Magnetic fields are generated by currents (charges in motion). However, we know that motion is relative. If I run along side a moving car at exactly the same speed, the car will appear to be stationary in that it is moving away from and I am not moving toward it. Now, since moving charges generate magnetic fields, if I see a charge zip by and I am standing still, I will measure a magnetic field due to the moving charge. Suppose that I run alongside the charge so that there is no relative motion between us. Will the magnetic field disappear (in my eyes) levaing only the electric field of the chrage? Yes. I can eliminate the effects of magnetic fields by appropriate motions. This was a big clue that electric and magnetic fields are related.

We know that there are magnets found in nature with no obvious sources of current. The sources for these fields are currents on the atomic or subatomic levels. We return to discussions of dia-magnetism, para-magnetism, and ferro-magnetism later. We first consider the generation of magnetic fields by macroscopic current flows.

Imagine a straight wire along which flows a current I. The strength of the magnetic produced by the wire is

B = mu I/(2 pi r)

mu(air) = 4 pi x 10^{-7} T-m/A

That is, the field are circles about the wire. If the current comes out of the paper, then the circles are in the CCW-sense. If the current points into the paper, then the circles are in the CW-sense.

Example

67. A long wire is stretched horizontally to a 1.0-m diameter tree, around it and back. Both ends are connected to the terminals of a battery and 10 A is drawn by the wire. If both lengths are parallel and the Earth's field is negligible, what force exists between each meter of the wires?

• The field due to one wire is
  B = (mu(air)/2 pi)I/r = (4 pi / 2 pi) x 10^{-7} 10 A / 1 m = 2 x 10^{-6} T
  If this is the left wire and the current is coming out of the paper, then the field circulates in the CCW-sense.

  

  The field of the right wire is the same except that it circulates in the CW-sense

• The Force on the right wire is F = l I x B. I x B points to the right and so, per unit meter, we have F/l = 10 A x 2 x 10^{-6} T = 2 x 10^{-5} N/m which tries to push the wires apart.

18. Two long horizontal straight parallel wires are 28.28 cm apart and each carries a current of 2.0 A in the same direction, namely due south. WHat is the B-field at a point that is a perpendicular distance of 20 cm from both wires?

If we stand at the south point on the horizon we see



The fields circle around each wire in the CCW-sense. So, a point which is 20 cm from both wires sits on the line which passes through the midpoint of the positions of the wiree



We see that the vertical components of B vanish so that there is only a leftward pointing field on the line. The magnitude of the field of one wire is

B = (mu/2 pi) I/r 14.14/20 = 2 x 10**(-7) 2.0 A/0.2 m x 0.71 = 1.4 x 10**(-6) T

The total field is then B = 2.8 x 10**(-6) T.

Suppose we wrap a wire around on itself to make a loop (a ring). What does the field look like? It is hard to argue on general grounds what it will look like (given the tools that we posses). The field looks like:

At the center of the loop

B = mu I /(2R)

Examples

Stacked Loops

Suppose we stack up N loops of current. By the Principle of Superposition, the effects simply add and we have that the strength of the field increases by a factor of N, i.e., B(center) = N mu I/(2R)

A helical wire. Imagine something like a compressed slinky through which current flows. Such a configuration is referred to as a Solenoid. This is slightly different from a set of stacked loops. In loops the current closes on itself and there is no net drift of charge from left-to-right or right-to-left. In a slinky, there is a slow drift of charge from end-to-end. The latter effect can be eliminated if one winds to the end and then doubles back to the begininning to close the loop. If you make a very long solenoid then

where the field at the center of the solenoid is

B = mu(air) n I