Magnetism: Fields, Forces, and Induction

Posted on Dec 4, 2024 in Physics

Magnetic Fields

A magnetic field exists in a region of space if a moving charge q experiences a magnetic force. Its value has been demonstrated experimentally. The intensity of the magnetic field depends on load, speed, and the angle between the velocity (v) and the magnetic field (B). The force is perpendicular to both v and B, and its direction is reversed if the sign of the charge is reversed. The force’s magnitude is F = qvB sin(θ) = q(v x B), measured in Tesla (T).

Field Lines

Field lines leave the north pole and enter the south pole. They are closed lines, meaning poles cannot be separated.

Magnetic Field of a Current

Rectilinear Current

The magnetic field (B) is proportional to the current (I) through the wire and inversely proportional to the distance (d): B = 2k’I / d = μ0I / (2πd). The field lines follow the right-hand rule.

Loop

B = μ0I / 2r. The field line direction is given by the corkscrew rule: the direction coincides with the loop’s axis, and the sense advances like a corkscrew moving in the current’s direction.

Solenoid

A solenoid acts as a magnet with north and south poles. To find the poles, use the right-hand rule: when your fingers curl in the current’s direction, your thumb points to the north pole. The magnetic field inside is B = μ0NI / l (where N is the number of turns and l is the length).

Magnetic Force on Moving Charges

A moving charge (q) with velocity (v) in a magnetic field (B) experiences a force (F). If the charge moves at an angle (θ) to the field, the force is F = qvB sin(θ) = q(v x B). A negative charge moves in the opposite direction to v x B (use the right-hand rule). A particle entering perpendicular to a uniform magnetic field follows a circular path.

Magnetic Force on a Current-Carrying Conductor

Rectilinear Current

In a conductor perpendicular to the magnetic field lines, electrons move with velocity (v). The force on a length (l) of the conductor is F = I(l x B).

Circular Loop

Similar to the rectilinear case, but it produces a couple of forces with a magnitude of ILB and torque.

Interaction Between Parallel Currents

Each conductor is in the other’s magnetic field. Two parallel conductors with current in the same direction attract; in opposite directions, they repel. One Ampere is the current flowing through two parallel conductors one meter apart in a vacuum, producing a force of 2 x 10-7 N per meter of length.

Magnetic vs. Electric Fields

  • Electric fields are created by static charges; magnetic fields by moving charges.
  • Electric field lines are open; magnetic field lines are closed.
  • Electric fields are conservative (have potential); magnetic fields are not.
  • Field strength is proportional to the source (charge or current intensity).
  • Both depend on distance, but magnetic fields also depend on the direction of movement.
  • Electric charges can be isolated; magnetic poles cannot.

Magnetic Flux

The number of field lines passing through a surface is proportional to the field intensity, the surface area, and the cosine of the angle between them. This concept is based on Faraday’s experiments.

Lenz’s Law

The induced current opposes the change in magnetic flux that produces it. For example, if a north pole approaches a coil, the induced current creates a magnetic field opposing the approaching field.

Faraday’s Law

The induced electromotive force (emf) is proportional to the rate of change of magnetic flux and the number of turns in the coil: emf = –N(ΔΦ / Δt). It is measured in webers.

Alternating Current Production

In a loop rotating with constant angular velocity (ω) in a magnetic field (B), if θ is the angle between the field and the loop’s normal, the magnetic flux is Φ = BS cos(ωt). The induced emf is emf = –BSω sin(ωt). For a coil with N turns, multiply by N.