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CHAPTER 28
SOURCES OF MAGNETIC FIELD
BASIC CONCEPTS
Magnetic field produced by moving charge.
Magnetic field of current element.
Ampere’s Law
Biot Savart Law
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Moving Charge
A moving charge produces a magnetic field.
The field will be perpendicular to the
direction of motion of the charge.
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Magnitude of magnetic field will be
Proportional to charge
Proportional to
Proportional to speed
Proportional to
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Look at figure and note direction of field
vector.
Field is always perpendicular to the
direction of motion and a line from the
charge to the point where we measure the
field.
Therefore
is a unit vector. It has magnitude 1.
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A wire carrying current has moving charge
so a wire will produce a magnetic field.
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Therefore similar to the argument for a
moving charge we have the field for a
section of wire carrying current:
Again is a unit vector.
This equation is the Biot‐Savart Law.
It can be used to find the magnetic field of
wires in various shapes.
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BIOT‐SAVART LAW
1. The vector dB is perpendicular both to dl
(which is the direction of the current) and
to the unit vector r directed from the
element to the point P.
2. The magnitude of dB is inversely
proportional to r2, where r is the distance
from the element to the point P.
3. The magnitude of dB is proportional to
the current and to the length dl of the
element.
4. The magnitude of dB is proportional to
sin , where is the angle between the
vectors dl and r.
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Book Example
Find B at P.
Biot‐Savart Law
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Therefore a segment of wire will
contribute to the field at P.
Using and
And
We get
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Integrate over length of wire to get total
field.
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This is for a wire with length equal to .
To find the magnetic field near a very long
wire let
.
Divide by :
And
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If the wire is very long compared to the
distance from the wire then
Or at a distance from the wire
we now know the magnetic field a distance
from a long wire.
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Consider the wire shown below with
current I into the page.
Consider path with radius r passing through
the point P.
ds is in the direction of .
Therefore
P
ds
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Integrate around the circle.
But we have seen that
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Therefore
Finally
Then we generalize to any closed path l.
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This is Ampere’s Law
Can be used to find B if geometry is right.
Otherwise have to use Biot‐Savart Law.
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Find the magnetic field at P.
Must use Biot‐Savart Law.
Contribution to due to small segment of
loop will be
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The component of B in the y direction will
add to zero.
The component of B in the x direction will
be
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Find the Magnetic Field inside (r < R) of a
conducting cylinder.
We can use Ampere’s Law because of
symmetry.
r
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Choose a circular path with radius where
.
Ampere’s Law
Integrate around path.
Everyplace on the circular path
And everywhere on the path B is the same,
a constant.
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Therefore
To find need current density, j.
Current enclosed
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The field inside the cylinder.
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Find the Magnetic Field inside a solenoid.
Here we can use either the Biot‐Savart Law
or Ampere’s Law.
Ampere’s Law is easier.
The solenoid has current I and n coils per
meter.
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If we view the solenoid from the far side a
cross‐section will be:
We will use Ampere’s Law and integrate
around the path from a to b to c to d and
back to a.
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Ampere’s Law
If the solenoid is long enough the field
outside is about zero.
The field inside is uniform and pointing to
the left (right‐hand rule).
The integral then is
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Or
90 0 · 90
So
The current enclosed will be the number of
coils enclosed time I.
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Therefore
Becomes
And