What do the symbols in the Lorentz Force Equation mean? And exactly how do Electric and Magnetic Fields influence Charged Particles placed within them?

Hey everyone, Parth here again with another equation explanation video. Today, we're looking at the Lorentz Force equation. This equation describes the forces exerted by both electric and magnetic fields on a charge particle found within these fields.

Named after Hendrik Lorentz, the equation can simply be split up into two parts. The force exerted by electric fields, and the force exerted by magnetic fields on charged particles.

In this video, we begin by seeing how electric and magnetic field lines are useful descriptions of how charges and magnets respectively interact with fields. Both can be described as vector fields, meaning we can assign a vector to each point in space. Therefore, we can also represent these fields as vectors in our mathematics.

When a charged particle is placed in an electric field, the field exerts a force on it equal to the charge of the particle multiplied by the electric field strength at that point in space. The direction of the force exerted is the same as the direction of the electric field at that point.

The magnetic component is a bit more complicated. When a charged particle is stationary in a magnetic field, the field does not exert a force on it. However if the particle moves relative to the field, then a force is exerted perpendicular to both the direction of the particle's motion, and the field lines. This is why the magnetic force is given by the particles charge multiplied by the cross product (vector product) between the particle's velocity vector and magnetic field lines.

In this video, we also see how to visualize vector products between two vectors. We see that if the original two vectors are perpendicular, then the cross product between them is maximal in size, but if they are aligned, its size is zero. The vector product is also always perpendicular to both the original vectors.

Additionally, we also see that a particle initially moving with a constant velocity in a constant magnetic field will undergo circular motion because the magnetic field exerts a force perpendicular to its velocity.

Finally, we see that the Lorentz Force equation is just a combination of the electric force and magnetic force experienced by a particle that finds itself in a region where both an electric and a magnetic field exist.

Timestamps:
0:00 - The Lorentz Force Equation, and its relation to Electromagnetism
0:31 - Electric and Magnetic Field Lines to Represent Vector Fields
3:00 - Charged Particles + Electric and Magnetic Parts of Lorentz Force
4:05 - The Force Exerted by the Electric Field on a Charged Particle (qE)
4:48 - The Force Exerted by the Magnetic Field on a Charged Particle (qvxB)
5:36 - Visualizing the Vector Product / Cross Product Between Two Vectors
6:45 - Circular Motion of Moving Charged Particles in Magnetic Fields
8:47 - Putting Everything Together into the Lorentz Force Equation

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