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The Magnetic Field
Concepts and Principles
Moving Charges
All charged particles create electric fields, and these fields can be detected by other charged
particles resulting in electric force. However, a completely different field, both qualitatively
and quantitatively, is created when charged particles move. This is the magnetic field. All
moving charged particles create magnetic fields, and all moving charged particles can detect
magnetic fields resulting in magnetic force. This is in addition to the electric field that is
always present surrounding charged particles.
This should strike you as rather strange. Whenever a charged particle begins to move a
completely new field springs into existence (distributing business cards throughout the
universe). Other charged particles, if they are at rest relative to this new field, do not notice
this new field and do not feel a magnetic force. Only if they move relative to this new field
can they sense its existence and feel a magnetic force. It’s as if only while in motion can they
read the business cards distributed by the original moving charge, and only while in motion
does the original charge distribute these business cards in the first place! Does it sound
strange yet?
Why the magnetic field exists, and its relationship to the electric field and relative motion will
be explored later in the course. For now, we will concentrate on learning how to calculate the
value of the magnetic field at various points surrounding moving charges. Next chapter, we
will learn how to calculate the value of the magnetic force acting on other charges moving
relative to a magnetic field.
Permanent Magnets
I claimed above that the magnetic field only exists when the source charges that create it are
moving. But what about permanent magnets, like the ones holding your favorite physics
assignments to your refrigerator? Where are the moving charges in those magnets?
The simplest answer is that the electrons in “orbit” in each of the atoms of the material create
magnetic fields. In most materials, these microscopic magnetic fields are oriented in random
directions and therefore cancel out when summed over all of the atoms in the material. In
some materials, however, these microscopic magnetic fields are correlated in their orientation
and add together to yield a measurable macroscopic field (large enough to interact with the
microscopic magnetic fields present in your refrigerator door). Although this is a gross
simplification of what actually takes place, it’s good enough for now.
1
The magnetic properties of real materials are extremely complicated. In addition to the orbital
contribution to magnetic field, individual electrons and protons have an intrinsic magnetic
field associated with them due to a property called spin. Moreover, even neutrons, with no net
electric charge, have an intrinsic magnetic field surrounding them. To learn more about the
microscopic basis of magnetism, consider becoming a physics major …
Electric Current
Moving electric charges form an electric current. We will consider the source of all magnetic
fields to be electric current, whether that current is macroscopic and flows through a wire or
whether it is microscopic and flows “in orbit” around an atomic nucleus.
The simplest source of magnetic field is electric current flowing through a long, straight wire.
In this case, the magnitude of the magnetic field at a particular point in space is given by the
relation,
B
0i
2r
where

0 is the permeability of free space, a constant equal to 1.26 x 10-6 Tm/A,

i is the source current, the electric current that creates the magnetic field, measured
in amperes1 (A),

r is the distance between the source current and the point of interest,

and the direction of the magnetic field is tangent to a circle centered on the source
current, and located at the point of interest. (To determine this tangent direction,
place your thumb in the direction of current flow. The sense in which the fingers of
your right hand curl is the direction of the magnetic field. For example, for current
flowing out of the page, the magnetic filed is counterclockwise.)
1
One ampere is equal to one coulomb of charge flowing through the wire per second.
2
Another common source of magnetic field is electric current flowing through a circular loop
of wire. In this case, the magnitude of the magnetic field at the center of the loop is given by
the relation,
B
0i
2R
where

Ris the radius of the loop,

and the direction of the magnetic field is perpendicular to the loop. (To determine
this direction, again place your thumb in the direction of current flow. The sense in
which the fingers of your right hand curl is the direction of the magnetic field. For
example, for current flowing counterclockwise around the loop the magnetic field
inside the loop points out of the page, and the magnetic field outside of the loop
points into the page.)
3
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The Magnetic Field
Analysis Tools
Long, Parallel Wires
Find the magnetic field at the indicated point. The long,
parallel wires are separated by a distance 4a.
+2i
-i
The magnetic field at this point will be the vector sum of the magnetic field from the left wire
and the magnetic field from the right wire.
For the left wire, I’ve indicated the direction of the magnetic
field. Remember, with your thumb pointing in the direction of
the current (out of the page), the direction in which the fingers
of your right hand curl is the direction of the tangent vector
(counterclockwise).
+2i
-i
The magnitude of the magnetic field from the left wire is:
Bleft 
Bleft 
Bleft 
 0i
2r
 0 (2i )
2 (2a ) 2  (a ) 2
0i
 5a
To determine the direction of this field, notice that
the magnetic field vector is at the same angle relative
to the y-axis that the line connecting its location to
the source current is relative to the x-axis. This line
forms a right triangle with  given by:
a
2a
tan   0.5
tan  

+2i
a

2a
  26.6 0
4
Thus,
Bleftx  
Bleftx
 0i
Blefty  
sin 26.6
 5a
i
  0 (0.45)
 5a
 0i
Bleftx  0.064
Blefty
 0i
Blefty  0.128
a
cos 26.6
 5a
i
  0 (0.89)
 5a
 0i
a
Repeating the analysis for the magnitude of the field from the right wire gives:
Bright 
Bright 
Bright 
 0i
2r
 0 (i )
2 (2a ) 2  (a ) 2
 0i
2 5a
With your thumb pointing in the direction of the
current (into the page), the direction in which the
fingers of your right hand curl is the direction of the
tangent vector (clockwise).

Notice that the magnetic field is at the same angle
relative to the y-axis as before, although this
magnetic field has positive x- and y-components.
a

-i
2a
Thus,
 0i
sin 26.6
2 5a
 0i
Brightx 
(0.45)
2 5a
 0i
0i
cos 26.6
2 5a
0i
Brighty 
(0.89)
2 5a
 0i
Brightx 
Brightx  0.032
Brighty 
Brighty  0.064
a
5
a
The resultant magnetic field at this point is the sum of the fields from the two source currents:
B x  Bleftx  Brightx
B x  0.064
B x  0.032
 0i
a
 0.032
B y  Blefty  Brighty
 0i
B y  0.128
a
 0i
B y  0.192
a
0i
a
 0.064
 0i
a
 0i
a
Thus, the magnetic field is predominately in the +y-direction with a slight leftward (-x)
component. The magnitude of this field could be determined by Pythagoras’ theorem, and the
exact angle of the resulting field determined by the tangent function.
Loops of Wire
The inner coil consists of 100 circular loops of wire with
radius 10 cm carrying 3.0 A counterclockwise and the
outer coil consists of 500 circular loops of wire with
radius 25 cm carrying 1.5 A clockwise. Find the
magnetic field at the origin.
The magnetic field at the origin will be the vector sum of the magnetic fields from the two
coils. Each loop of the inner coil produces:
B
0i
2R
(1.26 x10 6 )(3)
B
2(0.10)
B  18.9 x10 6 T
Since there are 100 identical loops of wire making up this coil, the total field produced by the
inner coil is:
B  100(18.9 x10 6 T )
B  1.89 x10 3 T
This field is directed out of the page.
6
Each outer coil produces:
B  N(
 0i
)
2R
(1.26 x10 6 )(1.5)
B  500
2(0.25)
B  1.89 x10 3 T
This field is directed into the page.
Since the two coils produce equal magnitude but opposite directed magnetic fields at the
origin, the net magnetic field at the origin is zero. (This does not mean, however, that the field
is equal to zero at any other points in space.)
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The Magnetic Field
Activities
8
Determine the direction of the net magnetic field at each of the indicated points. The wires are long,
perpendicular to the page, and carry constant current either out of (+) or into (-) the page.
a.
+i
-i
+i
+i
b.
9
Determine the direction of the net magnetic field at each of the indicated points. The wires are long,
perpendicular to the page, and carry constant current either out of (+) or into (-) the page.
a.
-i
-i
+i
-2i
b.
10
Determine the direction of the net magnetic field at each of the indicated points. The wires are long,
perpendicular to the page, and carry constant current either out of (+) or into (-) the page.
a.
+i
+i
+i
+i
-i
-i
+i
b.
11
Determine the direction of the net magnetic field at each of the indicated points. The wires are long,
perpendicular to the page, and carry constant current either out of (+) or into (-) the page.
a.
+i
+i
+i
+i
b.
+i
-i
-i
12
Determine the direction of the net magnetic field at each of the indicated points. All figures are planar and
the wires are long and carry constant current in the direction indicated.
a.
i
i
b.
2i
i
c.
3i
i
d.
4i
2i
e.
3i
2i
13
Determine the direction of the net magnetic field at each of the indicated points. All figures are planar and
the wires are long, insulated from each other, and carry constant current in the direction indicated.
a.
b.
i
2i
i
i
c.
d.
2i
i
2i
3i
14
Determine the direction of the net magnetic field at each of the indicated points. The points are in the
parallel or perpendicular plane passing through the center of each circular hoop.
a.
b.
i
i
c.
d.
i
i
i
15
2i
For each of the current distributions below, indicate the approximate location(s), if any, where the magnetic
field is zero. The wires are long, perpendicular to the page, and carry constant current either out of (+) or
into (-) the page.
a.
+i
+i
+i
- i
+2i
+2i
+2i
-4i
-2i
+8i
b.
c.
d.
e.
16
For each of the current distributions below, indicate the approximate location(s), if any, where the magnetic
field is zero. The wires are long, perpendicular to the page, and carry constant current either out of (+) or
into (-) the page.
a.
-i
-i
-i
-i
b.
+i
-i
-i
+i
17
For each of the current distributions below, determine and clearly label the regions where the magnetic
field points into and out of the plane of the page. All figures are planar and the wires are long, insulated
from each other, and carry constant current in the direction indicated.
a.
b.
i
2i
i
i
c.
d.
2i
i
2i
3i
18
A pair of long, parallel wires separated by 0.5 cm carry 1.0 A in opposite
directions. Find the magnetic field directly between the wires.
+i
-i
Qualitative Analysis
On the graphic above, sketch the direction of the magnetic field at the requested point. Explain why the
magnetic field points in this direction.
Mathematical Analysis
19
Four long, parallel wires with spacing 1.5 cm each carry 350 mA.
Find the magnetic field at the center of the wire array.
-i
+i
-i
Qualitative Analysis
On the graphic above, sketch the direction of the magnetic field at the requested point. Explain why the
magnetic field points in this direction.
Mathematical Analysis
20
+i
a.
Find the magnetic field at each of the indicated points. The
long, parallel wires are separated by a distance 4a.
+i
d.
Mathematical Analysis
21
b.
-i
c.
a.
Find the magnetic field at each of the indicated points. The
long, parallel wires are separated by a distance 4a.
+2i
d.
Mathematical Analysis
22
b.
+i
c.
The long, parallel wires at right are separated by a distance 2a. Find the
magnetic field at all points on the y-axis.
+i
+i
Mathematical Analysis
Questions
For all points on the y-axis, what should By equal? Does your function agree with this observation?
At y = 0, what should Bx equal? Does your function agree with this observation?
Sketch Bx below. Indicate the value of the function at y = 0.
B
y
23
The long, parallel wires at right are separated by a distance 2a. Find the
magnetic field at all points on the y-axis.
+i
-i
Mathematical Analysis
Questions
For all points on the y-axis, what should Bx equal? Does your function agree with this observation?
Sketch By below. Indicate the value of the function at y = 0.
B
y
24
The long, parallel wires at right are separated by a distance 2a.
Determine the location(s), if any, where the magnetic field is zero.
+i
Mathematical Analysis
25
+2i
The long, parallel wires at right are separated by a distance 2a.
Determine the location(s), if any, where the magnetic field is zero.
-2i
Mathematical Analysis
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+i
The long, parallel wires at right are separated by a distance 2a.
Determine the location(s), if any, where the magnetic field is zero.
-2i
Mathematical Analysis
27
+3i
Find the magnetic field at the origin. The inner radius R coil consists of N
loops of wire carrying current i counterclockwise and the outer radius 3R
coil consists of 2N loops of wire carrying current i clockwise.
Mathematical Analysis
28
Find the magnetic field at the origin. The inner radius R coil consists of
3N loops of wire carrying current i counterclockwise and the outer radius
3R coil consists of N loops of wire carrying current 2i counterclockwise.
Mathematical Analysis
29
A long wire carrying 2.0 A is bent into a circular loop of radius 5.0 cm as
shown at right. Find the magnetic field at the center of the circle.
Mathematical Analysis
30