The Lorentz force is the force exerted by a magnetic field on an electric charge in motion.
If you place a stationary charge (𝑣 = 0) inside a magnetic field, absolutely nothing happens. The magnetic field is invisible to it. But the instant the charge moves, the magnetic field exerts a lateral force on it.
This interaction is the basis of all motors, loudspeakers, and particle accelerators in the world.
The Lorentz Law (Microscopic Level)
Hendrik Lorentz formulated the law that describes the force felt by an individual particle (such as an electron) when crossing a magnetic field.
The rigorous vector expression is:
Where:
- F: Force (Newtons).
- q: Charge (Coulombs).
- v: Velocity of the particle (m/s).
- B: Magnetic Flux Density (Teslas).
- ×: Cross Product.
Direction of the Force
We know how much it pushes, but… in which direction does it push? Upwards? Downwards? To find out, engineers use the famous Fleming’s Left-Hand Rule.
Place the three fingers of your left hand perpendicular to each other:
- Index (
): Point in the direction of the magnetic field. - Middle Finger (
): Point in the direction of the conventional current. - Thumb (
): Indicates the direction of the force.
This rule allows us to design motors and know which way they will turn before connecting them.
The Laplace Law
In everyday life, we don’t usually deal with individual electrons. We use wires through which trillions of electrons flow.
If we sum the Lorentz force of all those electrons confined in a wire, we obtain the Laplace Force.
It is the force exerted on a Conductor carrying current inside a magnetic field.
Where:
- I: Current intensity (Amperes).
- L: Length vector of the conductor (meters, direction of the current).
- B: Magnetic Field (Teslas).
In scalar form, if the wire is perpendicular to the field (ideal motor case):
This formula tells us that to get more force in a motor you have three options:
- Put more current (𝐼).
- Make the motor bigger or with more turns of wire (𝐿).
- Use stronger magnets (𝐵).
From the Straight Conductor to Motor Torque
A cable shooting off in a straight line is fine for making a railgun or a loudspeaker, but not for moving a washing machine. We want it to rotate.
How do we convert that linear force into rotation? We make a loop with the wire, creating a Coil.
Think of a rectangular coil inside a magnetic field:
- On the left side of the coil, the current goes “that way”. Applying the left-hand rule, the force pushes UPWARD.
- On the right side of the coil, the current comes back “this way” (opposite direction). Applying the same rule, the force pushes DOWNWARD.
Result: We have one force going up and another going down, separated by a distance. We have created a Force Couple (Torque). We have just created a motor.
