Finding magnetic field at a specific location

Question

A long, straight wire carries a current of 2.5 μA. An electron moves parallel to the wire with a constant speed of 3.6x10⁴ m/s at distance "d" below the wire. What is the magnitude and direction fo the magnetic field at the location of the electron?

Answer 1

Hi Gnarls (I liked "Crazy," by the way):

The trick in this kind of problem typically is what they do not tell you, because it seems like there may not be enough information. The key, though, is in the phrase "parallel to the wire." A charged particle in a field should usually curve. If it does not, there must be some other force on it balancing out the magnetic force. In this case, it is the force of gravity on the electron.

Therefore, the magnetic force on the electron needs to be "up" (i.e. toward the wire, since the electron is said to be below). And its magnitude must equal mg for the electron (whose mass can be easily looked up).

Since the electron is traveling parallel to the wire, and the magnetic field of the wire exists in concentric circles around the wire, the particle must be traveling perpendicular to the field as it move parallel to the wire (by geometry). Therefore, the magnetic force on it is qvB. Since v is given and the charge q of the electron is also a commonly known value (easy to look up), this allows solving for the strength of the magnetic field at the electron's position.

There is a standard formula for the magnetic field a perpendicular distance r from a long, straight wire:

B = μ_oi/2πr, where r is the perpendicular distance and i is the current (and μ_o is (magnetic) permeability (of free space), another universal constant)

This will allow you to solve for r, the distance from the wire.

The direction of the current is a choice between "in the same direction as the particle" and "opposite the direction of the particle." You need to use the right-hand rule for the force on a moving charge to determine the direction of the field to get an upward force on the electron below the wire (remember that the electron has a negative electric charge). One you determine the direction of the field below the wire, then determine (by a different right-hand rule) the direction the current needs to go in the wire to get the magnetic field in the proper direction below it.

For comparison, I got the distance to be 323 m and the direction of the current to be opposite the direction of the electron's travel (I do not make any guarantees about the correctness of these results, but I checked my work a couple times and got the same answer; if I think of an error later, I will repost).

I hope this helps!