(i)(a): candidates derived r but did not conclude $r \propto n^{2}$ .
Do this: End with "all other quantities are constant, hence $r_{n} \propto n^{2}$ ".
Usually costs: 1 mark.
Mistake: Taking v ∝ n
Students assume bigger orbits are faster.
Do this: $v_{n} \propto \frac{1}{n}$ : outer electrons move more slowly.
Usually costs: 1 mark.

Exam tips

Number the equations in a derivation and say which you substitute into which.
State Bohr's postulates in words, then in symbols.
Give the Bohr radius in nm (the syllabus uses nm, not Å).
Quote the final proportionality the question asks for.

Practice questions on Bohr's Model: Radius and Speed of Orbits

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
According to Bohr, the angular momentum of an electron in the nth orbit is:
(a) nh
(b) $\frac{nh}{2 π}$
(c) $\frac{h}{2 π n}$
(d) $\frac{2 π n}{h}$
Hint
Second postulate.
Answer
(b).
Q2 · 1 mark · Short answerBoard level
Find the ratio of the radii of the fifth and second Bohr orbits of hydrogen.
Hint
r ∝ n².
Answer
25 : 4.
Q3 · 1 mark · Assertion-reasonHarder board level
Assertion (A): An electron in a Bohr orbit of hydrogen does not radiate energy.
Reason (R): Bohr postulated that orbits with angular momentum $\frac{nh}{2 π}$ are stationary, non-radiating states.
(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.
Hint
Postulate 1 and 2.
Answer
(a).
Q4 · 2 marks · Short answerHarder board level
Find the speed of the electron in the fifth orbit of hydrogen and express it as a fraction of c.
Hint
v ∝ 1/n.
Answer
≈ $4.38 \times 1 0^{5}$ m/s ≈ $\frac{c}{685}$ .
Q5 · 1 mark · Short answerHarder board level
Which orbit of hydrogen has radius 2.597 nm?
Hint
$n^{2} = \frac{2.597}{0.053}$ .
Answer
n = 7.
Q6 · 2 marks · Short answerHOTS / competency
The total energy of an electron in the n = 4 orbit of hydrogen is −0.85 eV. Find its kinetic and potential energies and explain the signs.
Hint
K = −E, U = 2E.
Answer
K = +0.85 eV; U = −1.70 eV. U is negative because the electron is bound by attraction; K is always positive.
Q7 · 3 marks · Short answerHOTS / competency
Show that the second Bohr orbit of hydrogen holds exactly two de Broglie wavelengths of the electron. ( $r_{2}$ = 0.212 nm, $v_{2} = 1.095 \times 1 0^{6}$ m/s)
Hint
Compare $2 π r_{2}$ with $λ = \frac{h}{m v}$ .
Answer
$2 π r_{2} \approx 1.332$ nm; $λ = \frac{6.63 \times 1 0 ^{- 34}}{9.1 \times 1 0 ^{- 31} \times 1.095 \times 1 0 ^{6}} \approx 0.665$ nm; $\frac{1.332}{0.665} = 2$ .
Q8 · 4 marks · Case-basedHOTS / competency
Case study: Suppose a hydrogen atom is excited to the n = 6 state.
(i) Find the orbit radius. (1 mark)
(ii) Find the electron's speed. (1 mark)
(iii) Find its angular momentum. (1 mark)
(iv) Find the time for one revolution, and check that it is 216 times the ground-state period of $1.52 \times 1 0^{- 16}$ s. (1 mark)
Hint
T ∝ n³.
Answer
(i) 1.908 nm. (ii) ≈ $3.65 \times 1 0^{5}$ m/s. (iii) $\frac{6 h}{2 π} \approx 6.33 \times 1 0^{- 34}$ J s. (iv) $T = \frac{2 π r}{v} \approx 3.28 \times 1 0^{- 14}$ s $= 216 \times 1.52 \times 1 0^{- 16}$ s.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Bohr radius?Show answer
≈ 0.053 nm.
2.How does $r_{n}$ depend on n?Show answer
$r_{n} \propto n^{2}$ .
3.How does $v_{n}$ depend on n?Show answer
$v_{n} \propto \frac{1}{n}$ .
4.Speed in n = 1?Show answer
≈ $2.19 \times 1 0^{6}$ m/s.
5.Relation between K and total energy?Show answer
K = −E.

Bohr's Model: Radius and Speed of Orbits: frequently asked questions

Why doesn't the electron radiate in a Bohr orbit?

Bohr simply postulated stationary orbits; classical radiation theory does not apply there. De Broglie later gave a reason: the electron's wave forms a standing wave in an allowed orbit.

Does Bohr's model work for other atoms?

Only for hydrogen and one-electron ions. It fails for many-electron atoms, line intensities and fine structure.

Why is the total energy negative?

Zero energy is taken at infinite separation; the bound electron needs energy to escape, so its energy is below zero.

What does the ISC 2027 syllabus need here?

Bohr's postulates (Z = 1) and expressions for orbital velocity, radius, kinetic, potential and total energy of the electron, with λ in nm.

How was this asked recently?

ISC 2024 Q19(i)(a) (3 marks) asked the derivation of the radius of the nth orbit; ISC 2025 Q19(i)(a) asked to show that r_n \propto n^2.