Electrodynamics/Lorentz Force

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We must at this point ask, if a charge is inserted into a system with E and B fields, what will be the effect of those two fields on the charge? In other words, what is the force exerted on a point charge by a combined electromagnetic field?

Force from E Field[edit]

\mathbf{F}_E = q\mathbf{E}

Force from B Field[edit]

\mathbf{F}_B = q\mathbf{v} \times \mathbf{B}

Notice that F_B is always perpendicular to the velocity and to the field, and that the force is zero when the particle is traveling along with the field lines. In general, the magnetic force cause the particle to move in spirals.

Notice also that in place of a charge q with a velocity, we can use an electric current. This means that a magnetic field will exert a force against a wire carrying a current as such:

\mathbf{F}_B = \int (\mathbf{I} \times \mathbf{B})\,dl

We will touch on the interaction between currents and magnetic fields again later when we discuss Faraday's Law.

Since F_B is always perpendicular to velocity, and since the displacement is always parallel to velocity, F_B \cdot s=0. The magnetic field never does any work!

Lorentz Force[edit]

\mathbf{F} = q\mathbf{E} + q(\mathbf{v} \times \mathbf{B})