Magnetic Field
A field that pushes only on moving charges.
The Magnetic Force
Moving charges feel a sideways push.
A charge at rest beside a current-carrying wire feels nothing. Set it moving and it is pushed. That speed-dependent push reveals a new field: the magnetic field.
A field that only acts on moving charges
A wire carrying current contains equal amounts of positive and negative charge in any small volume, so it is electrically neutral and makes (almost) no electric field. A charge q held at rest near it feels no force. But throw a positive charge parallel to the current and it curves toward the wire. Something there acts on moving charges only: the magnetic field B⃗, and the push it gives is the magnetic force.
Defining the field B
Fire a charge through a point P in different directions and at different speeds. You find:
- Along one particular line through P there is no force at all. B⃗ is defined to lie along this line.
- Moving at angle θ to that line, the force is proportional to v sin θ.
- The force is perpendicular both to the field and to the velocity.
- The force is proportional to q and reverses for a negative charge.
All four are captured by one vector product, which also fixes the sense of B⃗:
The SI unit of B is the newton per ampere metre, called the tesla (T), the same as the weber per square metre. The gauss is also used: 1 T = 10⁴ G. A tesla is large: the Earth’s field is about 10⁻⁵ T, and laboratory superconducting magnets reach about 10 T. (B used to be called “magnetic induction”.)
A proton moves east at 4.0 × 10⁶ m/s through a uniform field of 2.5 mT pointing vertically up. Find the force on it and its acceleration.
A 3.0 µC charge moves at 1.0 × 10⁶ m/s along +y through B⃗ = (0.30î + 0.50k̂) T. Find the force.
Electric and magnetic fields are one thing
Observer S sees a charge moving at v beside a wire: no electric field, but a magnetic force pulls the charge toward the wire. Observer S₁ moves along with the charge, so for S₁ the charge is at rest and no magnetic force can act on it. Yet S₁ must see the same attraction (the two observers don’t accelerate relative to each other, so they agree on accelerations). So S₁ must see an electric field.
- 1Magnetic force acts only on moving charges: F⃗ = qv⃗ × B⃗ (34.1).
- 2No force along B; maximum at 90°; perpendicular to both v⃗ and B⃗.
- 3Unit tesla (T) = N/(A m) = Wb/m²; 1 T = 10⁴ G; Earth ≈ 10⁻⁵ T.
- 4E and B fields depend on the observer's frame: one electromagnetic field.
Check yourself
Circles & Helices
Bent into circles, never sped up.
Because the magnetic force always acts sideways, it can bend a charge’s path but never speed it up. In a uniform field the result is a circle, or a helix.
No work, constant speed
The magnetic force is perpendicular to the velocity, so it does no work. The particle’s kinetic energy, and so its speed, never changes.
Diagrams show a field into the page as ⊗ (the tail of an arrow) and out of the page as ⊙ (its tip). Let a charge q move at speed v perpendicular to a uniform field. The force qvB is constant in size and always perpendicular to the motion, so it acts as a centripetal force:
The period does not depend on the speed: a faster particle runs a bigger circle in the same time. The frequency ν is called the cyclotron frequency.
A particle of charge 50 µC and mass 5.0 mg is projected at 10 m/s perpendicular to a uniform field of 40 mT. When does it first return to its starting point?
Singly charged neon-20 ions with kinetic energy 1.5 keV enter a 0.50 T field at right angles. Find the radius of their path, and the radius for neon-22 ions of the same energy. (Take the ion mass as A × 1.67 × 10⁻²⁷ kg.)
Velocity at an angle: a helix
If v⃗ isn’t perpendicular to B⃗, split it into v∥ along the field and v⊥ across it. The force is perpendicular to B⃗, so v∥ never changes; meanwhile the particle circles with radius r = mv⊥/qB. Together: a helix wound round the field lines. In one period it advances a distance called the pitch, v∥T.
Protons at 5.0 × 10⁵ m/s enter a 0.40 T field at 30° to the field direction. Find the radius and pitch of their helix.
A particle of charge 10 µC and mass 10 µg circles with radius 4.0 cm in a 0.50 T field. At one point an electric field is switched on, after which it moves in a straight line at constant speed. Find E.
- 1The magnetic force does no work: speed is constant.
- 2v ⊥ B: circle with r = mv/qB (34.2), T = 2πm/qB (34.3), ν = qB/2πm (34.4), independent of speed.
- 3General angle: helix with r = mv⊥/qB and pitch v∥T.
Check yourself
Force on a Current-Carrying Wire
Currents are pushed by fields.
A current is moving charge, so a magnetic field pushes on a wire that carries one. That push turns every electric motor and loudspeaker.
Adding up the forces on the electrons
Take a small length dl of a wire carrying current i in a field B⃗. Its free electrons drift with velocity v⃗d opposite to the current, and i = nevdA. Each feels an average force −ev⃗d × B⃗, and the element holds nA dl of them:
where dl⃗ points along the current. Using i = nAevd:
A straight wire 50 cm long carries 4.0 A at 30° to a uniform 0.20 T field. Find the force on it.
Closed loops in a uniform field
Bend a wire into an equilateral triangle carrying current, in a field perpendicular to its plane. Each side feels ilB, directed perpendicular to that side, all pointing inward (or all outward). Three equal forces at 120° cancel. A side parallel to B feels nothing, while sides at angles feel forces into and out of the plane that again cancel. In general:
A rod on rails
Two horizontal rails 20 cm apart lie in a vertical field of 0.50 T. A 50 g rod across them carries 3.0 A. (a) What is the least friction coefficient that holds it still? (b) If μ = 0.30, find its acceleration.
- 1dF⃗ = i dl⃗ × B⃗ (34.5); straight wire F⃗ = i l⃗ × B⃗ (34.6).
- 2Derived by adding the forces on drifting electrons.
- 3A closed loop in a uniform field feels no net force.
Check yourself
Videos, simulations & further reading (2)
Torque on a Current Loop
Loops twist to line up with the field.
A current loop in a magnetic field feels no net force, but it does feel a twist. That twist is what spins a motor and swings a meter needle.
A rectangular loop
A rectangular loop PQRS (sides l and b) carries current i in a uniform field B⃗, with the loop’s plane perpendicular to B⃗. The forces on opposite sides are equal, opposite and along the same line: no net force and no torque.
Now rotate the loop by θ about the line joining the midpoints of QR and SP. Sides PQ and RS (length l) still feel ilB each, unchanged in direction, but their lines of action separate: each now has moment (b/2) ilB sin θ about the axis. The forces on the other two sides stay equal, opposite and collinear. Total torque:
These results hold for flat loops of any shape, not just rectangles.
A circular coil of 50 turns and radius 3.0 cm carries 2.0 A. Find its magnetic moment, and the torque on it in a 0.40 T field at 30° to its axis.
A 200-turn coil 2.0 cm × 5.0 cm carries 0.50 A in a 0.80 T field. Its area vector makes 30° with the field. Find the torque.
An orbiting electron is a magnet
An electron circling at speed v on a radius r passes any point v/2πr times a second, so it is a current i = ev/2πr. Its magnetic moment:
The ratio of magnetic moment to angular momentum is the same for every orbit: a link between atoms and magnetism that later chapters build on.
- 1Torque on a loop: Γ⃗ = μ⃗ × B⃗, μ⃗ = niA⃗ (34.7–34.8); no net force in a uniform field.
- 2A⃗ by the right-hand rule; torque turns μ⃗ toward B⃗; zero when aligned.
- 3An orbiting electron has μ = evr/2 and μ/L = e/2m.