Energy Stored in a Magnetic Field

1
Energy Stored in a Magnetic Field
To derive a quantitative expression for the
energy stored in a magnetic field consider the RL
circuit previously discussed. The voltage
equation is re-stated
If we multiple both sides by i, we obtained the
rate at which the battery delivers energy to the
circuit.
Rate at which energy appears as thermal energy
The rate dUB/dt at which energy is stored in the
magnetic field
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Energy Stored in a Magnetic Field
So we can write the rate of energy stored in the
magnetic field on the inductor as
The dt cancels out and we are left with
Integrating yields
which represents the total energy stored by an
inductor
3
LC circuits
Consider a simple circuit consisting of only
inductor L and a capacitor C . Using the loop
rule, we get the following
L di/dt q/C 0
Since dq/dt i, the loop equation becomes
L d2q/dt2 q/C 0
This is the differential equation that describes
the LC circuit. The general solution has the form
of
where Q is the magnitude of the charge variation,
? is the angular frequency of the electromagnetic
oscillations which is equal to
? is the time phase shift
4
LC circuits
The charge on the either plate of capacitor
oscillates sinusoidally, being positive for half
a cycle and negative for the other half. The
current also oscillates, being in one direction
for half a cycle and in the other direction for
the other half. The energy also oscillates
between the capacitor and inductor. The energy
stored in the electric field on the capacitor
is where q is the charge on the capacitor
and
The energy stored in the magnetic field in the
inductor is where i is the current through the
inductor
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LC circuits
Eight stages in a single cycle in a LC circuit.
The bar graph by each figure show the stored
magnetic and electric energies as well as the
magnetic field lines of the inductor and the
electric field lines of the capacitor.
(a) Capacitor with maximum charge, no current (b)
capacitor discharging, current increasing (c)
Capacitor completely discharged, current is
maximum (d) capacitor charging but with polarity
opposite to (a), current decreasing (e)
capacitor with maximum charge having opposite
polarity to (a), no current (f) capacitor
discharging, current increasing with direction
opposite to that in (b) (g) capacitor fully
discharged, current maximum (h) capacitor
charging, current decreaing
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LC Circuits
It should be noted that the equations of motion
for a oscillating block-spring system and the
oscillating LC circuit are identical using the
below correlation between quantities in both
systems.
These correspondences leads us to the individual
energies as shown below
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Magnets
The external magnetic field of a bar magnet can
be closely approximated by a field that would be
produced by a theoretical positive monopole
(north pole) at one end and theoretical negative
monopole at the other. However, the existence
of monopoles have not been confirm and the field
in magnets arises solely from magnetic dipoles
associated with bounded electron motion in atoms.
Suppose we break apart a bar magnet. It would
seem we could isolate a monopole in this process,
but regardless of the size, each fragment of
magnetic has a north pole and south pole.
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Gauss law for Magnetic Fields
Gauss law for magnetic fields is a formal way of
saying that magnetic monopoles do not exist.
The law states that the net magnetic flux ?B
through any closed surface must be zero.
For comparison, below is Gauss law for electric
fields
In both equations, the integral is taken over a
closed Gaussian surface. Gauss law for electric
fields says this integral is proportional to the
net electric charge q enclosed by the surface.
Gauss law for magnetic fields says there can not
be a net magnetic flux through the surface since
there can be no net magnetic charge enclosed by
the surface
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Induced Magnetic Fields
We have discussed two ways in which a magnetic
field can be produced. One way is an electric
current and another is from dipole moments of
atoms in materials such as iron. There is 3rd way
by induction.
We have studied Faradays law of induction which
relates a changing magnetic flux to an induced
electric field.
The electric field induced along a closed loop by
changing magnetic flux ? through the loop.
Because symmetry occurs so frequently in physics,
we might ask whether induction can occur in the
opposite sense. Can a changing electric flux
induce a magnetic field ??
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Induced Magnetic Fields
The answer is yes !!
The equation governing the induction of magnetic
field is almost symmetric with Faradays Law. Its
often call Maxwells law of induction.
The magnetic field is induced along a closed loop
by the changing electric flux in the region
encircled by that loop
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Induced Magnetic Fields
Consider the charging of a parallel plate
capacitor with circular plates. Assume the charge
on the capacitor is being increased at a steady
rate by a constant current, therefore the
electric field between the plates must also be
increasing at a steady rate.
The figure on the far right shows that the
electric field is directed into the page.
Consider the circular loop inside the parallel
plates with radius r. Since the electric field
through the loop is changing so is the electric
flux which therefore induces a magnetic field
around the loop.
Experiments has verified these results. These
observations demonstrate that the magnetic field
magnitude is constant and tangential around a
loop between the plates and outside the plates as
well.
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Induced Magnetic Fields
Although these two equations are similar, there
are two differences
1st Maxwells law has two extra factors ?0 ?0
2nd Maxwells law lacks the minus sign meaning
the induced electric field and the induced
magnetic field have opposite directions when they
are produce in similar situations.
To see this opposition, compare the direction of
induced E-field on the left to the induce B-field
on the right. In both cases the changing flux is
increasing and into the screen. The induce
E-field is counter-clockwise while the induce
B-field is clockwise.
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Induced Magnetic Fields
The left side of Maxwells Law also appears in
Amperes Law
where ienc is the current encircled by the closed
loop. Therefore, these two equations specify the
magnetic field produced by a current and by a
changing electric field and give the field in the
exact mathematical form. We can combine these two
into a single equation which is called Ampere
Maxwell Law
14
Displacement Current
If we compare the two terms on the right side on
the above equation, we will see that scaling the
first term by ?1/?0 produces a quantity that has
a dimension of current. Historically, this
quantity has being treated as being a fictitious
current called the displacement current id, we
can re-write the above equation as
where the displacement current id is defined
as
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Displacement Current
Lets return to a charging capacitor with
circular plates. The current i that is charging
the the plates changes the electric field between
plates. The displacement current id between the
plates is associated with the changing field E.
Lets attempt to relate these two currents.
The charge q on the plates at any time is related
to the magnitude E of the field between the
plates by the relationship derived in the
capacitor chapter
where A is the plate area. To get the current, we
differentiate the above equation.
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Displacement Current
To determined the displacement current id, we use
the definition
Assuming the electric field E between the two
plates is uniform, we can replace the electric
flux ?E in the above equation with EA. The above
equation becomes
Comparing this with the current, we find that
they are identical
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Displacement Current
We can consider the displacement current to be
simply a continuation of the current from plate
to the other. Although no charges are transported
cross the plates, the displacement current can
help us to quickly determined magnitude and
direction of induce magnetic field.
We can find the direction of the magnetic field
produce by a current i by using the right-hand
rule. We can apply the same rule to find the
direction of an induced magnetic field produced
by a displacement current id as shown in the
figure to the right.
We can also use id to find the magnitude of the
induce magnetic field by simply using Amperes
Law as we did in chapter 30 and replacing a
current carrying wire with a displacement current
carrying capacitor. We get the results
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Maxwells Equations
We have studied the four fundamental equations of
electromagnetism, called Maxwells equations. The
four equations explains a diverse range of of
phenomena. They are the basis for the functioning
of many electromagnetic devices from electric
motors, TVs, radios, electric generators just to
name a few. Maxwells equations are the basis
from which many of equations we have encounter
since chapter 22.