Introductory Physics Volume Two

5.4 More Examples 105
distance r from the center the current density is J(r) =
I
2πRr . Note
that this is not a very realistic current density but it is easy to work
with. Find the magnetic field strength both inside and outside of this
wire.
§ 5.3 Force Between Parallel Wires
Now that we know that the strength of the magnetic field produced
by a long straight current carrying wire we can find the force between
two parallel wires.
In the figure above the field produced by current B in the region of
current A is shown. Using the right hand rule we can see that the
direction of the force on wire A is toward wire B. We can also see that
the field and the current are perpendicular so that the magnitude of
the force on a length L of wire A is
F = I A LB = I A L µ 0I B
2πd = µ 0I A I B L
2πd
where d is the distance between the wires.
Theorem: Force Currents
Two long parallel wires have a force per length between them of
F
L = µ 0I A I B
2πd
where d is the distance between the currents.
⊲ Problem 5.6
An electrical cable carries 45 amps in each of two wires that are a
distance of 4mm apart. The currents are in opposites directions.
(a) Is the force between the wires attractive or repulsive?
(b) What is the force per length between the wires?
§ 5.4 More Examples