A CurrentCarrying Coil as a Magnetic Dipole

1
A Current-Carrying Coil as a Magnetic Dipole
We have previously seen that a coil (circular
wire) with current behaves like a magnetic
dipole, in that, if we place it in an external
magnetic field B, a torque given by
acts on it. ? is the magnetic dipole moment of
the coil and has the magnitude NiA, where N is
the number of turns (or loops), i is the current
in each turn, and A is the area enclosed by each
turn.
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Magnetic Field of a coil
z
We know how the magnetic forces act on a coil
with current in a external magnetic field. But,
what is the resulting magnetic field produced
from a current in single coil.
i
Unfortunately, a coil does not have the proper
symmetry to make Amperes Law useful, so we must
apply the law of Biot and Savart.
For simplicity, we only consider observation
points on the central axis of the loop, which we
define as the Z-axis.
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Magnetic Field of a coil
The figure shows the back half of a circular loop
of radius R carrying a current i.
Point P is on the axis of the loop a distance z
from the loop center.
Biot and Savarts law will applied to a length
element located at the left side of the loop.
The vector ds is the length element on the loop
and points perpendicularly out of the page.
The angle ? between ds and the vector r is
independent of z, therefore ? 0 for all values
of z
The plane form by ds and r is perpendicular to
the plane of figure, therefore the differential
field dB at point P must lie in the plane of the
figure and perpendicular to r as shown in the
figure.
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Magnetic Field of a coil
dB is broken up into two components dB along
the z axis dB? perpendicular to the z axis
From the symmetry, the vector sum of all
perpendicular components dB? from all the loop
elements are zero. So we only have to consider
the axial component dB
For the element ds in the figure, the law of
Biot-Savart tells us that magnetic field at a
distance r is
And the parallel component is
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Magnetic Field of a coil
So combining these two relations, we obtained
r and ? are related to each other by
Making these substitutions, the above equation
becomes
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Magnetic Field of a coil
The variables R and z are constants for all
elements ds, therefore the integral becomes quite
simple
Since the integral is circumference 2?R of the
loop, we have
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Magnetic Field of a coil
It should be noted that the direction of the
magnetic field B of the loop is the same
direction as the magnetic dipole moment ?
For axial points very far away, where zgtgtR, we
can approximate the above equation to
Recalling that ?R2 is the area A of the loop and
extending out results include a coil of N turns,
we have
Since B and ? have the same direction and ? NiA
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Magnetic Field of a coil
There are two ways to view a current carrying
coil as a magnetic dipole
(1) It experiences a torque when it is placed in
a magnetic field
(2) It generates its own magnetic field that
interacts with the external field
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Quick Quiz
10
Induction and Inductance
Two Symmetric Situations
(A room full of mirrors ?)
As previously discussed, if we put a conducting
loop with a current in a magnetic field, magnetic
forces create a torque to turn it.
Then what would happen if we turned the current
off and turned the loop by hand ?? Will the
opposite occur, that is, will a current appear in
the loop ?
Yes !!! This is called Faradays law of induction.
The magnetic torque is the basis for the electric
motor, where as, Faradays law is the basis for
the electric generator.
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Faradays Two experiments
First Experiment If magnet is moved toward a
closed loop of wire, current is induced in the
wire. As the magnet is withdrawn, current is
again induced. The direction of the current is
reversed when the direction of motion is reversed.
1. A current appears only if there is relative
motion between the magnetic the current
disappears when relative motion stops.
2. Faster motion produces a greater current
3. Moving the magnets north pole toward the
loop causes counter-clockwise current, then
moving the north pole causes clockwise direction.
Whereas the south pole movements produce the
opposite directions.
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Faradays Two experiments
Second Experiment If two loops of wire are close
together and a changing current is produced in
one of the loops, then current is induced in the
second loop.
If the switch is closed, to turn on the current
in the right loop, the current meter suddenly and
briefly registers a current in the left loop.
Then the current in the left loop stops. But when
the switch is open, the current meter suddenly
deflects and then returns to a zero reading.
Current is left loop is observed ONLY when
current in the right loop is changing.
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Faradays Two experiments
In each of these two experiments, a changing
magnetic field through a closed loop induces an
EMF around the loop and the EMF generates a
current. The EMF is induced only when the
magnetic field through the loop is changing or
else when the loop is moving or rotating in a
magnetic field.
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Faradays Law of Induction
Faraday realized that a EMF and a current can be
induced in a loop, as in his two experiments,
changing the amount of magnetic field passing
through the loop.
He further realized that the amount of magnetic
field can be visualized in terms of the magnetic
field lines passing through the loop.
An EMF is induced in the loop at the left for
both figures below when the number of magnetic
field lines that pass through the loop is
changing.
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Faradays Law of Induction
To apply Faradays Law, we need a way to
calculate the amount of magnetic field that
passes through a loop.
As with the electric field, we define a magnetic
flux. A loop enclosing an area A is placed in a
magnetic field B, then the magnetic flux through
the loop is defined by the integral
over the loop area. dA is a vector of magnitude
dA that is perpendicular to a differential area
dA on the loop area
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Faradays Law of Induction
If the field is uniform over the surface, then B
can be brought out of the integral. If the field
is perpendicular to the plane of the loop. BdA
BdA and the above equation becomes
The SI units for magnetic flux is called a weber
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Faradays Law of Induction
Now that we have quantified the magnetic flux, we
can state Faradays in a more useful way.
As we will find in the next section, the induced
EMF tends to oppose the flux change, so Faradays
Law is formally written as
with the minus sign indicating the opposition. If
we change magnetic flux through a coil on N
turns, the total EMF induced is
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Faradays Law of Induction
We can change the magnetic flux through a coil by
3 different ways
1. Change the magnitude B of the magnetic field
within the coil
2. Change the area of the coil that lies within
magnetic field
3. Change the angle between the direction of the
magnetic field B and the area of the coil.
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Lenzs Law
An induced current has a direction such that the
magnetic field due to the induced current opposes
the change in the magnetic flux that induces the
current.