Magnetic field and forces

A particle enters a magnetic field and starts moving in a circle at constant speed. A force acts, yet it does no work. To see how this is possible, start with the field’s force on a current and then consider a single moving charge.

Bring a compass near a current-carrying wire: its needle changes direction. Moving charges and magnets create a magnetic field, detected through its action on moving charges and currents. The magnetic flux density vector B describes the field’s direction and strength; it is tangent to a field line. Field lines help represent the field, but they are not strings pulling on the conductor.

Place a current-carrying conductor between magnet poles and rotate it. The magnetic force on the current is greatest across the field and vanishes along it. Current and field strength alone are therefore insufficient: orientation matters. The active length l counts only the part inside the field being considered. Lengthening a wire outside this region does not increase the magnetic force on the segment inside it.

For a straight segment in a uniform field, F_A=IlB\sin\alpha. With I in amperes, l in metres and B in teslas, force is in newtons; α is the angle between current and B. Doubling current or active length doubles the force in the same field. At α=0° the sine is zero; at α=90° it is one. The formula gives the magnitude. Determine direction separately from the orientations of current and field.

How can the same experiment compare two magnets’ fields? Use the same current and length, with the conductor perpendicular to each field. Then B=F_A/(Il), giving 1\,\mathrm{Тл}=1\,\mathrm{Н}/(\mathrm{А}\cdot\mathrm{м}). A one-tesla field exerts one newton on a one-metre active segment carrying one ampere in this orientation. A smaller force after turning the conductor does not mean that the field itself weakened.

Bend the conductor into a loop. Forces on different sides can produce a torque even when the net force is zero. For one turn, B=M_{\max}/(IS), where S is area in m² and Mmax is the maximum torque over all orientations, in newton metres. Doubling area at the same current and field doubles this maximum torque. Use the maximum: another orientation would give a smaller torque with the same B.

A current consists of moving charges. For one particle, the magnetic part of the Lorentz force has magnitude F_L=|q|vB\sin\alpha, with q in coulombs, v in metres per second and α between velocity and field. Doubling |q| or speed at the same angle doubles the force. A stationary particle experiences no magnetic force. The full Lorentz force can also include an electric part, but here we consider only the magnetic field’s action.

For a positive charge, use v×B: curl your right-hand fingers from v towards B through the smaller angle; your thumb gives the force direction. For a negative charge, reverse it. The force is perpendicular to both velocity and field. Imagine changing the charge sign while keeping v and B: the particle bends the other way, although the force magnitude stays the same.

Over each infinitesimal displacement, magnetic force is perpendicular to the motion, so it does no work. It can turn the velocity vector without changing its magnitude or the kinetic energy, as in uniform circular motion. If velocity is exactly along B, force is zero. If an electric field also speeds up the particle, the energy gain cannot be attributed to the magnetic part of the force.

Let initial velocity be perpendicular to uniform B, with no other forces. Magnetic force continuously turns velocity while keeping the same magnitude, producing a circle. Newton’s second law gives |q|vB=mv^2/R, with m in kilograms and R in metres. The actual magnetic force supplies the required centripetal action: no additional force towards the centre is needed. This treatment assumes nonrelativistic motion.

Rearranging gives R=mv/(|q|B). A larger mass or speed widens the circle; a larger field or |q| narrows it. The period in seconds is T=2\pi R/v=2\pi m/(|q|B). As v increases, distance per revolution increases in the same proportion, so T stays unchanged. This compensation explains the model’s speed-independent period. Decreasing B increases radius and period; at B=0 there is no circular motion caused by the field.

Add a velocity component along B to the component across it. Magnetic force does not change the parallel component, so circular motion combines with uniform forward motion to form a helix. Use perpendicular speed, not total speed, in the radius formula. Near the speed c, ordinary mv no longer gives momentum correctly. The period then depends on speed and a relativistic model is required.

A straight conductor 0.5 m long carries a current of 2 A perpendicular to a uniform 0.4 T field. Find the magnetic force magnitude.

This is a straight current-carrying conductor, so use the magnetic force on a current. The active length is l = 0,5 м, the current I = 2 А and the flux density B = 0,4 Тл.

The conductor is perpendicular to the field: α = 90°, sin α = 1. For these B, I and l, the force magnitude is greatest.

F = BIl = 0,4 × 2 × 0,5 = 0,4 Н.

The units Тл·А·м give Н. Turning the conductor along the field would reduce the force to zero, even with the same current and flux density.

Magnetic force magnitude: F = 0,4 Н. This is the maximum for the given current, field and active conductor length.

Magnetic action depends on field strength and on the direction of current or velocity relative to B.

A force perpendicular to velocity can bend the path without doing work or changing kinetic energy.

The radius and period of circular motion follow from Newton’s second law. Their simple expressions apply to nonrelativistic motion.

Next, explore how a changing magnetic field produces an electric effect.