Lorentz Force Law

The Lorentz force law contains two contributions. The first term in the equation tells us that an arbitrary (positive) charge q moving in an electric field feels a force in the same direction as the electric field E. This means that if the electric field is uniform in space and constant in time, the force exerted on the charged particle will be constant, and therefore the particle will move with a constant acceleration in the direction of the electric field. A big charge feels a stronger force than a small charge, and will feel a stronger acceleration if it has the same mass. Indeed, electric fields can be used to accelerate particles. This force also leads to the attraction between two oppositely charged particles: the one charge feels the force of the electric field caused by the other, which may result in them forming a bound pair.

Now the second term, describing the force resulting from a magnetic field B, is of a very different nature. Note that the force is again proportional to the charge q, but also to the velocity v of the particle. Maybe surprisingly, this means that a particle at rest does not feel the magnetic field. The 'cross' product of the two vectors v and B means that the force is perpendicular to both the velocity of the charge and the magnetic field, as indicated in the figure. The magnitude of the cross product equals the product of the magnitudes of v and B times the sine of the angle between them. So if the two vectors are parallel, their cross product is zero (because sin 0 = 0).

If the particle moves through a constant magnetic field, then it will feel a constant force perpendicular to its velocity all the time. It will therefore undergo an acceleration of constant magnitude perpendicular to its velocity. If the velocity is perpendicular to the field to begin with, then the upshot is that the charge will move in a circular orbit in a plane perpendicular to the direction of the field. If the velocity is not perpendicular to the field, the component of the velocity in the direction of the field will not change, while the one perpendicular will cause a circular motion. Combining the two velocity components, one finds that a charge will move in a spiral motion around the magnetic field lines.

Questionnaire:

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