Magnetic field along a current-carrying wire (conceptual question)?

[lillybeans]

Homework Statement

Is the magnetic field around a point in a current-carrying wire radially outward (spherically) like the electric field? Or is it a bunch of concentric circles extending in one plane?

I've always thought of it as a bunch of concentric loops in the same plane, however... After I learned Biot-Savart's law, which is:

I wasn't so sure anymore. Why? see diagram

My Question: If the magnetic field around a segment of wire, ds, is a bunch of concentric loops extending in one plane (YZ plane), HOW can point P, which is OUTSIDE of the magnetic field's plane at that point (it's in the XY plane), experience the magnetic field due to a segment far away? Also, how can a segment far away generate a magnetic field at P, like in case 2, where P is not directly above a current-carrying wire?

 

[Harrisonized]

The magnetic field is found using a cross product. The following is the Biot-Savart's law

B = μ₀/(4π) ∫(i.dl×r)/|r³|, where bold denotes "vector".

Let's decompose what this means. Let's pretend that we're only taking the field from a single point contribution. Then we may as well rewrite i.dl, the as q.dv, where q is the charge on a single point particle and dv is the microscopic velocity of this point particle. Makes more sense, right?

B = μ₀/(4π) ∫(q.dv×r)/|r³|

The integral is a path integral over the space that the charge moves. Let's consider the point charge and it's velocity at a single time. Hence, let dv = v₀ at time t₀, so that we can remove the integral. Then

B = μ₀/(4π) (q.v₀×r)/|r³|

Essentially, this is

B = K.v₀×r, where K is a constant and K=μ₀/(4π) q/|r³|

Now if you want to visualize this, put it on a coordinate system. Let v₀≝(0,0,v₀) and calculate the field that comes from this.

Point | Field
(1,0,0) | K.v₀(0,1,0)
(1,1,0) | K.v₀(-1,1,0)
(1,-1,0) | K.v₀(1,1,0)
(0,0,1) | 0

It should be clear now the field is cylindrical with decreasing strength as one leaves the origin. Also, along the z-axis, the field is 0.

What sphere?

 

[lillybeans]

The magnetic field is found using a cross product. The following is the Biot-Savart's law

B = μ₀/(4π) ∫(i.dl×r)/|r³|, where bold denotes "vector".

Let's decompose what this means. Let's pretend that we're only taking the field from a single point contribution. Then we may as well rewrite i.dl, the as q.dv, where q is the charge on a single point particle and dv is the microscopic velocity of this point particle. Makes more sense, right?

B = μ₀/(4π) ∫(q.dv×r)/|r³|

The integral is a path integral over the space that the charge moves. Let's consider the point charge and it's velocity at a single time. Hence, let dv = v₀ at time t₀, so that we can remove the integral. Then

B = μ₀/(4π) (q.v₀×r)/|r³|

Essentially, this is

B = K.v₀×r, where K is a constant and K=μ₀/(4π) q/|r³|

Now if you want to visualize this, put it on a coordinate system. Let v₀≝(0,0,v₀) and calculate the field that comes from this.

Point | Field
(1,0,0) | K.v₀(0,1,0)
(1,1,0) | K.v₀(0,1,-1)
(1,-1,0) | K.v₀(0,1,1)
(0,0,1) | 0

It should be clear now the field is a superposition of
1. a field that points toward the charge
2. a field that points around the velocity of the charge (the cylinder-like field that you usually draw around a wire)

Also, along the z-axis, the field is 0.

What sphere?

Thank you for the detailed explanation, though my question remains unanswered. If you look at the diagram I drew, at ds (the red segment), the field only extends outward in one plane. Point P does not experience the field due to this segment simply because it is not in the plane of the field generated. (i.e. none of the green circles I drew will ever touch the blue point, they will always remain parallel to each other) So I don't see how point P can be affected by the field due to any segment that it is not directly below P, since magnetic field is NOT like the electric field, which can expand radially outward in all directions (like a sphere).

So a segment of wire does not only produce a magnetic field immediately surrounding it, but also at a distance?

 

[Harrisonized]

I answered your question sufficiently. The field exists everywhere except along the axis of the wire. However, I just realized that I made a mistake in my post earlier. The shape of the field is always cylindrical, and I edited my previous post just now to reflect that.

 

[lillybeans]

I answered your question sufficiently. The field exists everywhere except along the axis of the wire. However, I just realized that I made a mistake in my post earlier. The shape of the field is always cylindrical, and I edited my previous post just now to reflect that.

I see now, thank you very much.