THE FIELDS OF ELECTRIC CHARGES - PART 2
RELATIVISTIC CIRCULAR
ORBITS.
Electric Field Line Plots of a Charge in Circular Orbit.
Electric field line plots provide a good means of obtaining an initial appreciation of the field distribution of an orbiting charge. By comparing the result with those of Tsien (ref 6) it is possible to provide further confirmation that the retarded potential Maple computer program is producing the expected results. The following plot is for a charge orbiting with a β (_{} ) of 0.95, which is a gamma (_{}) of 3.2, and this can be directly compared with Fig 6 in Tsien’s paper:-.
FIG 1
In these field line plots the charge is positioned at the location of the + symbol at the time the plot is taken and the orbit is clockwise as marked by the green circle. The figure is in very good agreement with the corresponding plot by Tsien. It is worth noting that the field pattern depends only on the charge velocity. The axes are scaled as a percentage of the orbit diameter as the identical plot will be obtained whatever the actual radius of the charge orbit provided the charge velocity is kept the same. The field magnitudes will of course be different but not the relative magnitudes at different parts of the plot. It is because field line drawings do not show actual field strengths, only relative ones that the plots will be identical.
A strong band of radiation is shown at the position of the kink in the field lines which is following a spiral pattern. As the charge continues its clockwise orbit the field lines will appear to rotate with it and it can be seen that the spiral pattern indicating the radiation pulse will travel outward. This is known as synchrotron radiation. Do not be misled into thinking that there are gaps in the radiation spiral where the field is weaker. The radiation pulse spiral is continuous and uniform along the spiral. If more field lines were plotted the apparent gaps would be seen to fill in. Using the Maple program it is possible to obtain data points for the field intensity through the radiation pulse. For this purpose a particular radius must be specified as this will affect the peak field amplitude. In this case a charge orbit radius of 0.7 x 10^{-13} metres has been used. Taking a point directly above the charge in Fig 1 at coordinates x=0 and y=0 and z at approximately 400% of orbit diameter, the precise z coordinate range to cover the radiation pulse is found to be 5.72 to 5.76 x 10^{-13} metres. Plotting the total field over this range gives Fig 2 below:-
FIG 2
Although this is the total field, which includes the induction field, the latter is smaller and has no noticeable affect on the shape of the pulse. At greater distances from the charge the radiation field will have no radial component but close to the charge this is not the case as the small Ez field component shows in the above plot. This is due to the radiation field being perpendicular to the charge retarded position, not its present position.
As Tsien’s paper contains plots for lower charge velocities just those at higher speeds will be shown here. For a gamma of 10 the electric field is:-
FIG 3
The detail of the synchrotron radiation pulse is now not distinguishable but a close up plot shows that it is still of the same form as before although the pulse is much sharper. This is because, as Tsien shows, the pulse width is approximately proportional to _{}. Finally as an example of a charge orbiting at a very high relativistic speed Fig4 is for a charge with a gamma of 137.
FIG 4
Whereas previously eight field lines per charge were used, in order to obtain Fig 4 it has been necessary to use only one field line coming from the charge. To show more would have cramped the lines so close together it would be difficult to see the pattern. Because the field lines all follow a similar path, apart from where extremely close to the charge, the resulting plot is still valid. The area close to the charge where the difference would be apparent is so small it cannot be seen on a plot such as this which covers the complete orbit. Although the lines are even more curved and closely spaced there is not otherwise a great deal of obvious difference between Fig 3 and 4.
Electric Field Amplitudes Calculated Across the Diameter of the Charge Orbit.
The calculation of the electric fields diametrically opposite a relativistic orbiting charge is a requirement in a number of situations. The information is contained in the above plots but field magnitudes are not easy to estimate from field line spacing. Instead the field amplitude has been calculated along a vertical line which is the diameter of the orbit. This was always done with the charge located at x=0 and z=+orbit radius, which places it at the top of the vertical diameter in the above field line drawings. The field plotted in all of this section is the total electric field, which is the sum of the charge induction and radiation fields. Errors can easily arise in calculations such as these if it is not clear in which frame the field is being calculated. All field values here are in the laboratory frame in which the moving charge is travelling in a circular orbit. The measuring line or diameter being used is stationary in the laboratory. If the measuring line were rotating at some angular velocity then entirely different field values would be obtained.
Simply plotting the field amplitudes directly is not particularly helpful as Fig 5, which is for a charge moving with a gamma of 137, shows:-
FIG 5
The height of the peak of the field strength at the charge location depends only on how close the data points are taken to the point charge. This high peak of variable height prevents the rate of the field decay away from the charge from being easily judged. The plot has been included though as it shows very clearly that the field strength falls rapidly on either side of the charge. This fact may not be so apparent in some future plots! It should be mentioned that because it is a plot of the log10 of the total electric field which is obtained by taking _{} it does not show the change of sign of the field which might be expected due to the change of field direction occurring from one side of the charge to the other.
A preferred method of presentation requires finding the field of a stationary charge at the same position as the moving charge and plotting the ratio of field values. This makes it very easy to see what increase in field strength occurs due to relativistic motion. The ratio will be the same for any radius of orbit provided the value of gamma is kept constant. To obtain the actual field value from such a plot at any position it is only necessary to multiply the field ratio by the field from a stationary charge at the same position as the moving one ie _{}, where distance is taken from the charge to the plot point. The result shown in Fig 6 below is again for a charge moving with a gamma of 137:-
FIG 6
Notice that the field very near the charge is increased by a factor of gamma (137 in this case) as would generally be expected, but moving towards the centre of the orbit the field drops very quickly. The fall is so rapid that unless the data point nearest the charge is positioned exceedingly close to it a false maximum field ratio would be obtained which would appear to be considerably less than gamma. However, whereas for Fig 5 the amplitude always increased the nearer to the charge the field was calculated in this case the field ratio becomes closer to a definite limit (137) the nearer to the charge we move.
Of particular interest is the field ratio at the exact centre of the orbit (position z=0) which is nearly unity ie the field from an orbiting relativistic charge is of similar magnitude to that from a stationary charge at the same location. On the far side of the orbit the field ratio has risen slightly but again is close in magnitude to that from a stationary charge. A number of writers have assumed that the electric field at these two locations will be gamma times that from a stationary charge so this is a contradictory and very interesting result. To allow a more accurate determination of the fields different sections of the above plot have been re-drawn over a smaller z distance. Firstly Fig 7 shows the area containing the lower electric field strength ratios. Because the field amplitude drops so quickly it is only necessary to start the graph a small distance further from the charge than previously to obtain this result:-
FIG 7
The blue field ratio curve in Fig 7 shows how close to the field of a stationary charge the moving charge field is between the centre and the far side of the orbit. The green line is a field ratio of exactly one which is included for comparison. The actual ratio is 0.966 at the far side of the orbit and 0.99997 at the centre. The results of calculations using lower charge velocities are:-
FIG 8
The electric field ratio has a minimum value of 0.798 at a charge gamma of 1.26 at the far side and 0.866 at a gamma of 1.41 at the centre of the orbit. For slower charge speeds than these the ratios increase again until they are, of course, both exactly unity when the charge is stationary. The ratio of 0.966 at the far side of the orbit obtained for a gamma of 137 is very nearly at a limiting value which is not exceeded at even exceptionally high gamma. The high gamma limit at the centre of the orbit is as close to one as can be calculated.
Although the electric field magnitude is nearly the same as that from a stationary charge the field direction is very much different. This can be seen very clearly from the previous field line plots. The angle has been calculated for a range of charge velocities and is plotted below. For a stationary charge the field would be pointing straight along the diameter at both the centre of the orbit and the far side and this angle is taken as the zero degree reference. As the charge velocity increases the field angle moves increasingly clockwise at these two positions as shown in the following plots. The angle is in degrees and is measured clockwise from the diameter.
FIG 9
The angle increases linearly with the charge velocity gamma initially but the rate of angle increase gradually reduces so that very little further angle change occurs beyond a charge gamma of 5. The graph has not been extended to very high gammas as this would prevent the low values being seen so clearly but the angle for a gamma of 137 is marked on the plot and is very little greater than for a gamma of 10.
The red line on Fig 7 was obtained by calculating what I have called the gamma along the diameter. This is based on the gamma of the radial velocity (ie _{}) at that location assuming that the calculation line or diameter were rotating at the same angular velocity as the charge. It isn’t rotating, as it must be stationary in the laboratory frame, but the value of gamma obtained follows fairly well the calculated field ratio in the part between the charge and the centre of the orbit as can be seen. This is particularly apparent even closer to the charge as Fig 10 illustrates:-
FIG 10
Notice how closely the blue field strength ratio line follows the red gamma along the diameter line for the part inside the charge orbit. They are so close that it is difficult to separate the two curves at this resolution. Also the field falls so fast as the measuring point moves away from the charge, inwards along the diameter, that in a distance representing only about 0.004% of the charge orbit diameter the field ratio has fallen from 137 to 70.
In this example the plot has also been continued to show the field ratio calculated outside the charge orbit. Surprisingly the ratio actually rises considerably, initially following the red gamma along the diameter line but dropping away below the red line as the latter starts to quickly climb to infinity. The field ratio reaches a peak of 282.88 (2.065 times the charge velocity gamma) before falling rapidly and although not shown on this plot actually rising again but to a peak of only about 35 and with the opposite polarity field. This result was somewhat unexpected. It is not due purely to the radiation component of the field either. A plot of just the induction field follows a similar curve although the peak in field ratio occurs sooner and reaches only about 175 before falling towards zero. It is, however, associated with the start of the formation of the induction/radiation field pulse. Also it is worth emphasising that it is only the field ratio which is rising. Because the field of the comparison stationary charge is dropping, following an inverse square law with distance, it is falling much faster than the ratio is rising. So as mentioned previously the relativistic charge field is actually falling in amplitude very rapidly outside the charge orbit despite the rise in field ratio.
The electric field strengths along the orbit diameter follow the same general shape for all velocities. Fig 11 is a similar plot to those above but for a charge velocity gamma of 3.2:-
FIG 11
At lower values of gamma a single plot such as this can show sufficient detail along the entire charge diameter and even outside it. The results still follow the gamma on the diameter line in a very similar manner to those previously given. As the charge velocity reduces the point at which gamma on the diameter goes to infinity occurs at a much larger radius.
Notice how deceptive the field line plots are in estimating the relative field magnitude between Fig1 and Fig 4. The field line density appears much greater in Fig 4 especially as only one field line was used compared to eight in Fig 1. However, the field values between the centre and far side of the orbit are actually nearly the same. This shows that the field line plots are only suitable for showing changes in field amplitude between different parts of the same plot. Nevertheless the field directions shown are accurate and they give an overall view of the field distribution which is most useful.
For a final example at fairly low relativistic speed consider a charge gamma of 1.05:-
FIG 12
The field within the orbit is as expected but the maximum field ratio outside the orbit is now only 1.35, whereas it was about twice the value of the charge gamma at higher velocities. Comparing the parameters relating to the peak field strength ratio for a few different charge velocities gives:-
PEAK ELECTRIC FIELD STRENGTH RATIO MAGNITUDE |
||
Charge Velocity Gamma |
Peak Electric Field Strength Ratio |
Peak Field Strength Ratio Charge Velocity Gamma |
137 |
282.88 |
2.065 |
3.2 |
6.421 |
2.007 |
2.294 |
4.469 |
1.948 |
1.05 |
1.35 |
1.29 |
These results indicate that the electric field ratio reaches a maximum magnitude of just slightly over two times gamma compared to a stationary charge at high charge velocities. Both of the ratios in the above table will obviously be one for a very slow moving (ie stationary!) charge However the fall in peak field strength ratio divided by gamma does not substantially decline from its high velocity value of approximately two until the charge velocity is less than a gamma of about 2.
Comparing the position of the peak field seems to indicate there is a relationship between the position where the value of gamma along the diameter goes to infinity and the position of the peak field ratio. The value of the following expression appears to be fairly constant:-
Peak Ratio Position = _{} = approx 0.25 (1)
Values for the previously considered charge velocities are:-
PEAK ELECTRIC FIELD RATIO POSITION |
|||
Charge Velocity Gamma |
Position of Infinite Gamma Along the Diameter |
Peak Field Strength Ratio Position |
Peak Ratio Position:- Column 3-Column2 Column2-50.000 |
Distance from Centre of Orbit as Percentage of Diameter |
|||
137 |
50.001332 |
50.001667 |
0.251 |
3.2 |
52.636 |
53.306 |
0.254 |
2.294 |
55.556 |
56.986 |
0.257 |
1.05 |
163.98 |
196.75 |
0.287 |
These results indicate that the peak ratio position is only approximately constant but this is sufficient to enable the position of the peak to be estimated. The peak field strength ratio position is about 25 to 29% further out from the charge position than the point where the gamma along the diameter goes to infinity.
So for a given charge velocity the point where the gamma along the diameter goes to infinity can be calculated and using the above eqn (1) formula the location of the field ratio peak can be found. Its magnitude is approximately twice that of the charge velocity gamma for high gammas but falls until it is eventually one for a stationary charge. There is probably a similar relationship for the position outside the orbit where the field reverses but this has not been investigated yet.
To summarise it has been shown that the gamma along the diameter provides a simple way to estimate the approximate electric field magnitude for any orbiting charge along the diameter between the charge and the orbit centre. Between the centre of the orbit and the far side the field magnitude is slightly less than that of a stationary charge although it is not in the same direction. The above plots have mainly examined the field magnitude as the field direction is easily seen from the field line plots and Fig 9. It is appreciated that the values obtained using these suggestions are only approximate but failing the availability of an exact formula they do provide an easy way to visualize and estimate the magnitude of the electric fields of a relativistic orbiting charge. Precise field amplitudes for any charge velocity can then be obtained using the Maple retarded potential program.
Magnetic Field At the Centre of the Charge Orbit.
In the plane of the charge orbit the magnetic field is always perpendicular to the plane. Consider a charge orbiting in the x, z plane as in the examples above. The magnetic field will be entirely in the y direction. A negative charge orbiting clockwise will produce an anticlockwise current and the direction of the field will generally be the same as that in which a normal right hand screw would be turned when driven in the same direction as the current. The positive y direction is into the screen and the field inside the loop is therefore out of the screen which is the –y direction. The formula for the field magnitude at the centre of the orbit is just:-
_{} (2)
Where:- H is the magnetic field intensity in amperes/metre.
f is the charge frequency of rotation in revolutions/second.
r is the radius of the charge orbit in metres.
q is the charge magnitude in coulombs.
v is the charge velocity in metres/second.
As _{}
Substituting for f in the equation for H gives the alternative relationship:-
_{} (3)
This formula applies at even the highest relativistic charge velocities. Although the magnetic field is directly proportional to the velocity there is not, just as with the electric field, any relativistic field enhancement at the centre of the orbit by a factor of gamma.
The magnitude of the electric field at the centre of the orbit is:-
_{} (4)
Where K is the moving/stationary charge field ratio plotted as the red line in Fig 8 above and is between 1.00 and 0.866 depending on the value of gamma.
The ratio of the electric to magnetic field (ie the field impedance) at the centre of the orbit is therefore given by dividing eqn (3) by eqn (4) to give:-
_{} (5)
The field impedance is independent of the radius of the charge orbit, depending only on the charge velocity. Comparing this with eqn (9) in Part 1 and putting sin(θ) =1 shows that the field impedance at the centre of the orbit is K times less for an orbiting charge than for a charge in linear motion. This is of course due to the E field being gamma.K times less and the H field being only gamma times less than for a charge in linear motion.
Magnetic Field Along the Diameter At the Far Side of the Charge Orbit.
To find the way the magnetic field at the far side of the orbit changes it is convenient to express it as a ratio to the field at the centre. Expressed in this manner the far side/centre magnetic field ratio varies with gamma but is independent of the charge orbit radius. The actual field values can easily be obtained if required just by multiplying the far side/centre magnetic field ratio by the field at the centre of the orbit as calculated from the above formula. The far side/centre magnetic field ratio is:-
FIG 13
At very low charge speeds the magnetic field at the far side of the charge orbit is exactly 0.25 of that at the centre. This is the result one would expect if the field is following an inverse square law with distance from the charge. For higher velocities the ratio rises slightly reaching a peak of nearly 0.2594 at a gamma of 1.12 before falling again to a lower limit of about 0.2415. As previously mentioned these ratios are independent of the charge orbit radius. The variation with gamma is only very slight so again the field is not proportional to gamma. To show how the field varies at positions between these two locations the following plot was produced:-
FIG 14
The two values of
gamma chosen represent the extreme variation of the far side/centre magnetic
field ratio. For comparison purposes the ratio for an inverse square law has
been added and although not in exact agreement the field is approximately
following the inverse square. For low charge velocities the magnetic field
ratio follows the inverse square law precisely.
Magnitude of Magnetic Field Along the Diameter Compared With Electric Field.
As the way the electric field varies along the diameter through the charge has been fairly well investigated above one of the easiest ways to show the magnetic field magnitude is to plot the field impedance (E/H) along the diameter. The value of the magnetic field intensity is just E divided by this impedance. The result for the higher charge velocities is:-
FIG 15
For the very high speed charge with a gamma of 137 the field impedance was calculated as 376.73 which is identical to that of free space to the accuracy quoted. It is constant at this value along the whole diameter apart from an almost imperceptible increase extremely near to the charge (It rose by only 0.01 at a distance of 1 x10^{-5} % of the diameter). This means that the magnetic field is falling at the same rate as the electric field. At lower velocities the field impedance is higher and rises noticeably nearer to the charge. For a gamma of only 1.05 the result is:-
FIG 16
The field impedance is now higher due to the lower charge velocity and consequently lower magnetic field amplitude as predicted by eqn (5) above for the field at the centre of the orbit.