THE FIELDS OF ELECTRIC CHARGES – PART 1
In classical electromagnetism frequent use is made of the concept of a simple point charge. The electric field from such a charge, when stationary, is uniform in all directions and varies in magnitude as the inverse square of the distance from the charge. The field direction is by convention that which a positive test charge positioned in the field would move. As opposite charges attract this means the field is directed radially inward towards a negative charge and outward from a positive one. The formula for the electric field magnitude in a vacuum in the MKS (metre-kilogram-second) system of units is:-
Er is the radial electric field strength (in volts/metre).
q is the magnitude of the charge in coulombs. For a single electron q is -1.601864 x 10-19 C.
e0 is a constant known as the permittivity of free space (value 8.854 x 10-12).
r is the distance from the charge (in metres).
In quantum physics the electron is considered to be a point charge, but if this is used for a simple classical model there is the well known problem that the field will go to infinity as the radius approaches zero, which is impossible. Most physicists would say that when measuring the field at very small distances ‘quantum’ effects occur and the classical model breaks down. Another difficulty is that the magnetic field of a classical point charge will be zero when it is stationary and an electron has spin, an associated angular momentum and a small magnetic field. All these are considered ‘quantum’ features. To account for all the electrons characteristics, particularly at very small distances, it can be seen that a better and more complicated classical model would be needed.
An example of a positive point charge is the positron which is an electron with a positive charge. It belongs to the family of particles known as antimatter so in many cases a free positron will soon be annihilated. A proton, which is found in the nucleus of atoms, (usually!) is also positive. It has a more complicated structure than the positron and is known to have a radius of about 0.8 x 10-15 metres. This is the same order of magnitude as the classical electron radius which is calculated to be 2.82 x 10-15 metres. Although the electron field appears to follow the inverse square law at distances less than this it provides a very rough guide length below which classical analysis requires to be done with some caution. With this proviso the simple point charge model has been found to be surprisingly useful and accurate in many applications for predicting the field of the electron and the other charged particles. This is certainly not intended to rule out analysis at shorter distances but rather to acknowledge the points which need to be considered and it is intended to return to these on a future web page. Because the charge of the electron is so small if you work out the field strength due to a single electron at a distance of just 1mm away from it using the above eqn (1) then the field strength is only 0.00144 volts/metre. However, at atomic distances the field is much larger. In the Bohr model of a hydrogen atom (that’s the model where the electron orbits the nucleus) for an electron about 5.3 x 10-11 metres from the nucleus then the field from the nucleus at the electron is 5.125 x 1011 volts/metre.
It is quite difficult to show a plot of the electric field near a charge because the magnitude varies so enormously with just a small change in distance. It is often necessary to use different plotting methods depending on the field being depicted. A frequently used technique involves plotting field lines. Very close to a stationary charge the electric field lines are evenly spaced around it. The line is then drawn so that at all points along its length it is in the direction of maximum field strength. As the force on a charge is in the direction of the electric field these lines represent the direction of force on a stationary charge and are sometimes called lines of force. The field strength is proportional to the density of the field lines. Widely spaced lines indicate low field strength and closely spaced lines a high field strength. The plot for a stationary charge is shown below where the cross indicates the charge position. The lines actually extend to infinity in this case although the field is extremely weak even at the distances shown on this plot. Ideally arrows should be shown on the field lines to indicate the field direction but these are omitted here:-
Fields of Charges Moving in a Straight Line At a Constant Low Velocity.
At small velocities the electric field of a point charge moving relative to the observer hardly changes from that of a stationary charge. The noticeable difference is that a magnetic field (H) becomes apparent, the magnitude of which can be calculated using the formula (see ref 3a for more details):-
H is the magnetic field intensity in ampere/metre.
E is the electric field in volts/metre. (To be precise this should be the electric field of the moving charge which is given in eqn (3) later but at low velocities the error is small if the field given by eqn (1) is used).
v is the charge velocity in metres/second
x is the vector cross product of v and E. (The meaning of this was explained in the Poynting Vector description)
uo is the permeability of free space (value 4.π.10-7).
c is the velocity of light in metres/second.
As the electric field is radial from the charge the direction of the magnetic field, given by the vector cross product (v x E), is that of concentric circles around the moving charge, the plane of the circles being perpendicular to the path of the charge as shown below:-
In Fig 2 only one set of magnetic field lines in the plane perpendicular to the direction of charge motion are shown. This particular plane passes through the charge but it would be possible to draw further field lines both in front of and behind the moving charge. As the field will be weaker at the corresponding distance from the direction of motion line the concentric circles will be wider spaced the further from the plane through the charge they are. Radially along the plane through the charge the equations show that the magnetic field is decaying inversely proportional to distance squared in the same way as the electric field. Along the planes in front of and behind the charge the magnetic field decay is more complicated as there is also a sin(θ) term introduced due to the vector cross product of v and E. (Here θ is the angle between v and E). At the corresponding distance from the charge path the magnetic field will be less due not only to the larger distance but also because sin(θ) is less than one. Directly in front of and behind the charge θ will be zero and so there is no magnetic field at all in these positions.
An alternative way of showing the magnetic field would be to for example plot its magnitude along a line parallel to the charge motion. If the charge is moving in the +x direction as in the above example and the measuring line is placed say 5.3 x 10-11 metres away along the +z axis it can be seen from Fig 2 that the field will be entirely in the +y direction. Assuming the charge is travelling at a velocity of 20 metres per second the following result is obtained:-
In this case also, moving away from the point perpendicular to the charge motion and passing through the charge, the fall in amplitude of the field is due to both the increasing distance and the declining value of θ. Provided the charge velocity is not too great the magnetic field amplitude is directly proportional to the charge velocity.
Electric Field of Charges Moving in a Straight Line At a Constant High Velocity.
For charges moving at high speeds the error in the electric field calculated using the eqn (1) stationary charge formula becomes too great. The radial electric field is no longer the same magnitude in all directions so it is often easier to use equations for the field components along the x, y and z coordinate directions. For comparison let’s first express eqn (1) for the stationary charge in component form where the charge is situated at the origin so x, y and z are the coordinate distances from the charge in metres:-
If the total field is calculated from these equations, which is just, using the fact that then eqn (1) is obtained.
For the fast moving charge, assuming that the charge is moving in the +x direction the corresponding equations for the electric field are:-
Eqns (4) also give the correct answer for a slow moving charge as in that case gamma is very nearly equal to one and eqns (3) and (4) are then identical. Comparing the two sets of equations (3) and (4) it is apparent that for the fast moving charge all field components are multiplied in amplitude by gamma due to the gamma term in the numerator. But there is also a (gamma)2 term in the denominator which is always associated with the co-ordinate corresponding to the direction of charge motion, which in this case is x. The presence of this term means that all fields are not always gamma times the magnitude of those from a stationary charge. In the plane through the charge and perpendicular to its motion where x is zero there are only Ey and Ez electric field components. These components are gamma times greater in amplitude here than for a stationary charge. However, moving directly in front of or behind the charge y and z are zero so from eqns (4b) and (4c) Ey and Ez are also zero and eqn (4a) simplifies to:-
Directly in line with charge motion field amplitude is _____ (6)
Due to the extra (gamma)2 term in the denominator the gamma term in the numerator is cancelled leaving the Ex field amplitude inversely proportional to both x2 and gamma2. An alternative way of looking at the effect of the (gamma)2.x2 term in the denominator is that it gives the appearance of distances being contracted in the direction of charge motion by a factor of gamma. A similar contraction effect of all the field components in the x direction is also apparent at any angle to the direction of motion. A corresponding field line plot to Fig1 but for a gamma of 5 would be:-
The field lines are still radial but instead of being evenly spaced around the charge they are now compressed together, towards the plane through the charge perpendicular to its motion, indicating that the field is stronger here. To show this better here is a plot of the electric fields parallel to the direction of charge motion along the same path used for Fig3:-
The above plots clearly show that the maximum value of the field in the plane perpendicular to the charge (Ez) is increased by a factor of gamma i.e. in this case five times that of a stationary charge. Note that the fields parallel to the direction of charge motion (Ex) are the same peak amplitude but the peak occurs nearer to the charge when it is moving fast. The amount of contraction is difficult to judge accurately from the plot but is actually exactly by gamma. This is an example of the Lorentz contraction which is so familiar from the special theory of relativity.
An alternative formula which is sometimes useful gives the electric field not in component form but in terms of the magnitude of the radial field at a particular angle to the direction of charge motion. This is:-
Where θ is the angle between the charge trajectory and the field measuring position.
This equation gives the identical field to eqns (4) but expressed in a different form.
Magnetic Field of a Charge Moving in a Straight Line At a Constant Velocity.
The magnetic field for the fast moving charge can still be derived from the electric field using eqn (2) although it is now important that the electric field used in the equation is that for a fast moving charge as given by eqns (4) or (7).
The magnetic field intensity can often be derived from the electric field and the ratio of E/H can be very useful. This ratio is most frequently used for a free travelling electromagnetic wave in a medium. For an electromagnetic wave in free space the relationship between E and H is:-
In free space (8)
has the value of 376.73 ohms and is known as the intrinsic impedance of free space.
If a similar ratio is taken for a charge moving in a straight line, using eqn (2) and gives:-
Around a charge in linear motion (9)
Where:- θ is the angle between v and E.
Where this ratio is used for the field around a charge it will be called the field impedance. By comparing eqns (8) and (9) it can be seen that for a charge travelling at almost the velocity of light the magnetic field perpendicular to the charge (where v=c, approximately, and sin(θ) =1in eqn (9)) will be the same ratio to the electric field as found in a travelling wave in free space. The magnetic field will be less than this by a factor of sin(θ) both in front of and behind the charge.
Fields of Accelerating Charges.
The situations considered so far have put constraints on the type of charge movement allowed in order to obtain simplified solutions for the charge field. When acceleration is allowed which includes change in direction as well as velocity things become more interesting but unfortunately also more complicated. One very important change is that there is an additional field component produced which depends on the component of acceleration of the charge perpendicular to the line of sight between the field measuring position and the charge position at the time the field was radiated by the charge. This will be the position of the charge at an earlier time than when the field is measured due to the time the field requires to travel at the speed of light from the charge to the measuring point. This earlier time is known as the retarded time and the position of the charge at that time is the retarded position. The field considered previously is sometimes called the induction field and this generally decays inversely with distance squared. The magnitude and direction of this field will also change with charge acceleration as will be illustrated in the next section. The radiation field decays more slowly than the induction field, just inversely with distance so at large distances from the charge the radiation field will dominate.
The method used to find the field when the charge is accelerating relative to the observer involves calculating the retarded potentials, which are frequently called Liénard-Wiechert potentials. These are the scalar (φ) and vector (A) potentials expressed in terms of the retarded charge position and time. The scalar potential from a charge at a given position and time, normally just called the present position and time, can be found using the formula:-
and the vector potential at the present position and time is just:-
Where:- denotes the scalar potential at the present time (t) and position (r).
denotes the vector potential at the present time (t) and position (r).
rret is the retarded charge position at the retarded time (tret).
vret is the velocity of the charge at the retarded position and time.
Knowing these potentials the electric field can be found using the relationship:-
Where grad refers to the gradient or rate of change with distance of φ(r,t).
is the rate of change of the vector potential with time.
The magnetic field can be obtained from this electric field using the relationship:-
Where:- ε is the unit vector from the retarded position to the field measuring point.
x is the vector cross product of ε and E.
There is a difficulty in applying eqns (12) and (13) as the scalar and vector potentials are expressed in terms of retarded values of position, velocity and time and they require to be differentiated with respect to present position and time. This can be done but not without introducing some mathematical complexity. I don’t wish to go into the detail but ref(7) covers this very well. The resulting equations finally obtained for the induction and radiation E fields are quoted here.
This retarded potentials method has the advantage that it is accurate for any charge movement and the calculated field is also given in a form that splits readily into a radiation and an induction component. Unfortunately the fields are still expressed in terms of the retarded position and time and it is not always possible to transform these symbolically to the charge present position and time which would be the most useful parameters. It is sometimes pointed out that the expression for the induction field in terms of retarded parameters is the same as that for a charge travelling with uniform speed. While this is true the fact that the field equations are in terms of retarded parameters must be taken into account and this effect is very significant. If it is wished to know the field distribution over even a small area the retarded time will be different for each measuring point location. A field distribution at constant retarded time will therefore be nothing like the field distribution at observer time. Also when a charge is accelerating the retarded charge position in relation to the measuring position is different and this affects the measured field value enormously compared with a charge in linear motion. As a result the induction field seen by the observer usually varies considerably both in magnitude and direction from that of a charge in uniform motion.
To overcome these difficulties a retarded potential computer program has been written using Maple 6 which solves the equations for the fields automatically. The answer is in observer time and charge actual position which is the most useful form. This program can be used to find the fields and potentials from any charge provided the equation of motion can be written (the program uses Cartesian co-ordinates) and Maple can solve the retarded time equation. It has so far been used for charges that are stationary, in linear motion, sinusoidally oscillating and in circular orbit. As Maple 6 could not solve the retarded time equation for circular motion symbolically, the numerical Maple solver has always been used for all charge motions. It is written so that the fields from a number of charges can be combined and has been used to obtain the plotting points for all the examples of fields from relativistic charges on this site. Only a simple program is included to obtain the field values at a single position as some of the plotting programs I have written are either over complex or still require some simple additional manual programming for all the various different displays needed. However, should anyone wish to have it, if they e-mail me at the address on the Site Index page I can include a basic plotting program which enables a number of simple plots to be obtained and can be further modified by the user.
To validate the program its results have been compared with those obtained using the above formulae for a stationary charge and a relativistic charge moving in a straight line. An entirely numerical computer program was written for calculating the relativistic fields of various accelerating charges which carried out numerical integration of the Liénard-Wiechert potentials and numerical differentiation to find E and numerically took the curl of A to find H. The mathematical techniques used by the two programs are entirely different but the results of the two programs agree. To help visualize why the induction field from an accelerating charge is so different from that of a charge in linear motion the following example is offered as an illustration.
An Example of Calculating the Electric Fields of a Relativistic Orbiting Charge.
For this example a method of working out the induction field will be used which is particularly well suited to displaying on a charge position plot. The data for the plot was initially obtained from some simple calculations using the results from the Maple retarded potential computer program but displaying it in this way enables the validity of the results to be more easily assessed. For this simple example the induction field may also be easily calculated manually for comparison. The new method uses the fact that the induction field at present time even for a charge in linear motion is not equivalent to the field from a charge at the retarded position as might be expected. Instead it depends on the field of a charge at the position the charge would reach if it continued to travel with the same velocity it had at the retarded time right up to the present time. This also applies to an accelerating charge (see ref 3a). So it is just necessary to work out the charge velocity at the retarded time, assume it continues with this same velocity for the duration of the retarded time and note its position. Call this the projected position. The field at present time is that which a relativistic charge at this position and moving in a straight line would have if travelling with the retarded time velocity. To illustrate the effect of this for a charge in linear motion compared to one in a circular orbit consider the following diagram:-
Fig 6 shows a charge coloured green, travelling in a straight line parallel to the x axis along the line z=0.7 x 10-3 metres. The red charge is orbiting the origin with a radius of 0.7 x 10-3 metres. Both charges have a velocity of 0.9797959 of the velocity of light which gives them a gamma of 5 and at the instant shown both are located at x=0, z=0.7 x 10-3 metres. It is required to find and compare the field from the two charges. The retarded position of the green charge is at x=-3.42928 x 10-3 metres. This is easily confirmed as the time for the charge to travel from the retarded position to the present position is:-
Green charge travel time = .
The time for the field to travel from the retarded position to the measuring point is:-
Green field travel time =
The two expressions above have the same value which shows that the retarded position is correct. Projecting the charge retarded position forward for this value of the retarded time places it at the actual charge position. The field calculation point is perpendicular to the charge path so the field from the charge, calculated at the origin, is gamma times the field of a stationary charge in this position as given by eqn (4) above. This field is shown in green as Ei on the drawing and on the scale being used for the fields is so great it actually extends from the origin off the diagram and would terminate at z=14.69 x 10-3 volts/metre.
For the red charge the time for the field to travel from the charge to the origin is always just seconds. This is the retarded time, in which the charge will travel a distance of:-
Red charge retarded distance = charge velocity . retarded time
=0.685857 x 10-3 metres
To find where the red charge is, it is possible to express the retarded distance in degrees of charge rotation of its orbit:-
Red charge retarded distance in degrees =
= 56.14 degrees
The red charge retarded position is drawn in Fig 6 retarded by 56.14 degrees. The projected position is found by drawing the tangent to the orbit at this retarded position and extending it for the previously found retarded distance length of 0.685857 x 10-3 metres in a straight line. This distance is the same as the length of the arc between the retarded position and the present position. The field at the origin is the same as that of a charge travelling in a straight line along this projected path and which is presently at the projected position. However not only is the projected position slightly further away from the origin than the actual position but the required field is now at an angle to the charge path. Eqn (7) indicates that the reduction in induction field strength compared to a stationary charge due solely to this angle is:-
Red charge angle field reduction =
There is a right angle triangle formed by the three points comprising the origin, the retarded position and the projected position. As the lengths of two sides of this triangle are known it is just simple trigonometry to find the length of the other side and the angles of the triangle. The value of θ, which is the angle between the charge trajectory and the measuring point, is arctan(0.7/0.685857) which is 45.58 degrees. So the expression for the angle field reduction evaluates to 0.1098 in this example. If θ were 90 degrees the angle field reduction factor would actually be an increase by a factor of 5 as occurs for a charge in uniform motion. This change in angle can therefore be seen to be mainly responsible for the large reduction in field strength in the case of the orbiting charge.
Again the distance to the projected position from the origin in the same right angle triangle is 0.7 x 10-3 /sin(45.58). The field reduction due to the increased distance is inversely proportional to the ratio of the distances squared:-
Red charge distance field reduction = = 0.5102
The total induction field reduction is the product of the distance and angle reduction ratios which is 0.1098 x 0.5102=0.056 compared with a stationary charge at the same position as the orbiting one. The comparison is made to a stationary charge instead of the green charge because the angle field reduction factor was compared with a stationary charge and not a relativistic one. I have emphasized this field reduction as it is sometimes said that the field at the centre of a relativistic charge orbit is gamma times greater than that of a stationary charge and this is certainly not the case. The very small induction field vector (red Ei) is shown on the diagram to the same scale as used for the linear motion green charge field and is just 0.164 x 10-3 volts/metre. Also included for comparison are the radiation field of the orbiting charge which is marked as Er (2.76 x 10-3 volts/metre) and the vector addition of Ei and Er which is the total field Et which is 2.88 x 10-3 volts/metre. Note that the radiation field is at right angles to the line between the charge retarded position and the measuring point and the amplitude is much greater than the induction field. The radiation field is the most significant field at many places inside the orbit of the charge. The change in field strength along the complete diameter of the charge orbit will be covered in further detail when the results of using the Maple program to analyse orbiting charges are presented in Part 2.
Field Line Plots of an Accelerating Relativistic Charge.
As a further check on the validity of the results and also to obtain some insight into the field distribution some field line plots were produced using data from the retarded potential Maple 6 program but using a separate field line plotting program. Some excellent examples of field plots of charges undergoing various different motions are available in the literature due to Tsien (ref 6). He obtained the formula for the field lines and plotted them directly from this and his results are therefore highly accurate. The plot in Fig 7 below was obtained for an oscillating charge, commonly known as an harmonic oscillator, with a maximum velocity of 0.9c:-
The path of the charge is represented by the short vertical green line and its present position at the instant of the plot is denoted by the + sign. If a is the amplitude of charge oscillation then the charge equation of motion is:-
The charge is shown at the instant of maximum deflection when t=π/2ω and so sin(ωt) = 1 and x=a. Differentiating the above expression with respect to time to find the velocity:-
The maximum velocity is when cos (ωt) =1. Putting gives:-
This equation allows the calculation of oscillation frequency to give a desired value for βmax. Fig 7 was drawn for a βmax of 0.9 which is the same as Fig 15 in Tsien’s paper. Although this plot covers a slightly larger area the two diagrams are very similar indeed confirming that the field data produced by the Maple program has the correct amplitude distribution and that the field vectors have exactly the correct orientation. Such a comparison on its own cannot of course eliminate the possibility that the program contains an incorrect amplitude scaling factor applied to all field values but it gives further confidence in the accuracy of the program.
The results illustrated in Fig 7 are not without interest in their own right. In a 3 dimensional plot the field would be the same about the vertical axis. Note the strong cone of radiation pulses travelling outward just around the vertical. Also a charge of opposite polarity to the oscillator, ejected exactly along the vertical line would be accelerated directly away from the charge by the field gradient without deflection by the field. Should time permit it may be worthwhile to analyse this field in further detail. Study of harmonic oscillation along an arc of a circle instead of a straight line and the field of simple combinations of such charges would make a good project.