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A question gives you the radius and speed of an electron in a hydrogen atom and asks for its magnetic moment, or for the magnetic field it makes at the nucleus. It looks like atomic physics, but it is a current-loop problem in disguise. This post builds the result step by step, gets the Bohr magneton from it, and works the numbers for hydrogen.
Where does this sit in the syllabus?
The JEE (Main) 2026 syllabus lists "current loop as a magnetic dipole and its magnetic dipole moment" in Unit 13 and the Bohr model in Unit 18. The NCERT chapter Moving Charges and Magnetism defines the magnetic moment of a loop as m = IA (Eq. 4.22). The NCERT chapter Magnetism and Matter notes, when explaining diamagnetism, that orbiting electrons "are equivalent to current-carrying loop and thus possess orbital magnetic moment". The 2026-27 reprint of these chapters does not work through the orbiting-electron calculation itself, so treat it as a problem that joins Chapter 4 to the Bohr model in Chapter 12. Check your exam year's syllabus.
How is an orbiting electron a current loop?
An electron of charge e moves in a circle of radius r at speed v. It passes any point on the orbit once every period T = 2πr/v. Current is charge per unit time, so
I = e/T = ev / (2πr).
The loop's area is πr², so the magnetic moment is
μ = IA = (ev / 2πr) × πr² = evr/2.
The angular momentum is L = mvr, so
μ = (e/2m) L.
The ratio e/2m depends only on the particle, not on the orbit. Because the electron is negative, its current flows opposite to its motion, so μ points opposite to L.
Where does the Bohr magneton come from?
Bohr's model, as given in the NCERT chapter Atoms, allows only orbits with L = nh/2π. Substitute:
μ = (e/2m) × nh/2π = n × eh/(4πm).
The quantity eh/(4πm), also written eħ/2m, is called the Bohr magneton, μ_B. NIST gives its value as 9.274 × 10⁻²⁴ J/T. A joule per tesla is the same unit as an ampere square metre, the unit NCERT uses for m = IA.
Worked example: hydrogen's ground state
NCERT's Example 12.4 uses r = 5.3 × 10⁻¹¹ m and v = 2.2 × 10⁶ m/s for the electron in hydrogen, with e = 1.6 × 10⁻¹⁹ C.
Frequency. f = v / 2πr = 2.2 × 10⁶ / (2π × 5.3 × 10⁻¹¹) ≈ 6.6 × 10¹⁵ Hz, as NCERT finds.
Current. I = ef = 1.6 × 10⁻¹⁹ × 6.6 × 10¹⁵ ≈ 1.06 × 10⁻³ A. About a milliampere, from one electron.
Moment. μ = I × πr² = 1.06 × 10⁻³ × π × (5.3 × 10⁻¹¹)² ≈ 9.3 × 10⁻²⁴ A m². The shortcut evr/2 gives the same.
Compare. This is close to μ_B = 9.274 × 10⁻²⁴ J/T. The small gap comes from rounding r and v. As a check, L = mvr = 9.11 × 10⁻³¹ × 2.2 × 10⁶ × 5.3 × 10⁻¹¹ ≈ 1.06 × 10⁻³⁴ J s, which is h/2π, so n = 1.
Field at the nucleus. At the centre of a circular loop, B = μ₀I/2r = (4π × 10⁻⁷ × 1.06 × 10⁻³) / (2 × 5.3 × 10⁻¹¹) ≈ 12.5 T.
For higher Bohr orbits the moment is nμ_B, so the n = 2 orbit has twice the ground-state value.
What about the electron's spin?
The electron also has its own magnetic moment, not caused by orbiting. NIST lists its ratio to the Bohr magneton as about 1.001 in size. Spin is not in the JEE Main or CBSE physics syllabus, though JEE Main chemistry includes the electron's spin quantum number.
Which mistakes cost marks?
Writing I = eT. Current is charge divided by period, I = e/T.
Using the diameter in the area. A = πr².
Mixing h and ħ. μ_B = eh/4πm = eħ/2m. Dropping the 2π gives an answer about six times too big.
Getting the direction wrong. For an electron, μ is opposite to L.
Forgetting n. In the Bohr model μ grows in proportion to n, since r grows as n² and v falls as 1/n.
A practice routine
Day 1: Derive μ = (e/2m)L and μ_B without notes.
Day 2: Repeat the worked example for n = 2, using r and v from the Bohr formulas in our hydrogen spectrum problems, and check you get about 2μ_B.
Day 3: Find the torque on an atomic dipole of 1 μ_B at 30° to a 1 T field, using τ = μB sin θ.
Day 4: Mix in two current-loop problems from Chapter 4.
Day 5: Do a closed-book test. Dunlosky and colleagues (2013) rated practice testing and spaced practice as the most useful techniques in their review.
If a concept such as angular momentum quantisation feels shaky, Fermi's Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, so you can revisit it before trying the problems again.
Tonight, redo step 5 for the n = 2 orbit and see how much weaker the field at the nucleus becomes.
