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The Magnetic Force

As we shall see later, the "Laws" of ${\cal E}$&${\cal M}$ are so symmetric between electrical and magnetic phenomena that most Physicists are extremely frustrated by the fact that no one has ever been able to conclusively demonstrate the existence (other than theoretical) of a "magnetic charge" (also known as a magnetic monopole ). If there were magnetic charges, the magnetic force equation would look just like the gravitational and electrostatic force laws above and this part of the description would be nice and simple. Alas, this is not the case. Static (constant in time) magnetic phenomena are generated instead by the steady motion of electric charges, represented by a current I (the charge passing some fixed point per unit time) in some direction $\Vec{\ell}$. Usually (at least at the outset) we talk about currents flowing in a conductor (e.g. a wire) through which the charges are free to move with minimal resistance. Then $\Vec{\ell}$ is a vector length pointing along the wire, or (if the wire is curved) $d\Vec{\ell}$ is an infinitesimal element of the wire at some point. We may then think in terms of a "current element" $I d\Vec{\ell}$.

One such current element $I_1 d\Vec{\ell}_1$ exerts a magnetic force d $\Vec{F}_{12}^M$ on a second current element $I_2 d\Vec{\ell}_2$ at a distance $\Vec{r}_{12}$ (the vector from #1 to #2) given by

$$d\Vec{F}_{12}^M \; = \; k_M \; {I_1 I_2 \over r_{12}^2} \; d\Vec{\ell}_2 \times (d\Vec{\ell}_1 \times \hat{r}_{12})$$ (17.4)

where k_M is yet another unspecified constant to make all the units come out right [just wait!] and again $\hat{r}_{12}$ is the unit vector defining the direction from #1 to #2.

  

Figure:  The magnetic force $d\Vec{F}_{12}^M$ on current element $I_2 d\Vec{\ell}_2$ due to current element $I_1 d\Vec{\ell}_1$.

This ugly equation (4) does give us some important qualitative hints about the force between two current-carrying wires: the force between any two elements of wire drops off as the inverse square of the distance between them, just like the gravitational and electrostatic forces [although this isn't much use in guessing the force between real current-carrying wires, which don't come in infinitesimal lengths] and the force is in a direction perpendicular to both wires. In fact, if we are patient we can see which way the magnetic forces will act between two parallel wires: we can visualize a distance vector $\Vec{r}$ from the first wire (#1) over to the second wire (#2); let it be perpendicular to both for convenience. The " RIGHT-HAND RULE" will then tell us the direction of $(d\Vec{\ell}_1 \times \hat{r}_{12})$: if we "turn the screw" in the sense of cranking through the angle from $d\Vec{\ell}_1$ to $\hat{r}_{12})$, a right-handed screw [the conventional kind] would move in the direction labelled $d\Vec{B}_{12}$ in Fig. 17.2. This is the direction of $(d\Vec{\ell}_1 \times \hat{r}_{12})$. Now if we crank $d\Vec{\ell}_2$ into $d\Vec{B}_{12}$, the turn of the screw will cause it to head back toward the first wire! Simple, eh?

Seriously, no one is particularly enthused over this equation! All anyone really retains from this intricate exercise is the following pair of useful rules:

1.

Two parallel wires with electrical currents flowing in the same direction will attract each other.

2.

Two parallel wires with electrical currents flowing in opposite directions will repel each other.

Nevertheless, electrical engineers and designers of electric motors and generators need to know just what sorts of forces are exerted by one complicated arrangement of current-carrying wires on another; moreover, once it had been discovered that moving charges create this weird sort of action-at-a-distance, no one wanted to just give up in disgust and walk away from it. What can we possibly do to make magnetic calculations manageable? Better yet, is there any way to make this seem more simple?