Combined Electric & Magnetic Force
When a charged particle moves through a region containing both an electric field and a magnetic field simultaneously, it experiences two distinct forces at the same time. The electric force acts along the direction of the field (or opposite, for negative charges), while the magnetic force acts perpendicular to both the velocity and the magnetic field. Together, these two contributions add as vectors to give the total electromagnetic force on the particle. This combined result is called the Lorentz force, named after the Dutch physicist Hendrik Lorentz who formalised it in the late 19th century. Understanding the Lorentz force is essential for explaining a huge range of phenomena — from how cathode-ray tubes steer electron beams to how particle accelerators and mass spectrometers work. A particularly elegant application is the velocity selector: by carefully balancing the electric and magnetic forces on a charged particle, engineers can filter out only those particles travelling at one precise speed, regardless of their charge or mass. This interplay between E and B fields is a cornerstone of electromagnetism and bridges static electricity with the dynamic world of moving charges. The Lorentz force is also foundational to classical electrodynamics. It tells us that electricity and magnetism are deeply linked — two aspects of the same underlying phenomenon, electromagnetism. Maxwell's equations describe how E and B fields are generated and how they change, while the Lorentz force law tells us how those fields act on matter. Together, they form a complete classical description of how charged particles and electromagnetic fields interact.
- The Electric Force Component
- The Magnetic Force Component
- Adding the Forces as Vectors
- The Velocity Selector — Forces in Balance
- Circular and Helical Motion
The Electric Force Component
A charge q in an electric field E experiences a force F_E = qE. This force is parallel to E for positive charges and antiparallel for negative charges. Crucially, this force does NOT depend on the particle's velocity — a stationary charge still feels the electric force. Think of it like gravity: it acts on the object whether it is moving or not.
The Magnetic Force Component
A charge q moving with velocity v through a magnetic field B experiences a force F_B = q(v × B). The cross product means the magnetic force is always perpendicular to both v and B. A stationary particle feels no magnetic force at all — the magnetic force only exists when the charge is moving. Use the right-hand rule: point fingers along v, curl them toward B, and your thumb points in the direction of v × B (for positive q).
Adding the Forces as Vectors
Because both forces act on the same particle simultaneously, they simply add as vectors. The total force is F = q(E + v × B). This is the Lorentz force law. In component form, each x, y, and z component of the electric and magnetic forces must be calculated separately and then added. The resultant can point in any direction in three-dimensional space, depending on the magnitudes and directions of E, B, and v.
The Velocity Selector — Forces in Balance
A velocity selector is a device where E and B fields are arranged so their forces on a charged particle are equal and opposite. For a positive charge moving to the right with speed v, if E points upward (force upward) and B points out of the page (force downward by right-hand rule), then equilibrium occurs when qE = qvB, giving v = E/B. Only particles with exactly this speed pass straight through — faster or slower ones are deflected. This works for any charge, since q cancels from both sides.
Circular and Helical Motion
If E = 0 and B is uniform, the magnetic force provides centripetal acceleration and the particle moves in a circle perpendicular to B. If the particle also has a velocity component along B, that component is unaffected (no magnetic force along B), so the particle follows a helix — like a corkscrew path. Adding a non-zero E field can accelerate or decelerate the particle along B, stretching or compressing the helix, creating complex but predictable three-dimensional trajectories exploited in devices like cyclotrons.
The Key Equation
- \mathbf{F} — Lorentz force (Newton (N)): The total electromagnetic force acting on the charged particle, combining both electric and magnetic contributions.
- q — Electric charge (Coulomb (C)): The charge of the particle; positive or negative sign determines the direction of the resulting forces.
- \mathbf{E} — Electric field (Volts per metre (V m⁻¹)): The electric field vector at the particle's location; produces a force parallel (or antiparallel) to itself.
- \mathbf{v} — Velocity (Metres per second (m s⁻¹)): The instantaneous velocity vector of the charged particle through the fields.
- \mathbf{B} — Magnetic flux density (Tesla (T)): The magnetic field vector; combined with v via the cross product to give the magnetic force component.
Worked Example
Problem. A proton (charge q = +1.60 × 10⁻¹⁹ C) travels at v = 3.0 × 10⁵ m s⁻¹ in the +x direction through a region where E = 4.0 × 10⁴ V m⁻¹ in the +y direction and B = 0.20 T in the +z direction. Calculate the magnitude and direction of the total force on the proton.
Answer. The total Lorentz force on the proton is 3.2 × 10⁻¹⁵ N in the −y direction.
Origins · 1 minHendrik Antoon Lorentz and how this was discovered
Lorentz derived the complete expression for the electromagnetic force on a moving charge — now called the Lorentz force — in the 1890s, unifying the electric and magnetic force into a single elegant law. He also developed the Lorentz transformation equations, which later became a cornerstone of Einstein's special relativity. He was awarded the Nobel Prize in Physics in 1902, jointly with Pieter Zeeman, for his theoretical explanations of the influence of magnetism on radiation. Einstein deeply admired Lorentz and considered him his most important scientific mentor, once saying that Lorentz meant more to him personally than anyone else he had ever met.
Related Concepts
- Magnetic Force on a Moving Charge — The magnetic component q(v × B) of the Lorentz force is derived directly from the force on a moving charge in a magnetic field, making this the essential prerequisite concept.
- Electric Fields and Field Lines — The electric component qE of the Lorentz force requires a solid understanding of electric fields and how they exert forces on charges.
- Circular Motion of Charges in Magnetic Fields — When E = 0, the Lorentz force reduces to the magnetic force alone, which causes charges to undergo circular or helical motion — a direct application of the combined force concept.
- Velocity Selector — The velocity selector is the most direct practical application of the combined Lorentz force, using balanced E and B forces to filter charged particles by speed.