Download Lecture #13, October 26

Lecture #4.5
Magnetic Field and Electric Current
Last time we have continued our discussion of Magnetism. Since magnetic
interactions are long range interactions we were able to introduce the concept of magnetic
field. We saw that both permanent magnets and electric currents are able to produce such
magnetic fields.
Electric current produces magnetic field and, at the same time, an external magnetic
field also affects current-carrying wires. We have shown that the absolute value of force
acting on a straight wire is
F  ILB sin 
(4.5.1)
The direction of this force obeys the same right-hand rule as the force acting on a moving
charge.
So, electric currents are affected by magnetic field and are producing magnetic
field at the same time. One of our tasks today will be to find the field produced by a
straight wire as well as the magnetic field at the center of the loop carrying electric
current.
It should be noticed that magnetic field obeys superposition principle in a same
way as electric field. So, if we want to calculate the field of a wire, we should break this
wire down into collection of infinitely small elements and perform vector summation of
all the fields due to each element. This is a difficult task which requires knowledge of
calculus. So, we will based our conclusions on experimental results
Let us start from the straight wire. We know that magnetic field forms closed loops
around this wire but we can also find direction of the field by placing magnetic needle in
different locations around the wire. The direction of magnetic field is connected with
direction of current in this wire by means of the simple right-hand rule: If you hold the
wire in your right hand with your thumb pointing in the direction of current, your fingers
curl around the wire in the direction of magnetic field.
In a same way as it was with electric field, the magnetic field is decreasing as you
move away from the wire. Experiment shows that magnetic field is proportional to the
current, I, in the wire and inverse proportional to the distance r from the wire, so
B
0 I
,
2 r
Constant 0  4 107
(4.5.2)
Tm
is known as permeability of a free space.
A
Since now we know that an electric current produces magnetic field, this means
that two different currents should interact by means of the magnetic force similar to two
charges interacting by means of electrostatic forces. Depending on the direction of
current these electric wires should repel or attract. This interaction is described by
experimentally discovered law, which sometimes is called Ampere’s law. It states that
Magnetic force acting between the two parallel wires is proportional to both
currents  I1 , I 2  , length of the wires L and inversely proportional to the distance r
between these wires.
F
2Lk I1I 2
,
r
where coefficient of proportionality k  
(4.5.3)
0
 1107 N A2 . This law follows directly
4
from combined equations 4.5.1 and 4.5.2. This force is the attractive force, when currents
are flowing in the same direction and it is repulsive force, when currents are flowing in
opposite directions.
Recall that we have never introduced any accurate definition for the SI unit of
charge, Coulomb. This is because it is not a fundamental unit but rather combination of
two other fundamental units amperes and seconds, where definition of 1 A is the amount
of current flowing in each of the parallel wires with length of 1 m and separated by
distance of 1 m that produces force on each wire of 2  107 N .
Electric current cannot exist just in a straight wire. Every time when we are
talking about electric current we have to have a closed circuit. In the most part of cases
this means that the current forms a loop. So, we have to figure out the magnetic field
produced not just by a straight wire only, but also by a closed loop carrying electric
current. For simplicity let us say that this loop has a shape of a circle. Since magnetic
field depends on the distance from the wire, it will be different at different points inside
of the loop. We shall only consider magnetic field at the center of the loop. It can be
shown that this magnetic field is equal to
B
N 0 I
,
2R
(4.5.4)
where N is the number of wire turns in the loop and R is the radius of the loop.
The analog of the Gauss’ law for magnetic interactions is known as Ampere’s law
and it states that Circulation of vector B along any closed path is equal to a constant 0
times the total current enclosed by this path. The concept of circulation is similar to the
concept of flux, which we have developed for electric field. By circulation we mean the
following: for any given closed path magnetic field may have two components. One
component B is in the direction parallel to a small element, L , of the path and another
component is perpendicular to this small element of the path. By definition circulation is
the summation of B L along the chosen closed path. Then we can write Ampere’s law
as
 B L   I
0 enclosed
,
(4.5.5)
Since we already know that magnetic field forms closed circles around the wire, the
equation 4.5.5 becomes extremely simple in this case. Because the field only has
components in the direction of circular path around the straight wire with current I we
then have
2 rB  0 I ,
which is the same as equation 4.5.2. Ampere’s law can also be used to find magnetic field
not just outside but also inside of a wire.
We have also discussed magnetic field produced by a loop of current. In practice
electric devices are not made as single loops but rather as solenoids, which is a long wire
wound into succession of closely spaced loops with geometry of a helix. This solenoid
produces a nearly uniform magnetic film inside the loops. This magnetic field inside of
the solenoid is parallel to its geometrical axis. We shall consider ideal infinitely long
solenoid, so the magnetic field outside of the solenoid is zero. To prove this fact we can
use Ampere’s law in a same way as we used Gauss’ law to show that electric field
outside of an infinite capacitor is zero. We have to consider circulation of magnetic field
vector around a rectangular loop formed in such a way that the two sides of this loop are
perpendicular to the axis of the solenoid and two sides are parallel to the axis. One of
them is inside of the solenoid and another one is outside. Since there is no field outside
and the other two sides are perpendicular to the field, the only nonzero contribution to
this circulation is due to the inside part of the loop which is BL , where L is the length of
the loop, so we have
BL  0 NI ,
B
0 NI
L
 0 nI ,
(4.5.6)
where n is the number of turns per unit of solenoid’s length. The similar approach can be
used to calculate the magnetic field of toroid.