Understanding Magnetic Force and Lorentz Law Principles

Magnetic Force on a Moving Charge: The Lorentz Law

The force exerted on an electric charge has the following characteristics:

• If the charge is at rest, no forces act upon it.
• If the charge moves with velocity v, it experiences a force proportional to the value of the charge q, perpendicular to the velocity v.
• The magnitude depends on the direction of the velocity vector v: if it has a certain direction, the magnetic force is zero; if the vector v is perpendicular to that direction, the magnetic force is maximal.

These properties are summarized in the Lorentz Law:

F = q(v × B)
The module is F = |q|vB sin(α), where α is the angle formed by B and v.

Magnetic Force and Trajectory

The magnetic force acting on a charge is perpendicular to the velocity of the charge, meaning it does not perform work on the charge. Because the magnetic force is perpendicular to the vector v, it cannot change the speed of the charge, but it can alter its trajectory.

If a positive charge q enters a uniform magnetic field with a velocity perpendicular to the field, the Lorentz force compels it to follow a uniform circular motion (F_c = F_m):

F = m · a_c = m · v² / R
qvB = mv² / R
From this, we derive the radius of the circle described by q: R = mv / qB.

Magnetic Force on a Current Element

The formula for a current element is:

dF = I(dl × B)
F = I · L · B sin(α)

Forces Between Parallel Currents

Two parallel currents in the same direction attract, while two parallel currents in opposite directions repel. The formula is:

F / L = (μ₀ · I₁ · I₂) / (2π · d)

Explanation of Natural Magnetism

Experiments confirmed that electric currents produce the same effects as magnets. Ampère observed that electric currents attract or repel each other and can attract iron filings. He suggested that natural magnetism was due to small closed currents inside matter.

Today, we identify these flows with the movement of electrons inside atoms:

• An electron spinning around the nucleus is equivalent to a current that produces the same magnetic effects as a small magnet.
• Electrons revolving about themselves produce additional magnetic effects.

All materials are formed by magnets or magnetic dipoles. In most cases, these dipoles are randomly oriented and their effects cancel out. Applying a magnetic field to the material causes the dipoles to orient in the direction of the field.

Relative Permeability (μ_r)

The constant μ_r is characteristic of the medium and is called the relative permeability of the material. Substances are classified based on μ_r:

• Paramagnetic substances: Have a relative permeability slightly greater than 1.
• Diamagnetic substances: Have a relative permeability slightly less than 1.
• Ferromagnetic substances: Have a permeability much greater than 1.