# Magnetic Field Vector due to Infinitely long wire

Hi everybody, I have quite a simple question here.

I know that for an infinitely long current carrying wire, the magnitude of the magnetic field is given by (mu_0*I)/(2*Pi*R) where mu_0 is the magnetic permeability of free space.

What I would like to know is what is the vector form of this equation? i.e. How can you determine which way the B-field points? I've looked all over the internet but can't find anything! Does there need to be a cross product with r-hat in there somewhere!

## The Attempt at a Solution

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dx
Homework Helper
Gold Member
The direction of the B field can be worked out by the right hand rule. If you point your right thumb in the direction of the current, the direction in which your right hand fingers curl is the direction of the field.

Thanks for the quick response! What I was trying to get at, though, is there a way of putting that into some kind of notation with vectors? A cross product of the current vector and r-hat will give the direction of the B-field (I think!) but as I have never seen this anywhere, is it right?

dx
Homework Helper
Gold Member
Yes, thats right.

...but as I have never seen this anywhere, ...
Look up the Biot-Savat Law.

The direction of the vector for a B magnetic field is figured out by knowing the direction of the flow of current and the direction of the vector that points towards the point analyzed (radius).

IE:

(direction of magnetic field ONLY, not for magnitude) B = L X r

Direction of magnetic field B is determined by the cross product of the direction of the flow of current (here denoted by L) by the direction of the point being analyzed from the wire here denoted by r.

So say for example, if you had a closed circular wire with a current flowing counter-clockwise and you want to figure out the direction of its magnetic field at the center of the closed wire. The direction of the field would point upwards, or with vector k.

(pick any arbitrary point on the closed circular wire and say for example it's on the plane z=0 with it's center at the origen. The direction of the current is always tangent to the circle, or perpendicular to it's radius. I pick an easy point on the circle to work with, (x,0,0), in cartesian coordinates and x represents a radius however long desired. Since it flows counter-clockwise L has the direction j. The direction of the vector from (x,0,0) to the origen, r, is simply -i. j X -i = k. Picking any other points would still produce k as the direction since L and r are coplanar.)

For an infinitely long wire it's the same thing and technically easier. The wire is always in one direction and you analyze the magnetic field at whatever point desired. Since here L and r are again coplanar, the direction, with respect to the plane that contains them, points either up or down depending of the directions of the two.

And when trying to find magnitudes, you work with the shortest r; ie, with a point on the wire that has the shortest distance to the point analyzed. (making L and r perpendicular). Which would make sense since |a X b| = a*b*sin(theita), or the length of the component of b which is perpendicular to a.

And the r hat you mentioned is essentially the same r that I use. So yes, for the L and r that I use, use unitary vectors, since you only want to figure out the direction. The magnitudes are down by actually doing the cross product. I hope this helps, although I know it's a bit late now, lol