Magnetic Force on Charges and Wires

A stationary charge in a magnetic field feels nothing. However, the moment a charge starts moving, the magnetic field exerts a force on it. This is the foundation of the Lorentz Force. The magnitude of this force depends on four factors: the amount of charge ($q$), how fast it is moving ($v$), the strength of the magnetic field ($B$), and the angle ($\theta$) at which it enters the field. The formula is written as $F = qvB \sin \theta$. The force is at its maximum when the charge moves perpendicular ($90^\circ$) to the field lines and drops to zero when moving parallel ($0^\circ$) to them.

Since an electric current is just a stream of many moving charges, a current-carrying wire also experiences a force when placed in a magnetic field. This is known as the motor effect. Instead of looking at a single particle, we look at the current ($I$) and the length ($L$) of the wire inside the field. The formula evolves into $F = ILB \sin \theta$. This principle is exactly how electric motors work: by using magnetic fields to push on wires, we convert electrical energy into mechanical motion.

Magnetic force is unique because it acts perpendicular to both the velocity of the charge and the magnetic field. To predict this 3D direction, we use the Right-Hand Rule. 1. Point your Thumb in the direction of the velocity ($v$) or current ($I$). 2. Point your Fingers in the direction of the magnetic field ($B$). 3. Your Palm now points in the direction of the Force ($F$) for a positive charge. Note: If the charge is negative (like an electron), the force acts in the opposite direction (out the back of your hand)!