Magnetic Force on a Moving Charge — F = qvB
A charge sitting still feels no magnetic force at all — it only appears once the charge moves, and only the part of its velocity that is not parallel to the field matters. This guide gives the full formula with its angle term, the radius and period of the circular path a charge follows inside a uniform field, and four worked numericals covering a perpendicular charge, an angled charge, and the geometry of the resulting circular motion.
The magnetic force formula
where q is the charge, v is its speed, B is the magnetic field strength, and θ is the angle between the velocity vector and the field vector. Two special cases are worth memorising directly: at θ = 90° (velocity perpendicular to the field), F = qvB is maximum; at θ = 0° (velocity parallel to the field), sinθ = 0 and F = 0 — a charge moving exactly along the field lines feels no magnetic force at all.
Direction — the right-hand rule
For a positive charge, point the fingers of the right hand along v, curl them toward B, and the thumb gives the direction of v × B — that is the direction of the force. For a negative charge (an electron, for instance), the force points exactly opposite to what the right-hand rule gives, because F = q(v × B) and q is negative.
Circular motion in a uniform field
Because the magnetic force is always perpendicular to the velocity, it can never speed a charge up or slow it down — it only changes direction. When v is perpendicular to a uniform B, this constant sideways push produces uniform circular motion:
Notice T does not contain v at all — every charge of the same q/m ratio takes the same time to complete one loop, however fast or slow it is moving. Only the radius depends on speed.
Worked example 1 — force on an electron moving perpendicular to B
An electron (|q| = 1.6 × 10⁻¹⁹ C) moves at v = 2 × 10⁶ m/s perpendicular to a field B = 0.5 T. Find the magnitude of the force.
F = qvB sin90° = qvB (since sin90° = 1)
F = (1.6 × 10⁻¹⁹) × (2 × 10⁶) × 0.5
1.6 × 2 = 3.2, then 3.2 × 0.5 = 1.6; exponents: 10⁻¹⁹ × 10⁶ = 10⁻¹³
F = 1.6 × 10⁻¹³ N — direction opposite to the right-hand-rule result, since the electron's charge is negative.
Worked example 2 — force at an angle to the field
A charge q = 2 × 10⁻⁶ C moves at v = 1 × 10⁴ m/s at 30° to a field B = 0.2 T.
F = qvB sinθ = (2 × 10⁻⁶) × (1 × 10⁴) × 0.2 × sin30°
2 × 1 = 2, exponents 10⁻⁶ × 10⁴ = 10⁻², so qv = 2 × 10⁻²
2 × 10⁻² × 0.2 = 4 × 10⁻³
4 × 10⁻³ × sin30° (= 0.5) = 2 × 10⁻³ N = 2 mN
Worked example 3 — radius of a proton's circular path
A proton (q = 1.6 × 10⁻¹⁹ C, m = 1.673 × 10⁻²⁷ kg) moves at v = 3 × 10⁵ m/s perpendicular to B = 0.8 T. Find the radius of its circular path.
r = mv/(qB)
Numerator: 1.673 × 3 = 5.019, so mv = 5.019 × 10⁻²² kg m/s
Denominator: 1.6 × 0.8 = 1.28, so qB = 1.28 × 10⁻¹⁹
r = (5.019 ÷ 1.28) × 10⁻²² ⁺ ¹⁹ = 3.921 × 10⁻³ m
r ≈ 3.92 mm
Worked example 4 — the period is independent of speed
Same proton and field as example 3. Find the time period of one full circle, and confirm it does not depend on v.
T = 2πm/(qB) = (2 × 3.1416 × 1.673 × 10⁻²⁷) ÷ (1.28 × 10⁻¹⁹)
2π × 1.673 = 10.51, so the numerator is 10.51 × 10⁻²⁷ = 1.051 × 10⁻²⁶
T = (1.051 ÷ 1.28) × 10⁻²⁶ ⁺ ¹⁹ = 0.8214 × 10⁻⁷ = 8.21 × 10⁻⁸ s (about 82 nanoseconds)
This value is the same whether v is 3 × 10⁵ m/s or ten times that — only r would change, exactly as the formula T = 2πm/(qB) predicts, since v does not appear in it.
Common mistakes that cost marks
- Forgetting the sinθ factor when the velocity is not exactly perpendicular to the field — using F = qvB even when θ ≠ 90°.
- Applying the right-hand rule directly to a negative charge. Find the direction for a positive charge first, then reverse it for an electron or any negative ion.
- Thinking the magnetic force changes a charge's speed. It cannot — the force is always perpendicular to v, so it does zero work and only bends the path.
- Inverting r = mv/(qB) — writing r = qB/(mv), which gives a radius that shrinks as speed increases instead of growing with it.
- Forgetting unit conversions for charge — μC and nC must become coulombs (× 10⁻⁶ and × 10⁻⁹ respectively) before substituting.
Where this appears in exams
| Exam | Typical use |
|---|---|
| CBSE Class 12 | Moving Charges and Magnetism — force, radius and period in a uniform field |
| JEE Main & Advanced | Combined electric-and-magnetic-field (velocity selector) problems |
| NEET | Direct F = qvB and r = mv/(qB) substitution questions |
| GATE (Engineering Physics) | Charged-particle motion in fields, cyclotron-type numericals |
Powers-of-ten arithmetic is the real challenge here. The Scientific Calculator's exponent and square-root keys make checking a step like (1.673 × 3) × 10⁻²² quick while you practise these numericals.
Open the ABC Chemistry Calculator Suite →Working through Class 12 magnetism for boards or JEE/NEET? ABC Chemistry runs Class 11–12 chemistry coaching at the Gurugram centre plus online classes across India — details at abcchemistry.in.