Physics
Magnetic field and Lorentz force calculator
Pick the relation and clear the unknown: force, charge, speed, field or angle. Clearing the angle tells you at what tilt the force takes that value.
Why the magnetic force never does any work
The magnetic force is the only one in elementary physics that depends on velocity: on a charge at rest it does nothing at all, and on a moving charge it acts at right angles to both the velocity and the field. One striking consequence follows: being always perpendicular to the displacement, it never does work. A magnetic field can bend a particle's path as much as you like, but it cannot speed it up or slow it down.
This gives the uniform circular motion of a charge in a uniform field, with radius r = mv/(qB). The surprise is the period: 2πm/(qB), which does not depend on the speed. A faster particle traces a bigger circle, but in the same time. That is the principle the cyclotron runs on, and the reason a mass spectrometer separates isotopes.
The fields produced by currents all share the constant μ₀ and differ only in geometry. A straight wire gives μ₀I/(2πr), falling off as 1/r; the centre of a loop gives μ₀I/(2R); a long solenoid gives μ₀nI, constant throughout its inside. The last is the most useful in practice precisely because it is uniform: it is the standard way to produce a known field in a laboratory.
Common mistakes
- Forgetting the sine of the angle: the force is largest only at 90°, and a charge moving parallel to the field feels no force at all.
- Using the wire formula as if the field fell off with the square of the distance: for a straight wire it goes as 1/r. The inverse-square law is for point charges, not for wires.
- Confusing the total number of turns with turns per metre in a solenoid: in B = μ₀nI, n is the winding density, that is N divided by the length.
Frequently asked questions
What is the formula for the Lorentz force?
F = q·v·B·sin θ, where θ is the angle between the velocity and the field. The force is perpendicular to both, so it does no work and does not change the speed.
Why does the magnetic force do no work?
Because it is always perpendicular to the velocity, and work is the force times the displacement along its direction. At 90° that product is zero: the particle's kinetic energy stays constant.
How do you work out the field of a current-carrying wire?
B = μ₀·I/(2π·r), with μ₀ = 4π × 10⁻⁷ T·m/A. Five centimetres from a wire carrying 10 A the field is 4 × 10⁻⁵ tesla, about the strength of the Earth's magnetic field.
What does the period of a charge in a magnetic field depend on?
Only on the mass, the charge and the field: T = 2πm/(qB). It depends on neither the speed nor the radius, because a faster particle traces a proportionally bigger circle.
How this calculation works
Lorentz force: F = q·v·B·sin θ, perpendicular to both v and B (right-hand rule). Force on a wire: F = B·I·L·sin θ. Field of an infinite straight wire: B = μ₀·I/(2π·r). Field at the centre of a loop of radius R: B = μ₀·I/(2R). Field inside a long solenoid: B = μ₀·n·I, with n = N/ℓ turns per metre. Circular motion in a uniform field: r = m·v/(q·B), period T = 2π·m/(q·B), cyclotron frequency f = q·B/(2π·m). Permeability of free space: μ₀ = 4π × 10⁻⁷ T·m/A. Force between two straight parallel wires: F/L = μ₀·I₁·I₂/(2π·d), attractive for currents in the same direction. Field inside a toroid, from Ampère's law: B = μ₀·N·I/(2π·r), not uniform across the bore. Magnetic moment of a loop: m = N·I·A, and the torque in a uniform field τ = m·B·sin θ with θ measured from the loop's normal. Hall effect: V_H = I·B/(n·q·t), with Hall coefficient R_H = 1/(n·q).