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Larmor's Equation
Name: Joe
Status: student
Age: N/A
Location: N/A
Country: N/A
Date: N/A
Question:
I have a simple question and I have tried to find the
answer in reference texts and your library, but have failed. Here is
my question: I know that accelerating electrons generate
electromagnetic radiation because of the changes in the static
electric field. I have seen that solution online. I believe it is
called Larmor's equation. However, electrons have both static
electric and magnetic fields. So, does an electron emit
electro-magnetic radiation when it is moving at a constant velocity,
and in a straight line? Both of these fields are changing but not
like in the case of particle acceleration. The two fields appear to
be independent of each other caused by different mechanisms. I have
not seen that case addressed in my textbooks and have not found a
discussion on line. I seems like this problem has a simple answer,
but I am struggling with the concepts of Maxwell's equations.
Replies:
Joe
First of all AC vs DC. Direct Current (DC) is a flow of charges
(current-amperes) that flows in one direction. Here the charges are flowing
in a "Static" electric field. Alternating Current (AC) is a flow of current
first in one direction, then in the other direction such that you can
measure the change in direction as a frequency (cycles per second - Hertz
(Hz)). Here the charges are flowing back and forth establishing a "Dynamic"
Electric field.
When charged particles move down a wire (a current), a magnetic field is
generated. If it is a DC current, a magnetic field of constant magnitude
develops, if it is an AC current, the magnetic builds, falls, and builds
again in the other direction according to the direction of the current.
That is what Maxwell's equations are telling us.
The equation is Voltage (V) = L * (di/dt)
An inductor that accumulates the magnetic field is also called a choke
because it builds and collapses to oppose the current. As the current
increases, the magnetic field builds (pulls energy out of the
current) to oppose the current, as the current increases; and the magnetic
field collapses and injects energy back into the circuit as the charges.
Now to field theory.
If you put a positive voltage at the end of one wire and a negative charge
at the other end of the wire, you are actually establishing an electric
field in the wire conductor. If the charges do not vary in time (DC), you
get a "static electric field". If the charges vary in time (AC) you get a
"dynamic electric field."
Along with the electric field, a magnetic field is developed at a 90 degree
angle to the electric field.
So, an electric field established along a conductor runs parallel to the
conductor, establishes a magnetic field that is perpendicular to the
conductor.
There is a convention called the right hand rule. Clench a fist on your
right hand, stick your thumb in the direction of positive current flow and
the fingers will point to the direction of the magnetic field. Notice that
the two fields are orthogonal to each other (that is pointing to different
directions by 90 degrees)
The same is true for an alternating field. The magnetic field is
perpendicular to the electric field, but it is out of phase with the
electric field by a quarter wave. The two fields are not independent of
each other because one induces the other and the other induces the one.
This is how electro-magnetic fields propagate through space. The electric
field generates a magnetic field that generates another electric field that
generates another magnetic field and on and on and on and on out into space
until your radiation dissipates to an undetectable level. It is
like a wave of water at sea.
So the parts of your questions I can answer are:
1) "does an electron emit electro-magnetic radiation when it is moving at a
constant velocity, and in a straight line?"
An electron loses potential energy but gains kinetic energy as it moves down
an electric field.
In the DC case, an electron moving down a conductor in a straight line
(linear electric field) at a constant velocity generates a constant
(magnitude and direction) magnetic field.
2) "Both of these fields are changing but not like in the case of particle
acceleration."
The E-M fields change dynamically only in the AC case. DC case (static) EM
fields are of constant magnitude and direction.
A sine wave function can be considered the path of a particle that is
subjected to a constant acceleration in the opposite (negative) direction of
the particle's travel.
In the AC case the EM fields build and collapse as the current travels in
one direction, decelerates, and starts moving in the other direction.
3) "So, does an electron emit electro-magnetic radiation when it is moving
at a constant velocity, and in a straight line?"
An electron
moving down a constant (DC) electric field generates a constant magnetic
field with it. A moving magnetic field will induce charge movement in an
electric field.
I looked at Larmour's equation at:
http://www.cv.nrao.edu/course/astr534/LarmorRad.html
It discusses E-M radiation of an "accelerating" charge. Seems to be
analogous to the AC situation I am talking about because a sine wave can be
defined as the path of a particle that experiences a constant acceleration
force in the negative direction it is traveling. So if a charge is
"accelerating" in one direction, the choke action of the magnetic field will
subtract energy from the system by building a magnetic field, and will
collapse the magnetic field when the charge is negatively accelerating.
4) "Both of these fields are changing but not like in the case of particle
acceleration."
The elctro-magnetic (EM) field only changes in the AC case. If a sine wave
is the function of a negative acceleration, this is the accelerating case.
5) "The two fields appear to be independent of each other caused by
different mechanisms."
The two fields are NOT independent of each other. One induces the other and
the other induces the other.
Sincere regards,
Mike Stewart
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Update: June 2012
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