# Why is the Magnetic Force Perpendicular to the Motion

Q:
My question is a bit light-weight compared to some of the others, but i have been curious as to why an electron will experience a perpendicular force to the direction of travel, in a magnetic field and why that direction? this sort of phenomena is used in particle accelerators but I have had trouble finding an answer.
- Matt (age 16)
Australia
A:

That's quite a deep question.  The superficial answer is simply that the Lorentz (magnetic) force is proportional to v×B, where v is the particle velocity and B is the magnetic field. Since the vector cross product is always at right angles to each of the vector factors, the force is perpendicular to v. To give a more explanatory answer, we have to say something about why this force exists with that form. We have to start with some deeper principles.

Here's how the argument is often made, e.g. in Purcell's book on Electricity and Magnetism. We start with special relativity, specifically the Lorentz-Fitzgerald contraction effect. If something  is in motion relative to you, it shrinks along the direction of that motion, compared to the dimensions it has according to someone at rest with respect to the object.

Now think of an electrically neutral wire, with positive charges moving to the right and negative ones moving to the left. That's a simple symmetrical way of describing a current, a source of a magnetic field. Look at a positively charged particle up a bit from the wire, standing still in the wire frame. It sees no force, since the wire is neutral. If it's set in motion in any direction perpendicular to the wire, it sees no contraction of either the positive or negative line of charges. What if it's moving a bit parallel to the wire, say to the right? The negative charge line is more contracted in its frame, since it's moving to the left, and the positive charge line is less contracted. So our charged particle sees a more concentrated line of negative charges. From its point of view, the nearby wire is negatively charged, and it will experience a net electric field and accelerate toward the wire.

Switching back to the frame where the wire is stationary, we have to account for why that moving particle is accelerating toward the wire even though in this frame there's no electric field. We invent a different field, one which only causes moving charges to accelerate. We call that the magnetic field. As you can see in this example, it causes acceleration at right angles to the motion. That property turns out to be general, regardless of the details of the source of the magnetic field.

Mike W. (posted without checking until Lee returns)

p.s. I took your question out of that old quantum thread, since yours doesn't involve quantum.

(published on 05/25/2013)

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