- #1
Rich L.
I've been trying to better understand the relativistic origin of
magnetism. I tried to do a slight variation on a common derivation
and got an unexpected result. I'm hoping someone here can point to an
error I made.
The problem is to calculate the electric field a test charge would
experience in its rest frame due to a relativistic charged particle
beam near it. To be definite let's call it a beam of protons
traveling in the laboratory frame at 80% the speed of light. Let's
have the beam aligned with the x-axis and be offset a distance "a"
along the +y axis, and aligned with the origin of the z axis. The
charges are traveling in the +x direction. I want to calculate the
forces on a test charge at the origin (that is, a distance "a" from
the line current).
First let's consider the electric field around the beam in the rest
frame of the beam charges. In this frame (I'll call it the beam
frame) the charges in the beam are stationary and uniformly
distributed along a line. Let's call the line density of charge
sigma_0 in this frame. By standard E&M calculations the electric
field is everywhere radial from this line. The magnitude of the
electric field on the x-axis (where the test charge will be) is
epsilon_0*sigma/2*pi*a and is entirely in the -y direction. It is
uniform along x and has no other components, especially along the x
axis.
In the laboratory frame, the test charge at the origin will experience
an electric field from the charges in the beam. For the issue here we
don't even need to consider that the test charge is moving, so we only
need be concerned with the electric fields as seen in the laboratory
frame. Because the charges in the beam are moving at 80% of the speed
of light, the linear charge density along the beam is increased by a
factor of gamma.
This is where I've departed slightly from the more common
derivations. The derivations I've seen at this point all calculate a
uniform charge density for the beam at the current "present" time.
This would undoubtedly be correct for normal wires where the charges
are moving very slowly. For reasons that will be clear shortly I
don't believe this is correct for this relativistic beam, however. We
need to be concerned not just with where the charges are "at present",
but with where they were when they were on the past light cone of the
test charge. For this next argument I need to describe a three
dimensional space time diagram, that is one with 2 space dimensions
and one time dimension. We will discard the z direction as it is not
important to this problem. If we consider the world lines of the
charges in the beam, they will always be on the line y=+a. Consider a
slice of this diagram on the y=+a plane. Since the charges are moving
in the +x direction, their world lines in a slice of this diagram
would look like:
+t
|
/ / / | / / / /
/ / / /| / / / /
/ / / / | / / / /
/ / / / | / / / /
/ / / / | / / / /
/ / / / |/ / / /
---------------------------------------------> +x
/ / / /| / / / /
/ / / / | / / / /
/ / / ******* / / /
/ / / ***/ | / ***/ / /
/ / /** / |/ / ** / /
/ * / / / * / /
/ */ / /| / / * / /
/ */ / / | / / */ /
/ */ / / | / / /* /
/ */ / / | / / / * /
/ */ / / |/ / / */
|
-t
Each diagonal line represents the world line of one of the charges in
the beam.
The test charge, at the origin (which is slightly in front of the
diagram above) only "sees" the charges on its past light cone. Since
the world lines from the beam form a flat sheet parallel to the xt
plane, and the past light cone has it's axis on the t axis, the
intersection of these two surfaces is a hyperbola. I've tried to
represent these on the text mode plot above as *'s.
Now it is apparent just looking at this that the charge density on
this hyperbola is no longer uniform, but is higher to the right side,
where the lines tilt toward the normal of the curve, than on the left
side where the lines tilt to be parallel with the curve. The test
charge does not see this hyperbola, of course, because it on a space-
time diagram. What is does see is the charges on the axis of the
beam, but in positions projected up from this hyperbola. That is, the
charges appear to be distributed along the beam as shown by the *'s on
the x axis:
+t
|
/ / / | / / / /
/ / / /| / / / /
/ / / / | / / / /
/ / / / | / / / /
/ / / / | / / / /
/ / / / |/ / / /
--*---------*-------*-----*----*---*---*--*--> +x
| / / | / | /| | / | /| |/ | /
|/ / | / |/ | |/ | / | | | /
| / | / ******* |/ | /| | /
/| / |/ ***/ | / ***| | / | |/
/ | / /** / |/ / ** |/ | |
| / * / / / * | /| /
| / */ / /| / / * |/ | /
|/ */ / / | / / */ | /
| */ / / | / / /* | /
/| */ / / | / / / * |/
/ | */ / / |/ / / */
|
-t
I've done this calculation more rigorously and derived formulas for
this distribution. The average charge density along the beam is just
gamma times the rest frame (of the beam charges) charge density, but
looking in the direction the charges are coming from the density is
reduced, and in the direction they are going the density is
increased. At large distances the densities are uniform, but lower
(to the left) or higher (to the right). The faster the charges are
moving, the greater the difference between the charge density on the
left and that on the right. If the charges could move at the speed of
light, the density on the left would drop to zero.
This asymetric charge density it seems to me would create an electric
field along the axis of the beam. This is what puzzles me. I have
never heard of such a thing. I suspect my analysis above has an
error, but I don't see it.
I've also tried to do this using EM tensors. For the location of the
test charge in the beam rest frame, the only component is the Ey
electric field. The "F" tensor therefor only has two non-zero
elements corresponding to Ey and -Ey. When this tensor is
transformed by the Lorentz transform into the laboratory frame, I get
the expected Bz terms, but not the Ex terms predicted by the above
analysis.
| 0 0 -Ey 0 |
| 0 0 0 0 |
F(beam) = | Ey 0 0 0 | ==>
| 0 0 0 0 |
| 0 0 -g*Ey 0 |
| 0 0 g*b*Ey 0 |
F(lab) = | g*Ey -g*b*Ey 0 0 |
| 0 0 0 0 |
where "g" is gamma and "b" is beta=v/c, and I'm using c=1 units.
So my questions are:
-Am I making an error in my analysis of the apparent charge density
along the wire?
-Are axial electric fields observed around relativistic beams, such
as in particle accelerators? I am skeptical because they would induce
currents in the beam pipes that would drain energy from the beam, and
I haven't heard of such an effect.
-Am I doing the tensor transformation correctly? I am taking F in
the beam rest frame and calculating Lambda^u_a*F^ab*Lambda^v_b where
Lambda is the Lorentz transform for a boost in the x direction and F
is the EM tensor above.
Any assistance is appreciated,
Rich L.
magnetism. I tried to do a slight variation on a common derivation
and got an unexpected result. I'm hoping someone here can point to an
error I made.
The problem is to calculate the electric field a test charge would
experience in its rest frame due to a relativistic charged particle
beam near it. To be definite let's call it a beam of protons
traveling in the laboratory frame at 80% the speed of light. Let's
have the beam aligned with the x-axis and be offset a distance "a"
along the +y axis, and aligned with the origin of the z axis. The
charges are traveling in the +x direction. I want to calculate the
forces on a test charge at the origin (that is, a distance "a" from
the line current).
First let's consider the electric field around the beam in the rest
frame of the beam charges. In this frame (I'll call it the beam
frame) the charges in the beam are stationary and uniformly
distributed along a line. Let's call the line density of charge
sigma_0 in this frame. By standard E&M calculations the electric
field is everywhere radial from this line. The magnitude of the
electric field on the x-axis (where the test charge will be) is
epsilon_0*sigma/2*pi*a and is entirely in the -y direction. It is
uniform along x and has no other components, especially along the x
axis.
In the laboratory frame, the test charge at the origin will experience
an electric field from the charges in the beam. For the issue here we
don't even need to consider that the test charge is moving, so we only
need be concerned with the electric fields as seen in the laboratory
frame. Because the charges in the beam are moving at 80% of the speed
of light, the linear charge density along the beam is increased by a
factor of gamma.
This is where I've departed slightly from the more common
derivations. The derivations I've seen at this point all calculate a
uniform charge density for the beam at the current "present" time.
This would undoubtedly be correct for normal wires where the charges
are moving very slowly. For reasons that will be clear shortly I
don't believe this is correct for this relativistic beam, however. We
need to be concerned not just with where the charges are "at present",
but with where they were when they were on the past light cone of the
test charge. For this next argument I need to describe a three
dimensional space time diagram, that is one with 2 space dimensions
and one time dimension. We will discard the z direction as it is not
important to this problem. If we consider the world lines of the
charges in the beam, they will always be on the line y=+a. Consider a
slice of this diagram on the y=+a plane. Since the charges are moving
in the +x direction, their world lines in a slice of this diagram
would look like:
+t
|
/ / / | / / / /
/ / / /| / / / /
/ / / / | / / / /
/ / / / | / / / /
/ / / / | / / / /
/ / / / |/ / / /
---------------------------------------------> +x
/ / / /| / / / /
/ / / / | / / / /
/ / / ******* / / /
/ / / ***/ | / ***/ / /
/ / /** / |/ / ** / /
/ * / / / * / /
/ */ / /| / / * / /
/ */ / / | / / */ /
/ */ / / | / / /* /
/ */ / / | / / / * /
/ */ / / |/ / / */
|
-t
Each diagonal line represents the world line of one of the charges in
the beam.
The test charge, at the origin (which is slightly in front of the
diagram above) only "sees" the charges on its past light cone. Since
the world lines from the beam form a flat sheet parallel to the xt
plane, and the past light cone has it's axis on the t axis, the
intersection of these two surfaces is a hyperbola. I've tried to
represent these on the text mode plot above as *'s.
Now it is apparent just looking at this that the charge density on
this hyperbola is no longer uniform, but is higher to the right side,
where the lines tilt toward the normal of the curve, than on the left
side where the lines tilt to be parallel with the curve. The test
charge does not see this hyperbola, of course, because it on a space-
time diagram. What is does see is the charges on the axis of the
beam, but in positions projected up from this hyperbola. That is, the
charges appear to be distributed along the beam as shown by the *'s on
the x axis:
+t
|
/ / / | / / / /
/ / / /| / / / /
/ / / / | / / / /
/ / / / | / / / /
/ / / / | / / / /
/ / / / |/ / / /
--*---------*-------*-----*----*---*---*--*--> +x
| / / | / | /| | / | /| |/ | /
|/ / | / |/ | |/ | / | | | /
| / | / ******* |/ | /| | /
/| / |/ ***/ | / ***| | / | |/
/ | / /** / |/ / ** |/ | |
| / * / / / * | /| /
| / */ / /| / / * |/ | /
|/ */ / / | / / */ | /
| */ / / | / / /* | /
/| */ / / | / / / * |/
/ | */ / / |/ / / */
|
-t
I've done this calculation more rigorously and derived formulas for
this distribution. The average charge density along the beam is just
gamma times the rest frame (of the beam charges) charge density, but
looking in the direction the charges are coming from the density is
reduced, and in the direction they are going the density is
increased. At large distances the densities are uniform, but lower
(to the left) or higher (to the right). The faster the charges are
moving, the greater the difference between the charge density on the
left and that on the right. If the charges could move at the speed of
light, the density on the left would drop to zero.
This asymetric charge density it seems to me would create an electric
field along the axis of the beam. This is what puzzles me. I have
never heard of such a thing. I suspect my analysis above has an
error, but I don't see it.
I've also tried to do this using EM tensors. For the location of the
test charge in the beam rest frame, the only component is the Ey
electric field. The "F" tensor therefor only has two non-zero
elements corresponding to Ey and -Ey. When this tensor is
transformed by the Lorentz transform into the laboratory frame, I get
the expected Bz terms, but not the Ex terms predicted by the above
analysis.
| 0 0 -Ey 0 |
| 0 0 0 0 |
F(beam) = | Ey 0 0 0 | ==>
| 0 0 0 0 |
| 0 0 -g*Ey 0 |
| 0 0 g*b*Ey 0 |
F(lab) = | g*Ey -g*b*Ey 0 0 |
| 0 0 0 0 |
where "g" is gamma and "b" is beta=v/c, and I'm using c=1 units.
So my questions are:
-Am I making an error in my analysis of the apparent charge density
along the wire?
-Are axial electric fields observed around relativistic beams, such
as in particle accelerators? I am skeptical because they would induce
currents in the beam pipes that would drain energy from the beam, and
I haven't heard of such an effect.
-Am I doing the tensor transformation correctly? I am taking F in
the beam rest frame and calculating Lambda^u_a*F^ab*Lambda^v_b where
Lambda is the Lorentz transform for a boost in the x direction and F
is the EM tensor above.
Any assistance is appreciated,
Rich L.