Lorentz Force

The magnitude is |F| = |q|·v·B·sin(α), where α is the angle between velocity and magnetic field. Insert the charge in coulombs, the speed in metres per second and the flux density B in tesla, then you get the force in newtons. An electron with v = 10⁶ m/s perpendicular to B = 0.1 T experiences |F| = 1.6×10⁻¹⁹·10⁶·0.1 = 1.6×10⁻¹⁴ N. For perpendicular motion sin(α) = 1 and the force is maximal. You find the direction with the three-finger or right-hand rule, and for negative charges it points in the opposite direction.

Because |F| = |q|·v·B·sin(α) contains the sine of the angle between velocity and magnetic field. If the charge moves parallel to the field, α = 0° and sin(0°) = 0, so the force vanishes completely, no matter how fast the charge is or how strong the field. The magnetic force arises only from the velocity component perpendicular to the field. At α = 90°, that is perpendicular motion, the force is maximal. That is why charged particles move freely along magnetic field lines, while perpendicular to them they are forced onto circular or helical paths.

The Lorentz force is always perpendicular to the velocity. A force perpendicular to the direction of motion does not change the magnitude of the velocity, only its direction, so it acts as a centripetal force. As a result the particle follows a circular path when it enters a uniform field perpendicularly. Equating the Lorentz force with the centripetal force, |q|·v·B = m·v²/r, gives the orbital radius r = m·v/(|q|·B). Heavier or faster particles fly on larger circles, stronger fields curve the path more tightly. This is exactly what mass spectrometers use to separate particles by their mass-to-charge ratio.

The right-hand rule and the three-finger rule apply directly to positive charges: thumb in the direction of motion, index finger in the field direction, middle finger shows the force. For negative charges such as electrons the force direction reverses, so you flip the result by 180°. Alternatively use the left hand for electrons. The magnitude stays unchanged, only the orientation turns. A typical exam mistake is to apply the right-hand rule to an electron and forget the reversal; then the numerical value is right, but the predicted deflection direction is exactly wrong.

The electric force F = q·E acts on every charge, whether at rest or moving, and points along the electric field (for a positive charge). The magnetic Lorentz force F = q·v×B, by contrast, acts only on moving charges and is always perpendicular to velocity and magnetic field. Therefore the magnetic force does no work and changes only the direction, not the speed. In the general case both combine into the full Lorentz force F = q·(E + v×B). In many problems you consider only the magnetic part, because no additional electric field is present.