ELECTROMAGNETIC FUNDAMENTALS FOR SIGNAL INTEGRITY - Advanced Signal Integrity for High-Speed Digital Design - page 44

dl = νdt in a magnetic field, the amount of If a charge Q moves an amount

work done is

dW m = F m · dl = Q(ν × B) · ν dt = 0 (2-81)

ν × B is perpendicular to the flow of the current, which follows the
path of dl . In other words, the work done by the magnetic field is zero because
the force is perpendicular to the moving charge. Therefore, the magnetic field
can change the direction of the moving particle, but it cannot speed it up or slow
it down. This concept may be confusing, especially when considering the simple
electromagnets that we all played with in high school, because we all know that
we can use an electromagnet to pick up a paper clip. Since we are moving the
mass of the paper clip against Earth's gravitational field, we know that work is
being done. However, if the work is not being done by the magnetic field, what
is doing the work? The answer is demonstrated in the following example.

Note that

Example 2-4 Consider a long wire carrying current I 1 in the presence of a rigid
rectangular loop carrying current I 2 , as shown in Figure 2-16. The long wire will
generate a magnetic field as calculated in Example 2-2:

I 1 µ 0
2 πr

B 1 =

Calculate the magnetic force.

x

B C

F

F

F

z

y

l 2

b

F

A D

F 1

a

l 1

Figure 2-16 Forces generated on a wire loop in the vicinity of a magnetic field generated
from a wire.